# Seminar Archive

2013-2014 2012-2013 2011-2012 2010-2011 2009-2010

2008-2009 2007-2008 2005-2006 Current

**BLOCK 8, 2014 - April/May**

**Friday May 9, 2:30pm**

**Tutt Science 229**

**Speaker:** Stefan Erickson**Title:** Computer Security in the Post-Snowden Era

**Abstract**: The Snowden leaks about NSA spying have sparked a public debate about cybersecurity. There are many questions that people are struggling to answer. What is privacy? What is security? What is the right balance between individual privacy and national security? What will the effect of "big data" have on Internet commerce and security? The answer to these questions will inevitably have a huge impact on the future of society.

In this talk, we will explore many aspects of cybersecurity. We will explain how the Target Point-Of-Service breach occurred, what the Stuxnet worm is and why it was important, and how the recent "Heartbleed" bug worked. We will see what information the NSA has been gathering and how they exploited a weak random number generator to break a commonly used cryptosystem. We also hope to start a dialogue about the proper role of computer security. The talk is mostly rated G with occasional flashes of PG-13 material.

90% Rated G, 10% Rated PG13

**Friday May 2, Noon**

**Tutt Science 122 - Lecture Hall**

**Speaker:** Jim Powell, University of Utah (visiting professor)**Title:** Mathematics of the Life-Impaired: How Disease Theory Predicts the Zombie Apocalypse

Abstract:

Abstract:

From movies to pop music ("If I were a zombie, I'd never eat your brain...), it seems the undead are already taking over the world. The usually staid Centers for Disease Control launched a tongue-in-cheek "Preparedness 101: Zombie Apolcalypse" public campaign in 2011 to drive home the importance of emergency preparation. Even Utah State University has been infected as evidenced by USU Housing’s wildly popular, campus-wide“Humans vs. Zombies” (HvZ) war this past fall.

Anthropologist Krystal D'Costa suggests zombies capture our imagination because they represent modern society and technology gone awry and offer the perfect metaphor for an unstoppable pandemic. USU professor Jim Powell expands the zombie metaphor to illustrate the concepts and results of mathematical epidemiology. Using storylines from such movies as "Night of the Living Dead," "28 Days Later," "The Walking Dead" and "I am Legend," as well as data from the USU HvZ games, Dr. Powell will show how mathematicians model diseases. He'll talk about how scientists predict the course and impact of epidemics, discuss how "herd immunity" (vaccination levels for disease eradication) works and apply some of these modeling strategies to the understand and predict the spread of Chronic Wasting Disease (Zombie Deer) in southern Utah.

**Rated PG**

**Friday April 25, Noon**

**Tutt Science 122- Lecture Hall**

**Speaker:** Gene Abrams, University of Colorado at Colorado Springs**Title:** Fibonacci's Rabbits Visit the Mad Veterinarian**Abstract:** Since its origin (more than eight centuries ago) as a puzzle about the number of rabbits in a fantasmagorically expanding) colony, the Fibonacci Sequence 1,1,2,3,5,8,13,... has arguably become the most well-known of numerical lists, due in part to its simple recursion formula, as well as to the numerous connections it enjoys with many branches of mathematics and science.

Since their origins (less than two decades ago) as puzzles about the number of animals in a(fantasmagorically strange) veterinarian's oce, the (not so well-known) Mad Vet Scenarios have provided a source of thought-provoking entertainment to internet gamers and math enthusiasts alike. In this talk we'll show how Fibonacci's puzzle about rabbits is naturally connected to the puzzles found in the Mad Vet's oce. Along the way, we'll show how an investigation into Mad Vet Scenarios has led to the discovery of some heretofore unrecorded properties of the Fibonacci Sequence.

**Rated G. **This talk is rated G, meaning that it is intended for the most General of audiences. No prior familiarity with Fibonacci's breeding rabbits, or with the Mad Veterinarian's transmogrication machines, or with any other type of fantasmagorical animal population dynamics, will be assumed.

Pizza wiill be served.

**BLOCK 7, 2014 - March/April**

**Wednesday April 9 AND Thursday April 10, 1:30-3:30****Tutt Science 122- Lecture Hall**

**Student Capstone Talks!!**

Light refreshements will be served.

...

**Friday March 28 at NOONTutt Science 122 - Lecture HallSpeaker:** Loren Cobb

**Title:**War, or Peace, or What? A Math Clinic Studies the 21st Century

**Abstract:**This talk will present the results of a semester-long mathematics clinic on predicting the fate of human society in the Twenty-First Century. This clinic, a required course for both undergraduates and graduate students in UC-Denver's Math Department, focused on what we can deduce about the coming century from the use of interlocking dynamic models of population growth, resource exhaustion (food, water, and fossil fuels), urbanization, economic growth, governance, education, and policy optimization. The results were surprisingly positive, although several very specific areas of severe conflict and human misery were identified.

**BLOCK 6, 2014 - February/March**

**Friday March 7 at NOONTutt Science 122 - Lecture HallSpeaker:** Marlow Anderson

**Title:**CYCLOID WARS: A case study in 17

^{th}century mathematics.

**Abstract:** The cycloid is a curve known as the “apple of discord”, because it produced so much controversy and debate among 17^{th} century mathematicians. We will examine this history, with particular attention paid to how these debates illustrate the evolving 17^{th} century answers to the following questions: What is geometry? What is a curve? What is a proof? Almost all of the great mathematicians of the century play a role, from Galileo and his students Torricelli and Cavalieri; to the French mathematicians in the Mersenne circle such as Roberval, Descartes and Fermat; to Leibniz, his mentor Huygens, and his students the Bernoulli brothers; and to Newton and his English predecessors Wren and Wallis.

**Rated PG **

**Friday February 28 at NOONTutt Science 122 - Lecture HallSpeaker:** David Brown

**Title:**The Power of Positive Feedback

**Abstract:** Many biological (and other) systems can be understood in terms of linked positive and negative feedback loops. Over the past 10 years, a common theme in much of my own research has been the role of positive feedback in driving interesting dynamics in biological systems. Here, positive feedback refers to self-reinforcing dynamical processes. I will show how feedback diagrams can be turned into mathematical models consisting of differential equations. I will then show how positive feedback can cause phenomena like bistability, rapid decision making, and spatial pattern formation. I will illustrate these ideas with examples from my research on predator-prey dynamics, chemical communication between bacteria, and the dynamics of tree line on Pikes Peak.

**Rated: PG/PG-13. ** Familiarity with differential equations is assumed, but the technical details will be kept to a minimum.

**BLOCK 5, 2014 - January/February**

**Friday, February 7 at Noon****Tutt Science 122 Lecture Hall**

**Speaker:**Ransanjalee Dissanayaka from Georgia State University

**Title: **Finite State Automata

**Abstract: **Automata Theory is an exciting, theoretical branch of computer science. Simply stated, automata theory is the study of abstract computing devices or "machines". Through automata, computer scientists are able to understand how machines compute functions and solve problems and more importantly, what it means for a function to be defined as computable or for a question to be described as decidable. Before there were computers in 1930's A Turing studied an abstract machine that model the power of real computers. Turing machine allow us to describe precisely what a computing machine could do and could not do. The simplest automata used for computation is a finite state automaton, also called a finite state machine. Finite automaton is a basic model of computational systems with ﬁnite memory. It is much more restrictive in its capabilities than a Turing machine. Finite state machines originally proposed to model brain functions, turned out to be extremely useful for variety of other purposes. They can be used to recognize a type of formal languages called regular languages and used for building certain types of software including lexical analyzers. This lecture will provide an introduction to theory of computation and finite state automata.

**Rating: **PG

**Friday, January 31 at Noon****Tutt Science 122 Lecture Hall**

**Speaker:**Ben Ylvisaker from Swarthmore College

**Title:** All Your Code Are Belong to Us

**Abstract:** Software today exists in a hostile environment. Vandals, thieves, hacktavists, and various government agencies are hard at work subverting computer systems of all sorts. Understand the common attack vectors and what can be done to defend against them is important for nearly everyone in modern society (to different depths, depending on one's role, of course). In this talk we will take a guided stroll through the Common Weakness Enumeration Top 25 software errors. We will cover the basics like stack overruns, script injections and public key cryptography or related topics, according to audience interest.

**Rating:**

20% G

60% PG

20% PG13

**Friday, January 24 at Noon****Tutt Science 122 Lecture Hall**

**Speaker**: Jim Boerkoel from Harvey Mudd College**Title:** Helping Human-robot Teams Stay Coordinated

**Abstract**: An emerging trend in robotics is the development of inherently safe, mobile robots that are designed to assist humans across a wide variety of tasks. Humans are good at intuitively perceiving and quickly adapting to the intentions and conventions of their human teammates with little or no explicit communications. Robots, on the other hand, face the challenge of optimizing and adapting their schedules to, what to them appears to be, their inherently unreliable, non-communicative, and irrational human teammates. A primary goal of my work is to equip robots with methods to quickly and robustly develop plans that (1) adapt to established conventions to successfully negotiate human-oriented environments and (2) augment the workﬂow of human workers to increase overall productivity, safety, and quality. In this talk I discuss a new, distributed robotic scheduling approach that builds on classical shortest-path algorithms to enable a robot to coordinate its activities with its human teammates while providing them flexibility and autonomy.

**Rated PG**: A little undergraduate mathematics assumed.

**BLOCK 4, 2013 - December**

**BLOCK 4, 2013 - December**

**Friday, December 13 at 2:30 pmTutt Science 229**

**Speaker:** Jonathan Portiz **Title:** Computation in Nearby Universes**Rated**: mostly PG with brief interludes of PG13/R (A little to a lot of undergraduate mathematics or computer science assumed.)**Abstract:**

When first introduced, the Turing Machine (TM) seemed an entirely non-physical formalization of computation, but insights from thermodynamics and quantum mechanics showed this was not entirely true. The good news was the discovery of algorithms for quantum TMs which were much faster than the best classical ones -- even breaking important modern cryptographic algorithms. I will describe some of these physical influences on computation, including a brief introduction to quantum computation (which could be useful since all signs point to the imminent arrival of actual quantum computers). I will then propose a way to vary the "structure group" underlying the physics of TMs, in essence to do computation on other universes. A surprising mathematical connection is the relationship of probabilistic computation with some beautiful classical linear algebra.

BLOCK 3, 2013 - November

**ber**

**BLOCK 3, 2013 - Novem**

**Friday, November 15 at 2:30 pmTutt Science 229**

**Speaker:** Charles Morgenstern, CC alum, and graduate student at the Colorado School of Mines.

**Title:** Acoustic Wave Scattering with FEM-BEM **Rated**: PG-13 (A lot of undergraduate mathematics or computer science assumed.)**Abstract:**

This talk is a brief investigation into numeric methods for approximating time harmonic acoustic wave scattering in two-dimensions by bounded inhomogeneities and impenetrable objects. First boundary element methods (BEM) and finite element methods (FEM) are considered alone, and then a new method which couples these two is presented. Using the coupled method global solutions can be computed for problems too complex for a boundary element method alone. Parallel and serial computing performance as well as convergence for the method will be discussed. This is a joint project with Prof. Mahadevan Ganesh at the Colorado School of Mines and is supported by the National Science Foundation.

***

**Friday, November 8 at 12 pmTutt Science, Kresge Lecture Hall (TSC 122)**

**Speaker:** Matthew Whitehead

**Title:** Can Robots Build Robots that Build Robots? **Rated**: G**Abstract:**

Imagine a large warehouse filled with spare electronic and mechanical parts. Is it possible to build a robot that goes into the warehouse and builds an exact copy of itself? If so, then those copies could build more and more robots until the building materials were exhausted.

This talk will explore the computational requirements of machine self-replication using cellular automata. After seeing several examples of simple self replicators, we will then consider replicating Turing-complete machines: machines that can compute any computable function.

***

**Friday, November 1, 2:30pm****Tutt Science 229Speaker: **Dr. Gerald de Souz, PhD from The University of Albany, Albany - New York

**: From Fourier Analysis to Wavelet Analysis**

Title

Title

**Rated: PG-13Abstract**: In this talk, we discuss the origin of Fourier and wavelet analyses and explain the connections between them. In fact, in this presentation we will discuss the following:

How much information is needed to** reconstruct **a situation (signal)? And how we accomplish it? If a situation (signal) is represented by a function f(x). How can we decompose this function f(x) into “simpler” functions and from these “simpler functions” recovery all necessary information about f(x)? – **Building block functions**.

Assume we have too much information about a given situation (signal). How can we throw away some of these information and still have a complete necessary information about the situation (signal)? **Compression**

This is an informative talk and it is suitable for a general audience including students. My friend and late Paul Halmos often advised colloquium speakers “to make the first 15 minutes of their talks accessible to the entire audience. Do you know how difficult that is?” *Notices of the AMS*, February 2009.

I will try to follow his advice.

“Mathematics contains ideas that can be, and deserve to be, communicated to the wider public – even if what is communicated is at the level of appreciation rather than practical knowledge.” – Barbara Burke Hubbard, *The World According to Wavelets*, 1996.

**A bit about our speaker**: Geraldo Soares de souza, PhD from The University of Albany, Albany - New York in the area of Harmonic Analysis. He is currently a full Professor of Mathematics at Auburn University in Auburn, Alabama. He has had several Master and PhD students and publications in his area of expertise. For over 25 years, he has led a series of conferences in Harmonic Analysis and Related Areas at Auburn University sponsored by NSF. He and his colleagues have led several international projects in Africa, South America, Central America, etc. Recently, he and some colleagues have been awarded nearly 500 thousand dollars from NSF and other agencies to create a more robust infrastructure to support mathematics education in Africa.

**BLOCK 2, 2013 - Oc**tober

Denali Molitor, Hanson Smith and Ravi Donepudi (three of our current students) will give a PG rated talk on their research.**Time: **2pm (Note this is not a usual time for us!) **Location:** Kresge Lecture Hall, Tutt Science 122 Reception to Follow. **Titles: **Fractal Analysis, Equal Circle Packing on Flat Tori, Rational Numbers and Function Fields: Similarities and Differences

*The real world is full of rough surfaces, which poses the challenge of how to represent and study these surfaces mathematically. Fractal analysis, although it still poses unrealistic structural restrictions on these rough surfaces, provides a first step toward this goal. By defining Laplacians on fractals, we can also then study how some functions behave on these rougher spaces. This presentation will focus on how space-filling Peano curves can be used to define Laplacians on certain classes of fractals by defining a way to continuously trace through subsequent graph approximations of the fractals.*

**Fractal Analysis (Denali Molitor):****The study of maximally dense packings of disjoint equal circles in di fferent types of containers is a problem in Discrete Geometry that has developed over the past forty years. The optimal densities and arrangements are known for packings of small numbers of equal circles in hard boundary containers, including squares, equilateral triangles and circles. In this presentation, I will explore packings of small numbers of equal circles in a boundaryless container called a flat torus. We will introduce the basic concepts (including the notion of a flat torus, an optimal packing and the graph of a packing), demonstrate many maximally dense arrangements for four equal circles, and outline a proof of their optimality. This research was conducted as part of the 2013 REU program at Grand Valley State University.**

*Equal Circle Packing on Flat Tori (Hanson Smith):***Function fields are interesting analogs of the familiar rational numbers. Although these two objects appear very different at a first glance, they have many properties in common. Many problems that are difficult to solve in the rational numbers become very simple in this new setting. Among the most famous of these is the Riemann Hypothesis, which we will prove in this setting. I also present work I did over the summer relating to extensions of these function fields. There should be something for everyone at this talk. Please pass the word along to anyone who might be interested!**

*Rational Numbers and Function Fields: Similarities and Differences (Ravi Donepudi)*:

**Rating:** PG

*************************

** **

**BLOCK 7, 2013 - **April

**April**

**BLOCK 7, 2013 -****Friday April 12th****Noon in TSC 122 Tutt Science Lecture Hall ****Colin Weir, University of Calgary****Title**: What the L? An Intro and Application to L-functions

**Abstract**: The main goal of this talk is to introduce Artin L-funtions, a generalization of the Riemann zeta function, by wandering various realms in the world of mathematics. These will include small secluded islands within the lands of number theory, abstract algebra, representation theory, and complex analysis. Our travels will eventually lead us to the promised land hidden in the very depths of math world - the fountain of eternal theorems. It is here where we will unravel the mysteries of L and harness the power to produce an infinitude of number theory results.

**Rating:** PG

*************************

BLOCK 6, 2013 - February & March

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BLOCK 6, 2013 - February & March

BLOCK 6, 2013 - February & March

**Monday March 11****Noon - TSC 122 Lecture Hall****Emma Norbrothen **(candidate for visiting professor position)

** Title: Number Systems Base P** Rational numbers can construct the real numbers by using the absolute value metric. Under different metrics, rationals can construct different types of numbers. In particular, the

Abstract:

*p*-norm evaluates how much a prime,

*p*, is a factor in a given rational. We will explore some consequences of the

*p*-norm and what kind of numbers it creates rom the rationals.

**Rating:** PG

***

**Monday March 4****Noon - TSC 122 Lecture Hall****Colin Weir** (candidate for visiting professor position)**Title**: When Primes Factor

**Abstract: **We learned in elementary school that prime numbers are numbers that only factor as 1 times themselves. However, this definition assumes that the only allowable factors are positive integers. What does it mean to be prime if we are allowed to write or ? How can a prime number factor if we extend our allowable factors beyond positive integers? We will explore these questions and many others, eventually leading us to areas of modern research in number theory and cryptography.

**Rating:** Light PG-13 (we will eventually review some abstract algebra)

**Tuesday March 5****2:30 pm- TSC 229 ****Colin Weir** (candidate for visiting professor position)**Title**: Constructing and Tabulating Algebraic Function Fields

**Abstract: **Research into the construction of certain low degree function fields has surged in recent years, in part due to the cryptographic significance of elliptic and hyperelliptic curves. However there is comparatively little data available for higher degree function fields, leaving open many questions about the number of function fields of fixed degree and given discriminant. We present a Kummer theoretic algorithm for constructing degree ℓ dihedral function fields over a finite field with prescribed ramification. We then use this in a tabulation algorithm to construct all cubic function fields over up to a given discriminant bound and compare the data to known asymptotics. We will also survey other applications where these techniques can be made useful, such as constructing curves with many points, or in implementing cover attacks on hyperelliptic curve cryptography. Moreover, we show how our construction techniques can be extended to characteristic zero to construct interesting curves over number fields.

**Rating: **R (some Galois theory will be assumed)

***

**BLOCK 5, 2013 - January & February**

**BLOCK 5, 2013 - January & February**

**Friday February 8****Noon - TSC 122 Lecture Hall****Taylor McNeill** (candidate for visiting professor position)**Title**: Twisted Sisters - How a knot becomes a braid**Abstract: **What do cancer cells, the sun's corona, and your kid sister's hair have in common? Knots! In this talk I'll give a brief introduction to knot theory, including a glimpse into its history and applications. After that we'll discuss braids and their correspondence with knots. Finally we'll show that every knot is a braid in disguise. So say "bye-bye" to those tangled earbuds and "hello" to low dimensional topology.

**PIZZA will be served.Rated:** G

*

**Thursday February 7****3:00pm - TSC 229****Taylor McNeill** (candidate for visiting professor position)**Title:** A new filtration of the Magnus kernel of the Torelli group.

**Abstract:** For a surface S, the mapping class group, Mod(S) is the group of orientation preserving homeomorphisms of S mod isotopy. A key subgroup of the mapping class group, the Torelli group admits a representation onto a linear group via the Magnus representation. For many years it was unknown whether the Magnus representation of the Torelli group is faithful. In recent years there have been many developments on this front including the result of Church and Farb that the kernel of the Magnus representation, denoted Mag(S), is infinitely generated. I show that, not only is Mag(S) highly non-trivial but that it also has a rich structure as a group.

Specifically, I define an infinite filtration of Mag(S) by subgroups, called the higher order Magnus subgroups, M_n. I show that for each n the quotient M_n/M_n+1 is infinitely generated.**Rating:** PG-13

*

**Friday January 252:30pm, **

**TSC 229**

**Kathy Merrill**

**Title**: "Simple wavelet sets in n dimensions" Rating PG

**Abstract: ** A wavelet set in R^n can be defined analytically as a set whose characteristic function is the Fourier transform of a wavelet. Alternatively, it can be defined geometrically as a set that tiles n-dimensional space under both translation by the integer lattice and dilation by an expansive integer matrix. In both contexts, it is somewhat surprising that such sets exist. However, for some dilations, such sets can be found that are geometrically quite simple, consisting of a finite union of convex sets. This talk will describe the generalization to arbitrary dimensions of a two dimensional construction of simple wavelet sets presented in a Fearless Friday in late 2011. In keeping with the title, the proof itself has become simple by making use of a little known combinatorial result. **Rating**: PG (some undergraduate mathematics or computer science understanding)

***

**BLOCK 4, 2012 - December**

**BLOCK 4, 2012 - December**

**Friday, December 7**** Noon, ****TSC 122****Craig Guilbault (University of Wisconsin-Milwaulkee)**

Proof or Swindle? The surprising effectiveness of sketchy technique

**Rating**: PG (some undergraduate mathematics or computer science understanding)

***

**BLOCK 3, 2012 - October**

**BLOCK 3, 2012 - October**

**Friday, November 16**** 2:30pm, ****TSC 122****Marc Chamberland (Grinnell College)**

The Computer's Role in Mathematical Discovery and Proof

**Rating**: PG (some undergraduate mathematics or computer science understanding)

***

**Friday, November 9**** Noon**, **TSC 122****Stefan Erickson (CC)**

Mathematics of Poker

**BLOCK 2, 2012 - October**

**BLOCK 2, 2012 - October**

**Friday, October 5**** 2:30 pm, TSC 229****Jane McDougall (CC)****Title: Harmonic Mappings and Minimal Surfaces (or what I would be doing if I were on sabbatical)** **Abstract**: We will examine briefly some of the highlights in the history of minimal surfaces, with a particular goal of setting in context the classical Scherk doubly-periodic surface and its generalization to the notion of a JS surface. We will then discuss harmonic maps (complex functions which are harmonic rather than analytic) and their connection with minimal surfaces. We explore the advantages of using hyperbolic geometry in constructing harmonic maps with specified properties. Time permitting, we will also consider the construction of harmonic maps onto hexagons that admit a JS surface. Technical aspects of this talk will also be communicated by an appeal to geometric intuition (i.e. there will be pictures). **Rating**: PG13

*****Friday, October 12****NOON TSC 122****Kathy Merrill (CC)****Title: Sunrise, Sunset**

**Abstract**: Although we expect the earliest sunset and latest sunrise to occur on the winter solstice, they do not. The mathematics behind this phenomenon is related to a curve called the analemma. This talk will explore the factors that complicate our view of the path of the sun, and develop a model that accounts for the difference between the solstice and and the extrema of sunrise and sunset times.

**Rating**: G/PG

***

**Thursday October 18****, 20126:30 pm, Packard Hall**

**Joel E. Cohen (The Rockefeller University and Columbia University, New York)**

**Sustainability, nations, globalization: can we have them all with 7 billion people and more?**

Whatever sustainability means, it includes protecting Earth's physical, chemical, and biological environments locally and globally for time periods of human concern, subject to natural variability. National governments usually view their primary responsibilities as lying within their own boundaries on the time scales of the next election cycle or, at most, the education of the next generation. Global businesses look for resources and markets wherever they offer economic advantage to globally dispersed stockholders and management in time for the next quarterly report or the next annual meeting of stockholders, or (rarely) over the coming decade. Since 1820, aggregate economic activity grew more than 74-fold, and now the material inputs and outputs of economic activity have magnitudes comparable to what Earth can yield and absorb. Some scientists warn of possible abrupt changes in vital Earth systems. Half of today's 7 billion people remain poor. Billions more people and massive urbanization and ageing are in prospect in the coming decades. In this new world, it is time to create the information, incentives and institutions needed to reconcile environmental sustainability, national governance, and economic globalization.

**Friday October 19, 2012 ****Noon****Joel E. Cohen, The Rockefeller University and Columbia University, New York**

**Title: **Taylor's power law of population variation, stochastic models of population dynamics, and abrupt change

**Abstract:**

Vital Earth systems have changed abruptly in the past. Some scientists, based on models and data, warn of possible abrupt changes in prospect. My research on mathematical models of population dynamics led recently to an unexpected theoretical example of a singularity in the exponent of Taylor's power law of population variability (or fluctuation scaling) in response to smooth changes in the power spectrum of an environmental factor like temperature. Abrupt changes can arise even where least expected. Are we ready?

** **

BLOCK 1, 2012 - September

**BLOCK 1, 2012 - September**

Friday, September 7

2:30 pm, TSC 229

Fred Tinsley (CC)

**Title: Noncompact Manifolds that are inward tame or "what I did on my sabbatical"****Abstract**: This project has its roots in the summer geometric topology workshop

at Colorado College in 2000. By 2002 Craig Guilbault (UW-Milwaukee) and I had

formulated a ‘10-year plan’ to completely understand the ends of open, inward-

tame manifolds in high dimensions. This effort has used extensively theories from

manifold homology and cohomology, combinatorial and geometric group theory,

algebraic K-theory, and manifold topology. I will describe the problem and report

on our progress. This talk should be accessible to any student who has completed

MA 375 Introduction to Mathematical Analysis. I even will pose a problem in

computer graphics for which a solution would be quite helpful.**Rated**: R**Friday, September 14**

1:00 pm, TSC 122 (Lecture Hall)

Matthew Whitehead (CC)

**Title: **The End of Privacy?

**Abstract: **Do you value your personal privacy? Are you concerned about the degradation of privacy in the Internet Age? Your answers may depend on your age and cultural background, but one thing is clear: technology and data mining are redefining our notions of privacy. In this talk we will discuss domestic surveillance using hummingbird-sized drones, how some of your anonymous data might not be quite so anonymous, and how people are tracked online and shown personalized advertisements chosen by machine learning systems.

**Rating:** G**Friday, September 21**** 2:30pm, TSC 229****Mike Siddoway (CC)**

**Title: **"A Wind Erosion Equation"

**Abstract:**Wind erosion is still a major problem in the Great Plains. If you see dust rising off an acre of land, this translates to about a dime's width of soil being liberated from the surface, approximately five tons of particulate. Beginning in the late 1950's, scientists in Canada and the US began an effort to study the factors behind wind erosion. The idea being: "The better we understand the mechanisms of soil loss to wind, the more we can do to prevent catastrophic losses, like those suffered during the Dust Bowl years." The "equation" (really an algorithm) that was devised in the mid-60's, though improved upon in many ways since, is still the most widely used around the world. The modeling of "wind erosion," the problem that first brought mathematics deeply into the study of soil processes, is still one of the most actively pursued areas in all of soil science.

**: G**

**Rated**

**BLOCK 8, 2012 - April - May**

Friday, April 27

2:30 pm, TSC 229

Steven Janke (CC)

**Title: Two Statistical Paradigms: Bayesian versus Frequentist**

Abstract: The vast majority of statistical analyses done these days is based on Frequentist methods. Certainly all undergraduate statistics courses feature this approach. Yet, the Bayesian method has been around for nearly as long and has many features to recommend it. This talk will give an overview of both methods using some specific estimation examples and then list some of the mathematical and philosophical points of contention between the two. Using some recent results, it becomes a little clearer how to proceed in refining statistical techniques.

Rated: PG-13

Friday, May 4

12:00 pm, TSC 122

Dr. Jugal Kalita (UCCS, Computer Science)

**Title: Automated Data Extraction: Retrieving Temporal and Geospatial Information from Text**

Abstract: This talk discusses a system that extracts and displays temporal and geospatial entities in text. The first task involves identification of all events in a document followed by identification of important events using a classifier. The second task involves identifying named entities associated with the document. In particular, we extract geospatial named entities. We disambiguate the set of geospatial named entities and geocode them to determine the correct coordinates for each place name, often called grounding. We resolve ambiguity based on sentence and article context. Finally, we present a user with the key events and their associated people, places, and organizations within a document in terms of a timeline and a map. For purposes of testing, we use historical Wikipedia articles, such as those describing wars, battles, and invasions. We focus on extracting major events from the articles, although our ideas and tools can be easily used with articles from other sources such as news articles.

Rating: PG

Friday, May 11

2:20 pm, TSC 229

Stefan Erickson (CC)**Title: Pairing-Based Cryptography**

Abstract: In 1985, Victor and Miller and Neal Koblitz independently suggested that the discrete logarithm problem for elliptic curves could be used as the basis for cryptosystems. Elliptic curves offered faster computations and less bandwidth for the same level of security as traditional cryptosystems such as Diffie-Hellman and RSA. Over the past 27 years, elliptic curve cryptography has grown into one of the most interesting applications of theoretical number theory.

A recent development in elliptic curve cryptography is the use of bilinear pairings. At first, pairings were used as an attack on supersingular elliptic curves. Later, constructive applications came in the form of tripartite key exchange, identity-based encryption, and short signatures. In order for pairing to be used, however, efficient implementation and pairing-friendly curves had to be developed. This talk will provide an overview of many of these exciting ideas and applications.

Rated PG13

**BLOCK 7, 2012 - March - April**

Friday, March 30

2:30 pm, TSC 229

Robin Wilson (Open University, UK)

**Title: 100 Years of Graph Theory**

Abstract: The years 1890-1990 saw a complete change in the subject now known as graph theory -- from a collection of individual results to a more coherent theory. In this talk I survey the century decade by decade, outlining some of the mossignificant advances. No previous knowledge of graph theory is required.

Rating: PG-13

Friday, April 6

12:00 pm, TSC 122 (pizza will be served)

Robin Wilson (Open University, UK)

**Title: Great Mathematicians**

Abstract: Last September 'The Great Mathematicians' was published. Written by Raymond Flood and myself, it is a coffee-table book of 200 pages and 350 pictures which is very inexpensive and everyone should buy a copy! In this illustrated talk, I present ten of the featured mathematicians, ranging from Pythagoras and Fibonacci to Florence Nightingale and Hilbert.

Rating: G

**BLOCK 6, 2012 - February - March**

Friday, Feb. 24

2:30 pm, TSC 229

Dr. Loren Cobb, U.C. Denver

**Title: Tracking Epidemics and Wildfires**

Abstract: This will be an overview, with lots of computer animations, of the recent research of the Data Assimilation Group at the University of Colorado Denver. We apply our newly developed very-high-dimensional filters to incoming data in real time, to track and forecast emerging epidemics, wildfires, weather fronts, hurricanes, and anything else that has nonlinear spatial dynamics. Of special interest may be our simulation of the first wave of the Black Death in Europe, from 1347 to 1350.

Rated: PG-13

**Wednesday, Feb. 29**

12:00 pm, Olin 1 *(note the special location, under the Olin "Fishbowl")*

(pizza will be served)

Dr. Marlow Anderson, CC

**Title: A Celebration of Leap Day**

Abstract: Why do we have a leap day, anyway? In this talk I shall answer this question and more, exploring the history, customs and background for Leap Day. We’ll have a look at the inadequacies of the Roman calendar, and see why and how Julius Caesar was motivated to introduce calendar reform. But the flaws in the Julian calendar became more and more troublesome as the centuries passed, especially for the Church fathers charged with computing the date of Easter, and eventually Pope Gregory XIII convened a commission in the 1580s to reform the calendar again. We’ll examine some interested proposed alternatives, including one by Omar Khayyam. In the process we’ll encounter some interesting astronomy and mathematics.

Rated: G

**Friday, March 9 **

12:00 pm

TSC 122

Speaker: Andrea Bruder, Colorado College

**Title: Period 3 Implies Chaos **

Abstract: In 1964, Sharkovsky introduced a new ordering of the positive integers and proved that if a continuous function mapping an interval to itself has a point of period *k,* then it has a point of period *r* for all *r* following *k* in this ordering. His result unfortunately did not receive the attention it deserved. A decade later, Li and Yorke published a special case of Sharkovsky's theorem in their well-known paper "Period 3 implies chaos", which coined the word chaos in the mathematical sense. We will see how Elaydi proved a converse of Sharkosky's theorem recently by constructing piecewise continuous functions with certain periods, and we will show how continuously differentiable functions with the same properties may be constructed.

Rated: PG-13

**BLOCK 5, 2012 - January - February**

Friday, January. 27

2:30 pm, TSC 229

Marlow Anderson, Colorado College

**TITLE: Cyclotomic Polynomials are Irreducible**

The cyclotomic polynomial for a positive integer n is the monic polynomial whose roots are exactly the primitive n-th roots of unity. This polynomial is just

(x^n-1)/(x-1) if n is prime, but is a polynomial of order phi(n) in general (where phi is Euler’s phi function). It is a common exercise in a beginning abstract

algebra course to prove that the cyclotomic polynomials for primes are irreducible – they can’t be factored. This is also true for non-prime n. I recently

reviewed a paper for the Monthly which consisted of a compendium of many different historical proofs of these results. In this talk I’ll give some general

background, and have a look at some of the most interesting and illuminating of these proofs, including proofs by such figures as Gauss, Eisenstein,

Dedekind and Landau. Along the way we’ll have the chance to revisit such interesting topics as roots of unity, symmetric polynomials, and Gauss’s Lemma.

We’ll try to avoid the irresistible temptation to do some Galois theory!

Rated PG13.

**Thursday, Feb. 2 **

12 pm, TSC 122 (pizza will be served)

Dr. Beth Malmskog, Wesleyan University

**Title: Beyond Pig Latin: Private Communication over Pubic Channels**

Abstract: Credit card numbers, legal documents, medical records, business correspondence, love letters: just a few examples of information that we want to share with certain parties but definitely do not want to share with others. Governments and corporations have similar needs for privacy in communication. How can information be shared privately when eavesdroppers overhear every word? In this talk, I will discuss a few of the methods that people have used to send secret information over public channels. We discuss a sampling of the history and mathematics of symmetric-key cryptography, Diffie-Hellman key exchange, and the big picture of public-key cryptography. If time permits, we conclude with a look at the future—how can we maintain privacy after quantum computers render our current techniques useless?

Rated: PG

**Friday, Feb. 3 **

3pm, TSC 229

Dr. Beth Malmskog, Wesleyan University

**Title: Automorphisms of a Family of Maximal Curves**

Abstract: Over the real or complex numbers, a system of equations can have infinitely many solutions. That means that curves over these fields can have infinitely many points. If we instead look for solutions over a finite field, there can clearly only be finitely many. The Hasse-Weil bound gives an upper limit for how many points can lie on a curve over a finite field. Curves that actually attain this upper bound are called maximal. Maximal curves have some very interesting properties, and can be used to create good error-correcting codes. They often have rich structure, meaning many symmetries (automorphisms). Recently, Rachel Pries (CSU), Robert Guralnick (USC), and I found the full symmetry group for the curves in a new family of maximal curves. I will discuss the background and context of our result and give an outline of the proof.

Rated: R

**Friday, Feb. 10**

12 pm , OLIN 1 (lecture hall under the "Fishbowl")

Dr. Radu Cascaval, UCCS

**Title: Control Mechanisms in Spatial Networks**

Abstract: The dynamic control of spatial networks presents numerous challenging problems, both theoretically and computationally. I will describe a study of the cerebral autoregulation (CA) mechanism in the cardiovascular network, in which both the heart rate (HR) variability and the peripheral resistance (PR) variability act as controls for the pressure and flow rates throughout the system, with the aim of maintaining constant flow in the brain during dynamic changes. The dynamics of the network is characterized by distinct patterns of the nonlinear pressure and flow distribution, and related inverse problems will be presented. Furthermore, we will discuss evidence that the complexity of the underlying network and the presence of different scales in the model influences the effectiveness of these control mechanisms.

Rated: PG -13

**BLOCK 4, 2011 - December**

Dec. 2, 2:30 pm in TSC 229.

Speaker: Kathy Merrill

**Title: Simple wavelet sets in R^2**

Abstract: A wavelet set in R^2 can be defined analytically as a set whose characteristic function is the Fourier transform of a wavelet. Alternatively, it can be defined geometrically as a set that tiles the plane under both translation by the integer lattice and dilation by an expansive integer matrix. In both contexts, it is somewhat surprising that such sets exist. However, for some dilations, such sets can be found that are geometrically quite simple, consisting of a finite union of polygons. This talk will present new results about which matrix dilations have such simple wavelet sets.

Rated: PG-13 (linear algebra and some analysis required)

**Monday, December 12 at NOON in TSC122**

Pizza will be served

Speaker: Matthew Whitehead, Colorado College**Title: Evolving Video Game Bots**

Abstract: Many modern video games have computer-controlled characters (bots) that are painstakingly crafted by hand to behave in believable ways. Unfortunately, this process is expensive, time-consuming, and often leads to bots that act in peculiar ways. In this talk we will look at the automatic creation of bot code using biological evolution for motivation. This process of growing code is called genetic programming and can be used to solve a variety of problems without a human programmer manually writing the source code.

Rating: PG

**Tuesday, December 13 at 2:30 pm in TSC229**

Speaker: Matthew Whitehead, Colorado College**Title: The Provenance of CBR Cases**

Abstract: Case-based reasoning systems routinely record the results of prior problem-solving, but not the provenance of new cases: the way in which the new cases were derived. In this talk we will investigate the value added by tracking case provenance information in CBR systems, especially when timely feedback may not be available. In particular, we will examine the effect of using provenance to propagate case maintenance in order to target maintenance effort. We will look at results of such a system on two well-known machine learning datasets.

Rating: PG-13

**Thursday December 15 at Noon in TSC122**

Pizza will be served

Speaker: Michael Hay, Cornell University

**Title: Public Use of Private Data**

Abstract: This talk explores the following question: How can an organization --

such as a hospital -- release sensitive data about individuals -- such as

patient medical records -- to the public without violating the privacy of

those individuals? In today's digital age, this problem is much harder

than you might expect. I will show some of the pitfalls of simple

solutions that rely on anonymity and then look at how randomization

(i.e., coin flips) can play a role in protecting privacy.

Rating: G/PG (basic probability and statistics are used, but most of the

talk is accessible without this background)

**Friday, December 16 at 2:30 pm in TSC229**

Speaker: Michael Hay, Cornell University

**Title: Analyzing Private Network Data**

Abstract: Many of our social interactions -- email, phone, online networking --

leave behind a digital trace. These traces are a rich source of data for

social science and present an opportunity to fundamentally advance our

understanding of many facets of society. However, these traces are also a

liability. The collected data can be highly sensitive, revealing intimate

details of personal relationships. Concerns about privacy make data

owners reluctant to share the data and threaten to disrupt a vast array

of new digital services, social science, and economic opportunities.

In this talk, I will describe two different approaches for addressing the

problem of enabling the accurate analysis of private network data. In

the first approach, the network data is "anonymized" prior to being

released to the public. While this is a natural approach, I will provide

evidence that the privacy protection is much weaker than one might

expect. In the second approach, I will describe algorithms for releasing

randomly perturbed statistics about the network topology. With this

approach, it is possible to release accurate approximations of key

network statistics while simultaneously guaranteeing rigorous privacy

protection.

Rating: PG-13

**Monday December 19, 2011**

Speaker: Ben Widermann, Harvey Mudd College

Time: 12 Noon

**Title: Keeping Secrets: Language-based Secure Information Flow**

Rated PG for mild programming (method calls and if statements) and for propositional logic.

Abstract:

We all have secrets. Sometimes we entrust those secrets to software. And sometimes software blabs. Maybe your phone posts your whereabouts to Facebook, or your favorite website exposes your credit card. Is there any way to guarantee that our software won't intentionally or accidentally leak our secrets? In this talk, we explore this problem by developing a small programming language with an interesting property: every program in this language is guaranteed not to leak. Along the way, we'll run up against some of the fundamental limits of computation.

Note: this talk will be interactive -- please come prepared to do a bit of problem-solving.

**Tuesday December 20, 2011**

Speaker: Ben Widermann, Harvey Mudd College

Time: 12:30pm

**Title: Modular and Efficient Database Access**

Rated R for moderate programming (Java + SQL; virtual method dispatch) and for some graduate-level computer science (abstract-interpretation-based program analysis).

Abstract:

All our major financial, health, and government enterprises rely on software systems that manipulate large datasets. These "enterprise applications" are notoriously difficult to design, debug, and maintain. Because enterprise applications specify business logic rules over large datasets, they often combine a general-purpose programming language with a database. The clean, effective integration of these two semantic domains has eluded researchers for over 30 years. As a result, today's programmers are forced to throw out their best programming practices for the sake of program efficiency.

This talk presents a different approach, namely that program analysis and programming language design can help programmers write enterprise applications that are both well-designed and efficient. The premise of our approach is that programmers perform premature optimization when they write SQL queries. Instead, programmers should describe computations over database data in the same way they describe computations over in-memory data. The compiler should differentiate the computations and automatically prefetch database values on behalf of the programmer. We describe two techniques that accomplish this goal: one technique based purely on static program analysis and another technique that combines static program analysis with elegant language design.

**BLOCK 3, 2011 - November**

Nov. 4, 2:30 pm in TSC229

Speaker: Dr. Zak Mesyan, UCCS

**Title: Groups where free subgroups are abundant **

SLIDES FROM TALK (PDF)

Abstract: About 20 years ago it was shown that in the group of all

permutations of the natural numbers "almost all" (in some topological

sense) finitely generated subgroups are free. Since then a number of

other groups with this property have been discovered. I will discuss

these examples and the recent efforts to study this phenomenon

systematically.

Rating: R

**Nov. 11, 12:00 pm in TSC122 (pizza is served)**

Speaker: Dr. Beth Schaubroeck, USAFA

**Title: Julia Sets and Surfaces**

Abstract: In this talk we will discuss Julia Sets and surfaces related to them (joint work done with Julie Barnes, Clinton Curry, and Elizabeth Russell). I will also talk about how mathematics research can advance in surprising ways. Although the topic uses complex variables, all definitions and terms will be described at a very introductory level, and the only prerequisite for understanding the concepts involved will be calculus.

Rating: PG

**Thursday, Nov. 17, 12:00 pm in TSC 229**

Speaker: Matthew Whitehead**Title: Information Filtering using Machine Learning**

Abstract: Internet searches often return millions of pages of results, many of which are marginally related (or completely unrelated) to the user's actual target. Google News lists dozens of news stories on its front page and each story often has thousands of related articles. Sometimes it can be quite difficult to process and filter through all this data to get to the information that you really want. In this talk we will discuss using various machine learning methods to help us deal with information overload. In particular we will look at the problems of dealing with ambiguous search queries and summarizing news articles.

Rated: PG-13

**Nov. 18, Noon, in TSC122**

Speaker: Dr. James Powell, Utah State

**Title: The Timing of Insect Development and Trajectory of Bark Beetle Outbreaks**

Abstract: Temperatures directly but nonlinearly influence the rates at which insects complete development in their various life stages and therefore the timing (phenology) of their emergence. The mountain pine beetle (Dendroctonus ponderosae Hopkins), is an aggressive insect which attacks living pine host trees (genus Pinus). These trees have significant defensive mechanisms, requiring the beetles to attack en masse to successfully colonize. Since the beetle larvae consume the phloem underneath the bark they exhaust their host, requiring dispersal to new hosts every year. Adult beetles must colonize in a relatively narrow window of time to take advantage of warm temperatures for oviposition but also allowing cold-hardened larvae to emerge before winter freeze. Thus the beetles exist in a precarious niche depending on carefully synchronized timing. Changing temperatures in Western North America have broadened that niche across vastly larger regions, leading to mountain pine beetle outbreaks across the West. Impacts due to bark beetles are currently larger than those due to fire in these ecosystems. This talk outlines the development and application of mathematical models for bark beetle phenology and how the dynamics of phenology influences the course and severity of outbreaks.

Rating: PG

**BLOCK 2, 2011 - October**

Friday Oct. 7, 2:30 pm,

TSC 229

Speaker: David Brown, Colorado College**Title: Quorum sensing in bacteria: Simple models and general results**

Abstract: Many bacteria have the ability to detect and respond to their local population density, a phenomenon known as quorum sensing. Over the past decade, a number of mathematical models have explored quorum sensing in several species. While they have shared some core predictions, the models have generally been system-specific, and have used a variety of incompatible assumptions. I will talk about my recent attempt to bring some coherence to the subject by carefully formulating and analyzing simple, generic models of quorum sensing. I will highlight the need to derive differential equations from first principles, show how the models predict key similarities and differences between different kinds of bacteria, and explore the role that spatial structure may play. My methods include qualitative analysis and numerical solutions of ODEs and PDEs.

Rating: PG-13

**Friday, October 14, Noon**

Kresge Lecture Hall (Tutt Science Center 122)

Speaker: Barry Balof, Whitman College

(Barry is a CC alum, and his talk kicks of our celebration of Homecoming Weekend.)

**Title: "Fibonacci and Friends: Famous Number Sequences and the Liberal Arts"**

Abstract: One of the most ubiquitous constructions to come out of mathematics is

the Fibonacci Sequence, which has applications to seemingly myriad

different fields, such as Biology, Art, Music, and even Contract

Bridge. But does the Fibonacci Sequence and its associated Golden

Ratio tell the whole story? Are there other, less heralded sequences

and constants that arise in other fields? In this talk we will look

at the Padovan Sequence, and its applications to architecture, as well

as Coordination Sequences and their application to geology.

Rating: G

**Friday Oct. 21, 2:30 pm, **

TSC 122 Lecture Hall

Speakers will be Elise Hellwig and Hannah Thompson, CC students who will speak about research in mathematical biology that they did last summer. Each talk will last 30 minutes, be rated PG-13, and involve population dynamics.

Speaker 1 : Hannah Thompson **Title: Foraging Behavior and Pattern Formation in a Predator-Prey Model **

Abstract: I will be presenting a model of ladybugs (*Coccinella* *septempunctata, Hippodamia convergens*) and their aphid prey (*Aphis* *helianti*), which are patchily distributed on *Yucca glauca *near Colorado Springs. Fieldwork has shown that predation by ladybugs causes high variation in aphid population levels over short spatial scales (several meters). Here, we use bifurcation analysis of a two-patch ordinary differential equation model to investigate how ladybug foraging behavior leads to this heterogeneity. We have found that diffusion and density independent immigration of ladybugs promote heterogeneity between the patches in our model.

Speaker 2: Elise Hellwig **Title: Metapopulation Modeling and Analysis with Demographic Stochasticity **

Abstract: As human expansion claims more land, previously connected habitats become fragmented, converting larger populations into smaller subpopulations connected through migration. Demographic stochasticity has been shown to have a significant effect in small populations. We explore how taking demographic stochasticity into account affects metapopulations which are typically made up of small subpopulations. We develop a density dependent model using Markov matrices to simulate a metapopulation and compare the predictions with those of a deterministic model. We analyze the asymptotic population size predicted by the deterministic model, and the mean time to extinction and the quasi stationary distributions predicted by the Markov model. We show that there is a significant difference between the two models under certain conditions and that the deterministic model often overestimates metapopulation persistence.

**BLOCK 1, 2011 - September**

Friday, Sept. 9; 2:30 pm

Kresge Lecture Hall (Tutt Science Center 122)

Speaker: Molly Maxwell, Colorado College

**Title: Combinatorial Identities: Binomial Coefficients, Pascal's Triangle, and Stars & Bars**

SLIDES FROM TALK (PDF)

Abstract: Binomial coefficients form the entries in Pascal's Triangle and appear frequently in combinatorial identities. In this talk, we'll look at several identities that appear in Pascal's Triangle and provide combinatorial proofs for some of these. Binomial coefficients also appear in a famous identity for the compositions of positive integers. We'll use a combinatorial "stars & bars" argument to prove this identity.

Rating: G

**Friday, Sept. 16, 12:00 pm (pizza served)**

Kresge Lecture Hall (Tutt Science Center 122)

Speaker: Mike Siddoway, Colorado College

**Title: The Field of Origami Numbers**

Abstract: In 8th Block of 2010 I gave a talk which included discussion of origami constructions and Groebner Basis techniques for determining whether certain figures could be "paper-folded" and if so, what sequence of moves could produce the desired object. In the questions following the talk, Marlow asked where the "Field of Origami Numbers" fit in the chain of fields beginning with the rational numbers. We'll answer this question in the "one-fold-at-a-time setting" and see where the field of paper-folded numbers sits in relation to the classical field of ruler and compass constructed numbers, and to fields arrived at using more exotic tools.

Rating: G

**Friday, Sept. 23, 2:30 pm**

Tutt Science Center 229

Speaker: Greg Oman, UCCS

**TITLE: An independent axiom system for the real numbers**SLIDES FROM TALK (PDF)

ABSTRACT: Since the birth of formal mathematical logic, mathematicians have considered it aesthetically pleasing to axiomatize theories (which are axiomatizable) irredundantly; that is, in such a way that no axiom may be proved from the remaining axioms. In this case, such an axiom system is said to be independent. A famous early example of an independent axiom system is Hilbert's (original) axioms for Euclidean geometry. Since then, independent axiom systems have been found for various mathematical structures, including fields and vector spaces. To illustrate, it is a fairly well-known exercise in basic algebra to prove that the commutative axiom for addition is redundant in the set of axioms for a ring with identity. In this talk, we present an independent axiomatization of the complete ordered field of real numbers. In the presence of the completeness axiom, we will see that a surprising number of axioms can be 'thrown away' (i.e. can be proved from the remaining axioms). This talk should be accessible to all mathematics faculty as well as advanced undergraduates with a background in real analysis.

Rating: PG-13

## Block 8 2011

**Friday of Week 1 (** **April ** **29** **, 2011** **) at 2:30 PM in ** **T****utt Science 229**

*Harmonic maps and Scherk surfaces Part II. * *by Jane McDougall* Rating PG13

ABSTRACT: This talk is a continuation (with some review) of the February talk with the same title. In this talk I will look into the geometry that underlies the relationships among domain and range of harmonic maps of a specific type, and the (second complex) dilatation as a “measure of non-conformality”. Hyperbolic geometry will be particularly relevant.

Audience: Some understanding of hyperbolic geometry helpful but not necessary. Some understanding of complex variables helpful but not necessary.

February 25th abstract: Minimal surfaces are embedded surfaces in three dimensional space with the property that they are locally area-minimizing. Scherk's doubly periodic surface was developed in 1831 by Heinrich Scherk (along with the singly periodic Scherk surface). Under appropriate parametrization, any minimal surface gives rise to a harmonic map, which is defined as a complex valued function that is harmonic and one-to-one on a complex domain. Using complex variables we can discuss the situations under which a harmonic map "lifts" to a minimal surface. We will focus on minimal surfaces related to Scherk's doubly periodic surface, and also consider the related harmonic maps as interesting functions in their own right.

**Friday ****of Week 2 (** **May 6, 2011** **) at noon in Tutt Science 122** (Kresge Lecture Hall) *The Kalman Filter: an algorithm for dealing with uncertainty.* by Steven Janke Rating PG

ABSTRACT: Virtually all processes and measurements in the world are victims of error, often called noise. A robot never goes exactly as far as it should and never turns to exactly the direction programmed. With some probability and a classic algorithm, the Kalman Filter, we can at least manage all this uncertainty. This talk introduces the ideas from probability necessary to understand how error propagates and how even noisy measurements can help control the situation.

**Friday of Week 3 (** **May 13****, 2011** **) at 2:30 PM in ** **T****utt Science 229**

*Apollonian Circle Packings*

*.*by Stefan Erickson Rating PG

Apollonius's Theorem states that given three mutually tangent circles, there are exactly two circles which are tangent to all three. Apollonian circle packings are produced by repeating the construction of mutually tangent circles to fill all remaining spaces. A remarkable consequence of Descartes' Theorem is if the initial four tangent circles have integral curvatures, then all of the circles in an Apollonian circle packing will have integral curvatures. This process results a sequence of integers with fascinating arithmetic properties.

In this talk, we will investigate the arithmetic properties of Apollonian circle packings. We will describe the Apollonian group action on the set of Descartes quadruples. We will talk about modular restrictions and density conjectures and theorems. Finally, we will show a correspondence between the root quadruples and reduced binary quadratic forms and answer an open question about finding the root quadruple of a given Descartes quadruple.

## Block 7 2011

**Friday of Week 1 (** **April 1, 2011** **) at 2:30 PM in ** **T****utt Science 229**

*CM elliptic curves, applications, and generalizations*, by Sunil Chetty: Rating PG-13 to R

**Friday ****of Week 2 (** **April 8, 2011** **) at noon in Tutt Science 122** (Kresge Lecture Hall) *Design and Construction of Internet Search Engines* by Matthew Whitehead: Rating PG

ABSTRACT: We all use search engines to find relevant information online. For almost any search query we nearly instantly receive page after page of interesting results. In this talk we will discuss the theory of search engine design as well as some of the details of implementation. We will also consider some of the problems search engines will have to deal with in the future to maintain good performance.

## Block 6 2011

**Friday of Week 1 (** **February 25, 2011** **) at 2:30 PM in ** **T****utt Science 229**

*Harmonic maps and Scherk surfaces,*by Jane McDougall Rating PG13

**Friday ****of Week 2 (** **March 4, 2011** **) at noon in Tutt Science 122** (Kresge Lecture Hall) *NOVA Series on Artificial Intelligence**, hosted *by Matthew Whitehead Rating G/PG

ABSTRACT: Matthew is teaching Artificial Intelligence this block. This particular NOVA is appropriate to his class. Although it is not a talk, you may still summarize it and turn in your summary as one the four required for graduation. Try to sit near the front. Matthew is unsure of the film quality :) .

**Friday of Week 3 (** **March 11, 2011** **) at 2:30 PM in ** **T****utt Science 229**

*Why is Everyone Interested in Quantum Graphs? *by Bob Carlson Rating: PG-13

ABSTRACT: There has been a recent explosion of research activity at the

intersection of graph theory, linear algebra, differential equations,

functional analysis, and applications. We will initially consider some of the

motivating applications and a bit of history. The interaction of

linear algebra and graph theory will be reviewed. Differential

equations will then make their appearance. Having set up the mathematical

models, some recent results and open problems will be discussed.

## Block 5 2010

**MONDAY, January 31, 2011, at Noon, in Tutt Science 122 (Lecture Hall) ***Bioeconomics and optimal control of the spread of invasive species,* by Dr. Alan Hastings, U.C. Davis

Rated PG

Abstract: This talk will focus on the development of approaches for designing optimal controls for invasive species, with an emphasis on the spread stage (after establishment). The importance of the details of the model will play a role. Examples will be drawn from Spartina alterniflora and other species.

**FRIDAY, ** **February 11, 2011**, **at 2:30 PM in** **T****utt Science 229**

*High-dimensional knots *by Fred Tinsley

Rating: PG-13

ABSTRACT: In Fred's recent work on the ends of high-dimensional surfaces, issues of high-dimensional knotted surfaces have arisen. Fred will discuss the similarities and differences between these knots and those in three dimensions that we know and love He also will show how these knots arise naturally in understanding ends of high-dimensional surfaces.

## Block 4 2010

**Friday, December 3, 2010, at 2:30 PM in Tutt Science 229**

*Primary Decomposition of Conditional Independence Ideals* by Amelia Taylor Rating: R

ABSTRACT: A conditional independence ideal is an ideal generated by binomials which describe the conditional independence relations of a statistical model. The primary decomposition of an ideal is a decomposition that is for ideals what a prime factorization is for integers. The primary decomposition of a conditional independence ideal is interesting as it can give an alternate description of the independence relations for the model or illustrate the failure of the intersection axiom, among other applications and interpretations. I will define conditional independence ideals and then describe what is currently known about their primary decompositions including a new result that is joint with Irena Swanson.

**F****riday, December 10, 2010 at NOON in Tutt Science 122** (Kresge Lecture Hall)*Making Movie Recommendations Using the Netflix Dataset* by Matthew Whitehead Rating: PG

ABSTRACT: Have you ever wondered how Netflix recommends new movies to you based on previous movies that you've seen and rated? These recommendations are made by taking into account movie ratings by other Netflix customers. This process is called collaborative filtering. In this talk, we will discuss several collaborative filtering algorithms and see how they perform on the publicly available Netflix ratings dataset. We will also discuss the results of Netflix's recent $1,000,000 collaborative filtering challenge.

**Friday, December 17, 2010 at 2:30 PM in tutt Science 229**

*Irreducible Cubics Modulo Five**: *by Marlow Anderson Rating: PG-13

ABSTRACT: When you consider the integers modulo 5, you get a field F_5 with five elements. There are exactly 600 (=4.5.5.5) cubic polynomials with coefficients from this field. It turns out that exactly 160 of these polynomials cannot be factored – they are irreducible. Recently, Hendrik Lenstra has published a short article in the American Mathematical Monthly which gives a cute proof of this fact. In this talk, we will look at several ways to perform this count. In the process, we will give a gentle and student-friendly introduction to the construction of finite algebraic extensions of F_5. We will then encounter a lot of the fundamental theorems about finite fields.

### Block 3 2010

**Thursday November 4, 2010, 3-4pm****Yulan Qing (CC Alumna)**

A talk for students, including those interested in learning more about graduate school experiences.

Title: The Geometry of Groups

Abstract: You know what a group is, but do you know what a group looks

like from far away? We will see a few examples of how groups, graphs, and other spaces can be connected, and literally "draw" a few conclusions about the group of interest.

**Friday, November 5 at 2:30 pm**

*From Bagels to Boundaries of Groups*by Yulan Qing (CC Alumna)

Rating: PG

Abstract: Take three bagels and splice them together to form a pretzel. A tiny ant walking along this pretzel explores the space and gets the idea that the space is flat, with vertical walls that it can climb along, and that the space is infinite. We give the ant the concept of a visual boundary, and show that if we change the geometry of the bagel, the topology of the boundary also changes. This is a particular example of a right-angled Coxeter group. It is an open question as to whether right-angled Coxeter groups have unique CAT(0) boundary. This example could potentially give a negative answer to that question.

**Friday, November 19 at 2:30 pm, TSC 229 **

*Congruent Numbers and Elliptical Curves*by Sunil Chetty

Rating: PG-13

## Block 2 2010

**F****riday, ****October 8** **, 2010 at noon in Tutt Science ** **122 **(Kresge Lecture Hall)

*Chaos and the Mandelbrot set* by Andrea Bruder

Rating: PG

ABSTRACT: The Mandelbrot set is certainly the most popular fractal and has even been called the most popular object of contemporary mathematics, the most beautiful and the most complex object which has ever been made visible. Benoit Mandelbrot conducted his experiment in 1979 and, realizing that he had discovered a mathematical gem, he pushed further to revitalize an area of mathematics that had been dormant for over 60 years.

**F****riday, ****October 15** **, 2010 at noon in Tutt Science 122** (Kresge Lecture Hall)*Introduction to Web Development* by Benjamin Thomas

Rating: PG

ABSTRACT: Ever wanted to know what happens when you load up facebook.com in your browser? Ever wanted to know what it would take to write the next Twitter? Then this talk is for you! This will give you a brief (mostly) non-technical introduction to the technologies involved in writing web sites. We'll talk about HTML, CSS, Javascript, PHP, Django and Ruby on Rails. There will be code shown, but it should be pretty straightforward.

**Friday, October 22, 2010 at 2:30 PM in Tutt Science 229**

*The Graph Menagerie: Abstract Algebra meets the Mad Veterinarian* by Gene Abrams (UCCS)

Rating: PG-13 (with 5 minutes of X at the end)

ABSTRACT: There are recreational math activities known as “Mad Vet Puzzles". I was first exposed to these during a workshop on Math Teacher Circles (at the American Institute of Mathematics in Palo Alto CA, June 2008), where these were used as the basis of group problem solving activities for middle school math teachers and students.

(See, for example. http://www.bumblebeagle.org/madvet/index.html )

After playing with these puzzles for an hour or so, something about them struck me as being eerily familiar. On reflection, I realized that they are intimately and highly non-trivially related to the research work I have been doing recently on ring-theoretic structures called Leavitt path algebras. (Go figure!) In this talk I'll describe Mad Vet puzzles, and how they naturally lead to interesting questions about semigroups, groups, and graphs. The talk will definitely be more-than-accessible to anyone who has seen the definition of a group, and basic concepts about such structures.

## Block 1 2010

**Friday, September 10, 2010 at 2:30 PM in Tutt Science 229**

*Modeling Dependent Random Variables using Copulas* by Steven Janke

Rating: PG13

ABSTRACT: First courses in probability concentrate on independent random variables with an occasional nod to arbitrary joint distributions. But much of probability is involved with various forms of dependency. Yet, the basic notion of dependence between several random variables is not easy to characterize. Correlation coefficients give a summary view, but not a detailed one. Copulas, introduced somewhat under the radar about 40 years ago offer a more systematic approach to modeling dependency. This talk will define copulas, explore some of their uses, and finally address a paper using copulas that Wired magazine claimed is to blame for the financial meltdown in the last few years.

**F****riday September 17, 2010 at noon in Tutt Science 122** (Kresge Lecture Hall)*A Problem from the game SET^{TM}* by Amelia Taylor

Rating: PG

ABSTRACT: The game SET^{tm} requires players to look at a collection of cards and select three cards which form a set, according to particular rules. The cards have different shapes, colors, shades, and numbers of objects on them and I call these categories parameters. A set is formed when a player finds three cards such that for each parameter the cards are either all equal or all distinct. Many combinatorial questions arise naturally from this game. One in particular has generated a wealth of interest in the mathematical community, including a new result from 2009, but not necessarily stated as a combinatorial question. Amelia will explain this problem, its interpretation in enumerative geometry and in algebra (as well as any algebra or geometry needed for this explanation). She will also talk about approaches to solving the problem in low dimensions and at mention a few other fun mathematical questions related to the game.

**Friday September 24, 2010 at 2:30 PM in Tutt Science 229**

*Manifold learning: *by Fred Tinsley

Rating: PG-13

ABSTRACT: A common phenomenon in multivariate statistical analyses is the following: *as the complexity of the statistical technique employed increases, the concern with the validity of that technique decreases*. A particularly common sin is the tacit assumption of linear relationships among the various variables. Fred will introduce the general idea of *dimension reduction* in which one determines whether data measures on *k* variables are secretly of dimension less than *k*. One very commonly used method is *principal components analysis* (PCA) where algebraic techniques identify* linear*subspaces that for all practical purposes contain the data. Even in the absence of a useful linear reduction, a non-linear one may be possible. Fred will discuss *Manifold learning, *a collection of techniques for smooth reduction to subspaces that are not necessarily linear. These subsume PCA and most other linear reduction techniques.

## BLOCK 8, 2010

** Friday April 23rd at 2:30**: Courtney Gibbons of the University of Nebraska, Lincoln

Title: The Determinant Trick. Rating PG13

Abstract: Somewhere, buried in your mathematical tool box, you have a mathematical Swiss army knife. Somewhere in that mathematical Swiss army knife, you have the determinant trick. Courtney will (re)introduce you to this handy tool and show you some cute commutative algebra applications. You don't need to have seen much past linear algebra to follow this talk, but having played with modules a little bit would be ideal.

** Friday April 30th at NOON with Pizza:** Mike Siddoway, Kresge Lecture Hall.

Title: "The 0th Watkins Lecture" (The Zeroth Watkins Lecture). Rating PG13

Abstract: Prof. John Watkins is retiring from our department this spring. Looking back over his widely-varying (and ongoing!) mathematical career I thought it natural to give a talk that calls on seemingly different but amazingly connected areas of mathematics, particularly from general fields that John contributed to. I'll start with some algebra (the computational variety which makes extensive use of Groebner bases) and move on to applications in graph theory, automatic theorem proving, and if time permits some origami theorem proving.

**Friday May 7th at 2:30:**Steven Janke

Title: Modeling Dependent Random Variables using Copulas. Rating: PG13 - NC17 (!)

First courses in probability concentrate on independent random variables with an occasional nod to arbitrary joint distributions. But much of probability is involved with various forms of dependency; the study of Markov processes is a good example. However, the simplest notion of dependence between several random variables is not easy to characterize. Correlation coefficients give a summary view, but not a detailed one. Copulas, introduced somewhat under the radar about 40 years ago offer an approach to modeling dependency. This talk will define copulas, explore some of their uses, and finally address a paper using copulas that Wired magazine claimed is to blame for the financial meltdown in the last few years.

## BLOCK 7, 2010

** Friday March 26th at NOON** in TSC 122 (Kresge Lecture Hall) with Pizza:

Title: How to draw a Tree (with a computer). By Steven Janke.

Rating PG

It is often difficult to draw organic forms on a computer, but early work in the theory of computation has led to systems (L-systems) for finding algorithms that produce pretty good trees. In fact the systems lead to an interesting model for tree growth and perhaps can answer questions concerning the shapes of trees. This talk will introduce L-systems, explain how they lead to algorithms for drawing trees, discuss some of the math involved, and show how to actually draw some nice trees on the computer screen.

**Wednesday March 31st 3:15-4pm **and **Thursday April 1st, 1:30 pm - ~4pm**, in TSC 122 (Kresge lecture hall)

Capstone Talks**WEDNESDAY****3:15-3:35 Sarah Fletcher**, “Theory and Practice of Bootstrapping with Applications in R”

**3:40-4:00 Leon Li,** “Ranks of Elliptic Curves over Number Fields”

*THURSDAY***1:30-1:50 Marina Gresham**, “(Hyper)Graphs of Zeta Functions and How They Related to Primes”

**1:55-2:15 Sam McDowell,** “Computer Vision and Robotic Guidance”

*2:15-2:30 Intermission*

**2:30-2:50 Pat Ramsey**, “A Design and Implementation of Distributed Generic Matching Framework”

**2:55-3:15 Ben Rogers**, “Using Leslie Matrices to Study Flammulated Owl Demographics”

**Friday April 2nd at NOON** in TSC 122 (Kresge Lecture Hall) with Pizza:

Title: Old Mathematics in the Modern World. By Michael Dorff of Brigham Young University.

Rating PG

Have you ever asked yourself "When will I going to use math?" If you search this question on Google, you learn that when "ordering lunch at the local fast food chain [you can] use math to count change." Hopefully, you don't need to study calculus to count change. So, there must be better uses for math. That's the topic of this presentation. We will discuss this question in several ways including movie clips that show how Hollywood sees math saving the world and serious news blurbs declaring the job opportunities for students who know mathematics. We will then discuss two or three topics that employ beautiful mathematics but are also used in many people's daily lives.

**Tuesday April 6th at NOON** in TSC 122 (Kresge Lecture Hall) with Pizza:

Title: Hedge Fund Manager Speaks

**Friday April 9th at 2:30 pm **in TSC 229

Title: Infinite Galois Theory. By Stefan Erickson.

For centuries, people have been interested in finding roots of polynomials. The quadratic formula, which has been known for millennia, explicitly finds the roots in terms of the coefficients of a degree 2 polynomial. Renaissance mathematicians produced formulas for the cubic and quartic equations, while the quintic equation eluded solutions until the early 1800s when Abel proved the unsolvability of the quintic. The question remained as to when a polynomial could be solved in terms of radicals. The answer was first found by a 20 year old French mathematician named Evariste Galois. His discovery of the connection between solving polynomials in radicals and symmetry groups is one of the most fundamental and beautiful ideas in all of modern mathematics.

In this talk, we will investigate the various amazing aspects of Galois theory. After giving several examples of finite extensions, we explore what Galois theory for infinite extensions might look like. This talk should be accessible to all math majors, although some background in abstract algebra is recommended.

## BLOCK 6, 2010

** Friday February 26th at NOON, in TSC 122,** with Pizza: Matthew Whitehead of Indiana University Bloomington.

Title: Mining User Opinions on the Web.

Rating G

The Web is full of user opinions about books, CDs, movies, restaurants, political ideas, and just about everything else imaginable. Opinions can be found on blogs, message boards, social networking sites like Facebook and Twitter, and personal webpages. Powerful machine learning models that can automatically classify these opinions can be used to aggregate this information so that it is accessible and understandable by humans. For example, a search engine can rank results from a local search for the query "restaurants" by the percentage of positive opinions posted about each nearby restaurant. In this talk we will discuss the details of mining online opinions and show how performance can be improved with the use of ensemble machine learning algorithms.

** Friday February 26th at 2pm, in TSC 229**: Matthew Whitehead of Indiana University Bloomington.

Title: Building Accurate and Efficient Machine Learning Ensembles.

** Friday March 5th at NOON, in TSC 122,** with Pizza: Gary Grunwald of UC Denver.

Title: Statistical models for heart surgery risk.

Rating PG

Heart surgery can be a risky business, better only than the alternative. Deciding which patients are highest risk, and during what periods, is a job for smart statistical models that can use data on dozens of variables and tens of thousands of surgeries and can handle complications in the data including non-normal outcomes, non-linear functional forms, differential follow-up, and risks that change over time. Dr. Grunwald will talk about some of the statistical models he has used, the mathematics behind them, and what this tells us about heart surgery. The talk will include a lot of graphs, and not to assume any statistical background. Dr. Grunwald will also discuss some other projects in Biostatistics at UC Denver, and careers in biostatistics in general.

## BLOCK 5, 2010

*Friday January 29th***at Noon**, **in Tutt 122**: Jennifer Young, PhD Candidate, University of North Carolina-Chapel Hill

Math Department Non-Tenure Track Candidate

Title: The Fish Tank Problem

An aquarium holds 6 gallons of water. Each gallon contains 0.01 gallons of salt. Fish can survive in the tank as long as the total salt level stays below 0.09 gallons. Each week one gallon of water evaporates from the tank (the salt stays), and a new gallon is added.

How many weeks will it be until the fish can no longer survive in the tank? Not very long! So the real question becomes how much water do we need to remove and then add to the tank each week to keep the fish alive? To answer this question, I will present a brief introduction on discrete dynamical models (basically models that describe how a quantity changes over time). Then we will construct a model to figure out how to keep the fish happy!

*Tuesday February 2nd***at Noon**, **in Tutt 122**: Kevin Woods, University of Michigan

Math Department Non-Tenure Track Candidate

Title: Cubing the Pyramid: or, Why We Need Calculus (and Measure Theory!)

Once you agree that a 1 inch x 1 inch square should have area 1 (isn't that the very definition of one square inch?), you can find the area of any polygon by cutting and rearranging the pieces. Can we do the same in three dimensions? Can you chop up a regular tetrahedron and rearrange the pieces into a cube? This was Hilbert's third problem, solved by Max Dehn, and the answer turns out to be "no". To find the volume of a regular tetrahedron, we (sadly? intriguingly?) need calculus.

But all of this analysis is based on an assumption: if you cut up an object and rearrange the pieces, the new object will have the same volume as the original. We will next challenge this assumption.

Weirdness ensues! A subject called measure theory comes to the rescue.

Talk with Pizza: Tutt Science Center 122

**Thursday February 4th at 2:30 pm: Andrea Bruder of Colorado College. Tutt 229. **

Title: Applied Left-Definite Theory: the Jacobi Polynomials, their Sobolev Orthogonality, and Self-Adjoint Operators. Rating PG13.

Abstract: It is well known that, for appropriate values of alpha and beta, the Jacobi polynomials are orthogonal on R with respect to a measure mu. However, for negative integer parameters alpha and beta, an application of Favard's theorem shows that the Jacobi polynomials cannot be orthogonal on the real line with respect to any positive or signed measure. But it is known that they are orthogonal with respect to a Sobolev inner product. Here, we will consider the special case where alpha=beta=-1. We shall discuss the Sobolev orthogonality of the Jacobi polynomials and construct a self-adjoint operator in a certain Hilbert-Sobolev space having the entire sequence of Jacobi polynomials as eigenfunctions. The key to this construction is the left-definite theory associated with the Jacobi differential equation, and the left-definite spaces and operators will be constructed explicitly. The results can be generalized to the case where alpha > -1 and beta=-1.

**Friday February 5th at NOON with Pizza: Andrea Bruder of Colorado College. Tutt Science 122 **

Title: Soap bubbles and mylar balloons: the calculus of variations. Rating G

Abstract: Soap bubbles and soap films have been studied mathematically for about 100 years, since the Belgian physicist Joseph Plateau showed that solutions to shortest path problems could be generated by dipping a wire framework into a soap solution. Most recently, the behavior of soap and water films has been studied on board the International Space Station in a zero gravity environment, with surprising results! We will explore the mathematics of soap films, in particular the calculus of variations, to understand the shape of soap films and mylar balloons.

## BLOCK 4, 2009

*Friday Dec 4th ***at 2:30 pm**: Fred Tinsley of Colorado College. Tutt 229.

Title: Algebraic Topology in a Non-compact Setting. Rating R

Abstract: In a world where pretty much everything of interest is compact, why would one bother studying non-compact objects? Unfortunately, analysis of compact (finite) complexes naturally requires understanding non-compact (infinite) ones, and generalizing the necessary techniques to this more complicated setting often runs into significant obstacles. Fred will illustrate some of these difficulties and discuss how they are affecting his current research.

** Friday Dec 11th at NOON** with Pizza: Jonathan Bredin of Colorado College. Tutt 122 Lecture Hall

Title: Introduction to Digital Audio. Rating G

Abstract: Do you want to learn about how your iPod and CD collection store your music?

Come to the Mathematics and Computer Science Department's block-4 lunch seminar series where Prof. Jonathan Bredin will illuminate how your digital music player can store thousands of songs, how music representation differs for iPods and CD collection, how to manipulate audio to change pitch and tempo or add reverb using free software, and why computer scientists and mathematicians care about digital audio.

The talk will cover basic signal processing, including interpretation of the Discrete Fourier Transform and filtering signals in the time domain.

No calculus, programming, or music-theory background will be assumed.

Pizza will be served.

## BLOCK 3 , 2009

** Friday Oct 30th 2:30 pm**: Andrea Bruder of Colorado College.

Title: Orthogonal polynomials and left-definite theory

Abstract: Orthogonal polynomials were first studied by Legendre and Laplace in their work on celestial mechanics around 1800. Over 100 years later, the same polynomials came up as solutions to the Schrödinger equation in quantum mechanics. I will talk about the classical orthogonal polynomials (i.e. the Jacobi, Hermite, and Laguerre polynomials) and their properties, and then turn to the general left-definite spectral theory developed in a recent paper by Littlejohn and Wellman (2002). It is best introduced from a differential equations point of view, and I will consider the Legendre equation (and the Legendre polynomials) as an example.

** Friday Nov 6th at NOON**: Zach Treisman of Western State College

Title: Images and patterns in hyperbolic geometry. Rating G

Abstract: Changing a basic assumption about parallel lines gives rise to a new and fascinating geometry, called hyperbolic geometry. We'll look at some of the models that one can make of this geometry, and use them to explore some of the ways in which it is unlike the geometry that we are used to. This talk will be gratuitously full of pictures and conspicuously short on equations.

** Friday Nov 13th 2:30pm**: Amelia Taylor of Colorado College

Title: The Geometry of Phylogenetic Invariants. Rating PG13

Phylogenetic trees are one key tool that biologists use to describe evolutionary relationships and phylogenetic invariants are polynomials, based on a given tree and model of evolution, that may be helpful in determining phylogenetic trees. Two major obstacles to using phylogenetic invariants directly in biology are the fact that they are difficult to obtain and the fact that the polynomials do not give a full geometric description of the statistical model. In particular, there are non-trivial polynomial inequalities that must be satisfied and are difficult to describe. I will describe some examples of both the polynomials and the inequalities and give some recent results by myself, co-authors and others in this study of the missing polynomial inequalities.

Familiarity with linear algebra and polynomials helpful

## BLOCK 2, 2009

* Friday Oct 2nd 2:30 pm:* Barbara Prinari of UCCS

Title: Solitons: A singular and beautiful phenomenon: beauty and novelty of nonlinear systems (rated PG/PG13).

Abstract: In 1834 lord Scott Russell described an unexpected phenomenon. He happened to observe a bell-shaped wave (solitary wave), produced by a boat in a narrow channel around Edinburgh, which continued its course along the channel for miles and miles, apparently without change of form or diminution of speed. In spite of this early observation, it was only in 1965 that waves of this type were identified as an expression of nonlinear phenomena (solutions of certain 'special' nonlinear Pies that were later on termed 'integrable'), and the word 'soliton' was coined.

In this seminar I will illustrate the history of solitons and soliton theory, and deal with some of the associated mathematics.

In particular, I will focus on the so-called Inverse Scattering Transform, an extremely powerful technique that can be considered as the nonlinear analogue of the Fourier Transform to solve an initial value problem for linear PDEs.

(Note: This talk will be largely PG with some PG13 material towards the end.)

** Friday Oct 9th at NOON:** Jane McDougall of Colorado College

Title: Ptolemy's Theorem, past and present (rated PG).

Abstract: Claudius Ptolemy of the first century AD is remembered for many things. In cartography he was first to use the coordinates of latitude and longitude to describe locations on the earth. He leaves us with the first known map projections and is first on record to have discussed the problems of projecting portions of the curved surface of the earth onto a flat map one would find in an atlas. He is known for one of the first trigonometry books, which superseded that of Hipparchus and was used for centuries thereafter. He is also remembered for his theorem which until recently was routinely taught at the high-school level.

This talk will address the origins of Ptolemy's theorem, its refinements, and applications to some of the topics listed above, and including characterizations of curvature in geometric spaces of modern analysis.

** Friday Oct 16th 2:30pm:** Sunil Chetty of Colorado College

Title: Rank and a generalized parity conjecture for elliptic curves (rated R).

Abstract: Following Mordell's proof that the group of rational points of an elliptic curve is finitely generated, the study of elliptic curves has largely centered around understanding the size, known as the rank, of a minimal set of generators for a given curve. The famous Birch and Swinnerton-Dyer Conjecture asserts that this rank exactly coincides with the order of vanishing of the L-function of the curve at the value s = 1. As a stepping stone, one attempts to understand this conjecture modulo 2, i.e. to determine if the parity of the rank matches the parity of the order of vanishing of the L-function. We will discuss a method of approaching this type of question via theories of analytic and arithmetic local constants.

## BLOCK 1, 2009

*Friday September 4th - 2:30pm, TSC 221: *

Title: Functions of Graphs and Hypergraphs by Stefan Erickson*- Rated PG*

Abstract: Zeta functions are one of the most important tools in number theory. Broadly speaking, zeta functions are built out of the "primes" of a mathematical object. In fact, zeta functions are the subject of the Riemann Hypothesis and the Birch Swinnerton-Dyer Conjecture, two of the seven million-dollar Millennium problems.

An interesting development in recent years is to define zeta function for graphs. One can hope that graph zeta functions behave similarly to other zeta functions. In this talk, we will investigate what a "prime" means in a graph, how to define the zeta function of a graph, and some nice properties of these functions. Along the way, we will draw many pretty pictures and explore the possibility of generalizing to hypergraphs. Some familiarity with infinite series and products would be useful.

*Friday September 11th - NOON, TSC 122: *

Title: Exploring Number Theory via Diophantine Equations by Sunil Chetty *(Riley Scholar)- Rated PG*

Abstract: Number theory is one of the oldest areas of mathematics, with many problems which are very easy to state and yet very difficult to solve. In so-called ''Diophantine Equations'', one wishes to find integral or rational solutions to polynomial equations in several variables.

Understanding the solutions to particular Diophantine equations allows one to make progress on some of the more easily stated number-theoretic problems. Our aim is to explore some Diophantine equations, their solutions, and some related elementary and advanced problems.

** Friday Sep 18th - NOON, TSC 122:**Shandelle Henson of Andrew's University to give a talk, title and abstract to be supplied.

## BLOCK 8 2009

**Friday, April 24th, at noon in TSC122: **

Putnam and Modeling Team Report, from CC students (Rated PG)

In this session, students who participated in the two important annual contests (the Mathematical Modeling Contest and the Putnam Exam) will discuss their experience, and describe some of the mathematics involved in the problems they worked on. Come join our CC participants!

**Friday, May 1st, at noon in TSC122:**

Tilings in Wonderland, by Kathy Merrill (Rated PG)

We in the math department are extremely pleased to have Kathy Merrill back in our department for block eight! Tilings, such as those that appear in M.C. Escher's art or in the Alhambra, have long been of interest to mathematicians, artists and scientists. The tiles we usually see have the property that identical copies of them can be moved around to cover the plane without overlap. This talk will focus on shapes that not only tile the plane by copies moved around, but also tile by copies expanded and contracted (like Alice nibbling the mushroom). Such a shape is called a wavelet set, and is related to the mathematical theory of image compression.

**Friday, May 8th, at 2:30 p.m. in TSC229: **

Brouwer's Fixed Point Theorem, by Stefan Erickson (Rated PG)

Fixed points of functions (points where f(x)=x) play an important role throughout mathematics. There are many applications, including general equilibrium theory in economics and the football college ranking system. Although there are many methods of finding fixed points, we would like to know when a function actually has a fixed point. The Brouwer Fixed Point Theorem provides one set of conditions when a function is guaranteed to have a fixed point.

After a brief introduction and motivation, Stefan will present a simple and elegant proof of the Brouwer Fixed Point Theorem. Along the way, he will prove and make use of a combinatorial result known as Sperner's Lemma. He will also discuss some of the neat applications of fixed point theorems.

The talk will be followed by a wine & cocktail party at Stefan’s house (1627 N. Franklin St.) starting at 4:00pm. Significant others and small children are encouraged to come!

## BLOCK 7 2009

**Friday, April 10th, 2009: at 2:30 p.m. in TSC229**Joseph Silverman's Elliptic Curves, by Stefan Erickson (Rated R)

Elliptic curves are one of the most mysterious and powerful objects in all of number theory. Elliptic curves were used to prove Fermat's Last Theorem, the most famous and longest unsolved conjecture. One of the $1,000,000 Millenium Problems is the Birch Swinnerton-Dyer Conjecture is about elliptic curves. They are also used in modern cryptography, including protecting President Obama's Blackberry.

Stefan will present a talk given by Joseph Silverman, a professor at Brown University and a leading expert in elliptic curves. His slides tells the tale of sublime nature of these curves, explains the amazing connection between the algebraic elliptic curves and analytic L-functions associated with those curves, and gives a heuristic reason why we should believe the Birch Swinnerton-Dyer Conjecture. Although this talk does not assume any knowledge of elliptic curves, some familiarity with number theory, algebra, and analysis is recommended.

**Friday, March 6th at 2:30 p.m., in Tutt Science 122**Dr. Robin Wilson, speaking on Lewis Carroll in Numberland.

**( This is a general interest talk which presumes no mathematical background.)**

Abstract: If Charles Dodgson (Lewis Carroll) had not written 'Alice's Adventures in Wonderland' and 'Through the Looking-Glass', he would probably be remembered as a pioneering photographer, one of the first to consider photography as an art rather than simply as a way of recording images. But his day job was as a mathematics lecturer at Christ Church, the largest college of Oxford University. But what mathematics did he do? How good a mathematician was he? And how influential was his work? In this illustrated talk Robin will answer these questions by describing Dodgson’s work in geometry, algebra, logic and the mathematics of voting, and on the lighter side Robin will present some of the puzzles and paradoxes with which he entertained his contemporaries

**Tuesday, March 10th at 12 Noon in Tutt Science Center 122:**

The mathematics of rock climbing, by ANDREA BRUDER, Baylor University (Rated PG)

Andrea will talk about some applications of mathematics in rock climbing. First order linear differential equations can be used to determine the optimal shape of camming devices (aka “friends”), and a second order equation models how a (dynamic) rope catches a falling climber. This has implication for how we climb and for the testing of climbing gear in terms of the maximal force that the rope exerts on the climber at the end of a fall.

**Friday, February 20th at 1 p.m. in Packard Hall: **

YEA, WHY TRY HER RAW WET HAT?

by Dr. Robin Wilson of the Open University in England (Rated G)

Our frequent visitor Dr. Robin Wilson will give an interesting and entertaining illustrated lecture exploring the connections between mathematics and music, as part of the Music Department’s celebration of emeritus professor Carlton Gamer’s 80th birthday. This talk will feature musical examples and commentary by CC composers. In order to understand the title, you will need to attend the talk! The first event in the Carleton Gamer Retrospective is a concert in Packard Hall at 7:30 p.m. on Thursday, February 19th, which is free and open to the public.

* Monday, February 23rd at 2:30 p.m., in TSC 122: *Projective Spaces (Rated PG)

by Amanda Beeson, a Ph.D. candidate at the University of California, San Diego; she is a candidate for a non-tenure-track job in our department.

What happens to the real plane if we add on a so-called line at infinity and declare that parallel lines intersect in a point on that line so that, in this new space, ANY two distinct lines intersect in exactly one point? We get what is called real projective two-space and, as you shall see, lots of interesting beautiful geometry ensues!

**Tuesday, February 24th at noon, in TSC 229:**

Groups of Special Units (Rated R)

by Amanda Beeson

We will introduce briefly the philosophy of the Stark conjectures about special values of L-functions, the work of Sinnott regarding unit groups, and Anderson's construction of the maximal almost abelian extension of the rational numbers. We will see that these three areas tie neatly together in the form of groups of special units, out of which we will produce new and surprising trigonometric and L-function identities.

** Friday, February 27th at noon, in Tutt Science Center 122: **CANCAN YOU KENKEN?

by John Watkins and Maija Benitz (Rated G)

One week ago The New York Times began carrying a numerical logic puzzle next to its daily crossword puzzle. The new puzzle, called KENKEN ®, which loosely means "wisdom squared," was invented in Japan in 2004 and, like Sudoku, it involves filling a square grid with numbers without repeating a number in any row or column. But, unlike Sudoku, doing a KENKEN ® puzzle requires arithmetic and sometimes a good deal of cleverness. Join John and Maija for an introductory session on “The World’s Most ADDictive Puzzle ™”. There will be free pizza available at the talk!

There will a third Fearless Friday talk this block too – watch this space for further details!!

## BLOCK 5 2009

** Thursday January 29th, at 2:30pm in TSC 122**Frames and the Uncertainty Principle in Time-Frequency Analysis (Rated PG)

by Jan Cameron, Riley Scholar Candidate

You may have heard of the Heisenberg uncertainty principle, or even studied the result in a physics or chemistry class. It is less known that a similar principle is at the heart of the "localization problem," one of the deepest mathematical problems in time-frequency analysis. Time-frequency analysis is the area of mathematics devoted to understanding how the frequency and spectral content of signals (e.g. sounds, images, electrical activity) evolve through time, and how this information can be stored digitally. In this talk, we will develop some basic tools of time-frequency analysis, and see how one tool in particular, frame theory, emerges as a means to address the localization problem. Frame theory is not only useful for applications in engineering and signal analysis, but also has deep connections to several areas of mathematics. As time permits, we will explore a few such connections, and discuss some current avenues of research in frame theory that may be of interest to you.

** Friday, January 30th, at noon in TSC 122 (featuring Pizza!!)**Web Security for Liberal-arts Majors (Rated G)

Business Logic Flaws are a large problem for the Computer Security industry. The security group, OWASP put out their top ten flaws in web applications. These include vulnerabilities such as leaving configuration scripts available to anyone after the installation of a product, SQL Injection, and Cross Site Scripting. All of these are problems that can be detected and fixed by code review and configuration management

Business Logic Flaws are much harder. These problems cannot be detected by normal means because they are not errors. They come about by the lack of understanding by software developers of the customer’s ability to do the unexpected.

We will discuss a group of these Business Logic Errors and how to fix each one, if they can be fixed. The lecture will be understandable by anyone who has used the Internet.

* Friday, February 6th, at 2:30 p.m. in TSC 229:*Constructing and Squaring Lunes Whose Fundamental Quotient is Algebraic, by Mike Siddoway (Rated R)

Hippocrates of Chios (5th century BC), with perhaps our first known proved theorem, showed that a special lune could be constructed with ruler and compass, and could also be “squared” using the same tools. This was the first “curved” figure that admitted such properties. Hippocrates would find two other such lunes, and several more were found about 2200 years later bringing the total to five. Using Galois theory and advanced techniques for finding solutions to polynomials, it was settled in the mid-20th century that these were the only five “constructible squarable lunes” given the classical strictures placed on the problem. In particular, the classical problem assumes (in part) that the quotient of the arcs defining the lune (the “fundamental quotient’) is rational. Using deep theorems of Hermite and Lindemann, Gelfond and Schneider, and Fields medalist Alan Baker it can be shown that no new “constructible squarable lunes” are gained by extending beyond the commensurable setting to lunes whose fundamental quotient is properly algebraic. Along the way many other stunning results play crucial roles including the Eisenstein irreducibility theorem and Ferrari’s method for finding the roots of quartic equations. This grand problem is a treasure trove of mathematical history whose solution spans two and a half millennia and relies on many beautiful theorems that came to light along the way. In this talk I’ll follow on my 4th block presentation “Call of the Lune,” with three basic goals. I’ll first provide some of the details that settled the classical question. Then we will answer the question raised by distinguished biology professor Werner Heim, “Are there any constructible, squarable lenses? (“Lenses” are lunes which bow out on both sides.) Finally, we’ll put to rest the other question raised at the last talk, “Are there any constructible, squarable lunes whose fundamental quotient is a non-rational algebraic number?”

## BLOCK 4 2008

** Friday, December 5th at 2:30 p.m., in TSC 221**: Looking back from a theorem of Kaplansky to a famous result due to Krull, and forward to a theorem on Bezout rings with (many) stops in a paper by Matlis along the way, by Mike Siddoway, Colorado College

(Rated R)

We will look at an old theorem due to Krull on valuation domains, and move on to a new view of this work by Kaplansky that led to a result for more general domains. We'll then consider non-domains (rings with zero divisors) and see how Kaplansky's idea provides a setting for work in reduced rings (special rings with zero divisors for which a^2 = 0 implies a=0). In particular we will be studying semihereditary Bezout rings and semigroups. (I'll define these in the talk). A Bezout semigroup S is called "semi-hereditary" if for each a in S there is an idempotent (e^2 = e) in S that generates the annihilator of a. Our main result states that a semigroup is a semi-hereditary Bezout semigroup if and only if it is isomorphic to a "semigroup of divisibility" of a semi-hereditary Bezout ring partially ordered by reverse inclusion. This theorem harkens back to Krull's famous theorem and is inspired by Kaplansky's work with Bezout domains, and Matlis's work with reduced rings.

**Friday, December 12th at Noon in TSC 122 (with PIZZA!)**

Title: The Life and Legacy of Alan Turing, by David Brown (Rated G)

In 1935, a young graduate student named Alan Turing lay in a field near Cambridge, thinking about a deep problem in mathematical logic. He imagined machines that were capable of carrying out calculations according to a set of rules. Turing’s daydreams eventually became the mathematical foundation for the invention of computers (to a computer scientist, your laptop and your iPhone are examples of “Turing machines”). In his remarkable and tragically short career, Turing helped lay the groundwork for computer science and artificial intelligence, and explained how patterns can form spontaneously in biological systems. He helped crack Nazi codes during WWII, but was later persecuted by his own government for his homosexuality. In this talk, I will provide a sketch of Turing’s life and some of his key mathematical ideas. I will also try to make the case that this shy, rather obscure English mathematician belongs in the short list of the greatest scientists of all time.

## BLOCK 3 2008

* Friday, November 14th, at 2:30 p.m. in TSC229: *Transforming results for monomial ideals to toric ideals, by Amelia Taylor

(Rated R)

In combinatorial and computational commutative algebra it is common that research is presented for monomial ideals with the caveat that "the work also holds for toric ideals." I will give examples of such research, including work of my own, and then talk about how to make the leap from a monomial ideal result to a result on toric ideals.

** Friday, November 7th, at noon in TSC122 (featuring PIZZA!): **The Call of the Lune: A 2500 year journey from perhaps our first proof, through Euler in the 1700s, and on to a complete solution in the 20th century, by Mike Siddoway

(Rated PG)

A loon is a large odd duck, whose eerie call makes Midwesterners wax nostalgic. A lune is a curved figure formed by the arcs of two circles that has similarly haunted the longings of mathematicians. Think of a crescent moon. The Greeks knew how to square arbitrary polygons. That is, they could construct a square with the area of any given polygon using simple tools. When a figure can be squared we mathematicians say it is quadrable. The big test was to construct an equal area square” for a figure with curved sides. The circle was the big prize. Could the circle be squared? Hippocrates of Chios (c. 470 - c. 410 BCE) found that he could square several lunes that had special properties, and it seems he even believed he had a proof for squaring the circle. One wonders. It turns out the proof is (very) flawed, and the search for squarable lunes went on. Euler (1707 -1783) took on the problem early in his life, and again when in his twilight years. He found several new lunes that admitted construction with classical tools, bringing the total to five quadrable lunes. In the decades that followed, revolutionary advances in algebra led to a proof that the circle couldn’t be squared. A consequence of this was that arbitrary lunes were not quadrable. But was Euler’s collection of five squarable lunes complete? Advances were made through the 19th Century, but it wasn’t until the mid-20th Century that several young Soviet mathematicians (Chebotarev and his student Dorodnov) proved that Hippocrates and Euler had found all the quadrable lunes. We’ll look at this grand story and marvel at the deep insights of Hippocrates, the technical acumen of Euler, and the beauty of modern methods that settled the problem once and for all.

** Friday, October 31st, at 2:30 p.m. in TSC229: **Three Short Talks About Bacterial Genetics, by David Brown

(Rated PG-13 for content, R for pace)

I’ll give brief and probably incoherent summaries of three of my research projects in varying stages of completion. First, I’ll talk about a project with Phoebe Lostroh in which we developed a method for reconstructing time series for gene expression dynamics from reporter protein data. This will be a review and slight extension of a talk I gave last fall. Second, I’ll talk about my ongoing attempt to model a set of gene interactions in the bacterium Pseudomonas fluorescens. This system is an example of quorum sensing, in which bacteria detect their population density and adjust their behavior in response. This part will be an extension of a talk I gave last spring. Finally, I’ll talk briefly about the next project I have in mind: a stochastic model of quorum sensing. The material is not overly challenging, but I will be going kind of fast.

## BLOCK 2 2008

** Friday, October 17, 2008, at 2:45 (note the special time), in TSC229: **Amelia Taylor

**Vanishing Ext and Tor**

*(Rated R)*

In commutative algebra, the vanishing of Ext and Tor modules often gives information about key properties of a ring, a module, or, in some of Amelia’s recent research, complexes. Amelia will start with defining a complex. Then she will work her way towards giving some of the motivation and recent results on the impact of vanishing of Ext (mostly, but maybe some Tor as well) of semi-dualizing complexes.

** Monday October 13th - Noon, TSC 122: **Fred Adler, University of Utah

**Title: Time lags and sex-biased prevalence in hantavirus **

Abstract: This talk examines two aspects of the population dynamics of Sin Nombre virus, the cause of hantavirus pulmonary syndrome in humans. First, we use population trajectories from the literature and mathematical modeling to analyze the expected time lag between deer mouse population peaks and peaks in deer mouse seroprevalence. Because the virus is not transmitted vertically, rapid population growth can lead initially to reduced prevalence, but the resulting higher population size may later increase contact rates and generate increased prevalence. Second, we investigate the role of sex-biased prevalence in persistence of the virus. Analyzing data on over 2000 individual mice captured, we determined that males are more likely to contract the infection due to higher susceptibility or higher encounter rates and that they live longer than females. Mathematical models predict that these differences can buffer the disease from extinction by concentrating it in the subgroup most likely to survive population bottlenecks.

** Friday October 10th - Noon, TSC 122: **John Watkins

**Teen Mathematics: a tour of amazing mathematical discoveries**

(

*Rated G)*

John will give an interesting report on mathematical discoveries made by the very young. Come here about some exciting mathematics, and enjoy pizza while you’re doing so. This talk is in celebration of Homecoming Weekend.

** Friday October 3rd - 2:30pm, TSC 229:** Associate Professor Michael Dorff from Brigham Young University

**Title: Harmonic mappings, minimal surfaces, and linear combinations.**

*(Rated PG-13)*

Abstract: Complex-valued harmonic mappings can be regarded as generalizations of analytic functions studied in a beginning complex analysis course. Minimal surfaces in R^3 are saddle surfaces whose mean curvature vanishes everywhere and can be visualized as the soap film formed on wire frames. In this talk, I will present some general background about harmonic mappings, minimal surfaces, and the connection between the two.

Then I discuss some new results on the convex combinations of mappings and surfaces. One application of these results is that they offer an easy way to construct harmonic mappings onto nonconvex polygonal domains and to construct the corresponding minimal graphs over these nonconvex domains.

This provides us with a method to construct new families of minimal surfaces that are generalizations of the Jenkins-Serrin surfaces. These results are related to work done by Peter Duren, Jane McDougall, and Beth Schaubroeck.

## BLOCK 1, 2008

*Friday September 12th - NOON, TSC 122: ***Research opportunities in the math department**, by Amelia Taylor (and Jonathan Bredin, David Brown, Stefan Erickson, and Mide Siddoway :) ) --*Rated G*

(Please click on the link to view the PDF presentation!)

ch by students takes many forms in the mathematics department. We will talk about some of the departmental student research projects from the recent past and most importantly advertise questions and opportunities for students to do research in the future. Problems will be presented by Amelia Taylor, David Brown, Jonathan Bredin, Mike Siddoway and Stefan Erickson. Please join us to find out about all the fun problems to work on!

This talk will be immediately followed by the fall math department picnic – there’ll be Mexican food available in the Skilling Commons (near the math department on the second floor)!

*Friday September 19th - 2:30pm, TSC 229: *

More Fantastic p-adics, by Stefan Erickson -- *Rated PG-13*

A standard question in analysis is how to construct the real numbers from the rationals. But what happens when we measure distances between rational numbers in different ways? We get the p-adic numbers, every bit interesting and exciting as the reals.

After a playful introduction, we will delve into some of the fascinating properties of the p-adic numbers. We will develop the p-adic metric, p-adic functions, and the local-global principle. Finally, we will explore some applications to number theory. This is a continuation of a talk given in Block 8 of the last school year.

## BLOCK 8, 2008 (April -May)

**Friday April 25th at 2:30 p.m., TSC229:**

A mathematical model of the Gac/Rsm signal transduction network in Pseudomonas fluorescens, by David Brown (Rated PG-13)

Living cells must make many decisions about what genes to express in response to the conditions in their environment. For example, the soil-dwelling bacteria Pseudomonas fluorescens detects its local population density and adjusts various processes like antibiotic production in response. This behavior is regulated by a network of genes that activate and inactivate each other, and control the production of antibiotics and signaling molecules. David will present a mathematical model of this system, consisting of a set of differential equations for the cell population and various molecular concentrations. He will show how the model can be tuned to match published data on gene expression dynamics, then explore the behavior of the system under various conditions.

**Friday May 2nd at noon, TSC 122: Putnam Exam Discussion (Rated PG)**

The Putnam Exam is a highly competitive national competition offered each December. Undergraduates from all over the country take it. Each year Colorado College fields a team, after spending the fall working on sample problems and problem-solving strategies. A group of CC students will report on the problems on this year’s exam, and discuss how they can be solved.

*Friday May 9th at 2:30 p.m., TSC 229:*

The Fantastic p-adics, by Stefan Erickson (Rated PG13)

A standard question in analysis is how to construct the real numbers from the rational numbers. But what happens when we measure distances between rational numbers in different ways? We get the p-adic numbers, every bit interesting and exciting as the reals.

After a playful introduction, Stefan will delve into some of the fascinating properties of the p-adic numbers. He will develop the p-adic metric, p-adic functions, and the local-global principle. Finally, he will explore some applications to number theory.

## BLOCK 7, 2008 (March-April)

**Thursday, April 3rd starting at 2:00 p.m. in TSC 122:STUDENT DISTINCTION TALKS!**

**Schedule of Talks**

Distinction**3:00**: Tra Ho, “Explicit Formulas for Real Hyperelliptic Curves of Genus 2 on the Projective Coordinate System”**3:30:** Jes Coyle, “The Effect of Fire on Ponderosa Pine Forest Structure”**4:00:** Matthew Swanger, “A Fault Tolerable Content Addressable Network”**4:30:** Jette Petersen, “The Initiation of Cancer via Two Mutations”**5:00:** Andrea Buchwald, “The Unit Group of Real Biquadratic Number Fields”

**Friday, April 4th at 2:30 p.m. in TSC 229:**

"The classification question for Leavitt path algebras", by Enrique Espino, University of Cadiz (Rated R)

Enrique will obtain isomorphisms between the Leavitt path algebras of specified graphs. From these isomorphisms he is able to achieve two ends. First, he shows that the K_0 groups of various sets of purely infinite simple Leavitt path algebras, together with the position of the identity element in K_0, classify the algebras in these sets up to isomorphism. Second, he shows that the isomorphism between matrix rings over the classical Leavitt algebras, established previously using number-theoretic methods, can be re-obtained via appropriate isomorphisms between Leavitt path algebras.

**Friday, April 11th at Noon. in TSC 122: **

"Counting the days", by Marlow Anderson, Colorado College (Rated PG)

The 16th century classicist J. J. Scaliger introduced the modern science of chronology, which is the careful correlation of historical dates. He combined three important calendric cycles (the Julian cycle of 28 years, the indiction cycle of 15 years, and the Metonic cycle of 19 years), to obtain a cycle of 7980 years, which is long enough to include all dates in written human history. Scaliger’s cycle happens to begin on Jan. 1 4713 BC. Modern astronomers have refined this to make a literal count of days, leading to the Julian Day Number for any date in historical times. The Julian Day Number is the tool used to translate from any one calendar to any other. There is some modestly interesting modular arithmetic involved in making these calculations. As a bonus, it’s easy to compute how many days old you are!

**Friday, April 11th at 2:30 in TSC 122:VENTURE GRANT RECIPIENT TALKS!**

**2:30:** Andrea Buchwald, “The Tradition of Geometry in Architecture: from Ancient Greece to the Ottoman Empire”

**3:00:** Melissa Miller, “Exploring Geometries and Symmetries of Greek and Turkish Mosaics “

*Tuesday, March 25th at 2:30 p.m. in Tutt Science Center 122:*

"Squaring the Square" by Molly Maxwell, PhD University of Minnesota, rated PG.

A squared rectangle is a dissection of a rectangle into non-overlapping squares, all of different sizes. The problem of finding a squared square caught the interest of Brooks, Smith, Stone, and Tutte in the 1930's while they were undergraduates at Trinity College in England. The "Trinity Four" used two approaches for finding squared rectangles. Their first approach involved solving a system of equations, while the second involved rephrasing the problem in terms of electrical networks. This talk includes a description of the Trinity Four's approaches and their resulting squared squares. Since this time, many more example have been found, and several artists have included these patterns in their work. The talk concludes with a discussion of several generalizations of this problem, including squared cylinders, squared tori, cubed cubes, and triangulated equilateral triangles. (Student Level talk)

*Wednesday, March 26th at 2:30 p.m. in Tutt Science Center 229:*

"Self-Dual Spanning Trees and Self-Dual Matroid Bases" by Molly Maxwell, PhD University of Minnesota, rated R.

In 1979 Tutte studied the properties of a class of graphs called central reflexes. In particular, he proved that the number of spanning trees is the square of the number of selfdual spanning trees. Several years later, Kalai defined higher dimensional spanning trees in the boundaries of simplices and proved analogues of Cayley's Theorem and the Cayley-Pr¨ufer Theorem for these trees. In addition, Kalai conjectured that an enumerator for these higher dimensional trees is the square of an enumerator for self-dual trees.

In this talk, we'll discuss a class of cell complexes that contains both Tutte's central reflexes and Kalai's boundaries of simplices. We'll look at k-dimensional spanning trees in these complexes and a result which shows that an enumerator for these trees is the square of a related enumerator for self-dual trees. This result gives a new proof of Tutte's theorem and solves Kalai's conjecture when k is odd. We'll also discuss a weighted version of Kalai's conjecture and a bijective proof of Tutte's theorem for odd wheels, which are a specific class of central reflexes. (Faculty Talk)

*Friday, March 28th at 2:30 p.m. in TSC 229:*

"An unusual polynomial, and Error-correcting Codes", by Rachel Pries of Colorado State University (Rated PG13)

Rachel will describe a polynomial that has special properties in number theory. She will then show how it is important for error-correcting data-transmission codes.

## BLOCK 6, 2008 (February - March)

**Friday, February 22nd at 2:30 p.m. in Tutt Science Center 229:**

"Special Low Dimensional One-sided h-cobordisms" by Fred Tinsley and Roberto Pelayo Rated R (with a little bit of X)

During the 1990's Fred and Bob Daverman proved that in high dimensions (ambient dimension larger than five), every one-sided h-cobordism admits a special Morse function. We obtained partial results for ambient dimension five but learned nothing about ambient dimension four. A cobordism is an n-manifold W with two boundary components M and N. A Morse function is a continuous map f:W ? [0,1] with f -1(0) = M, f -1(1) = N. To be special, we require in addition that f -1(t) be a closed (n-1)-manifold for each t Î (0,1). In other words, W is a generalization of a product.

Fred and Roberto have embarked upon a study of the ambient dimension-four case in which we plan to combine Roberto’s knowledge of three-dimensional topology with Fred’s knowledge of high-dimensional topology. They will talk about the differing techniques required for the differing ambient dimensions and then present strategies for attack.

*Friday, February 29th at noon in Tutt Science Center 122 (featuring free pizza!)*

"The Other Polyhedra" By Steven Janke (rated PG)

Nearly everyone has seen the five Platonic polyhedra (tetrahedron, cube, octahedron, dodecahedron, and iscosahedron) and probably many of us know a few things about them. However, there are a lot more polyhedra that are interesting for various reasons. This talk will review some of the geometry of the Platonic polyhedra and then look at four more that should be included in the same category. And then there are another 66 that we shouldn't dismiss. With the help of software, the talk will examine some of these polyhedra visually, trace a little history, outline some of the main mathematical results, and perhaps uncover what makes these shapes aesthetically pleasing.

## BLOCK 5, 2008 (January- February)

Tenure Track Candidate Talks!**Stephanie Treneer**, whose Ph. D. is from the University of Illinois; she is presently at Dartmouth College.

*Monday, February 11th at 2:30 p.m. in Tutt Science Center 122:*

"Exploring Partitions of Numbers" (Rated PG)

A partition of a positive integer n is a sequence of positive integers that sum to n. The partition function p(n) counts the partitions of n without regard to order. This deceptively simple function has led to a rich theory. We'll look at two elementary methods for analyzing partitions: Ferrers graphs and generating functions. We'll then briefly discuss the connection between partitions and modular forms, which has led to some recent surprising results about p(n).

*Tuesday, February 12th at noon in Tutt Science Center 229:*

"Arithmetic of the Fourier Coefficients of Modular Forms" (Rated R)

In the last decade, the theory of modular forms has provided an important framework in which to study questions in arithmetic number theory. Modular forms act as generating functions for many partition functions, divisor functions, and other combinatorial functions. We will review the role modular forms have played in establishing congruence properties for such functions, and give general results about congruences for the coefficients of weakly holomorphic modular forms. We will also discuss new work involving harmonic Maass forms.

**Stefan Erickson**, whose Ph. D. is from the University of California at San Diego; he is presently a member of our department.

**Thursday, February 7th at 2:30 p.m. in Tutt Science Center 122:**

"Secret Codes and Algebra: How to Use Curves to Take Over the World " (Rated PG)

Since ancient times, codes and ciphers have been used to transmit vital information without it falling into the wrong hands. The advent of the computer age has produced a strong demand for fast and secure cryptosystems. Today, cryptography is found in military, economic, and personal applications, including Internet commerce and digital signatures.

Most modern cryptosystems are based on computationally hard problems from mathematics, such as the factoring of the product of two large prime numbers in RSA. Recently, there has much interest in using elliptic curves as the basis for cryptosystems. After describing elliptic curves and motivating their use, we will demonstrate a variety of applications. We will also introduce hyper-elliptic curves and describe a number of open questions related to their arithmetic and suitability for cryptography.

*Friday, February 8th at 2:30 p.m. in Tutt Science Center 229:*

"Computational Evidence of the Stark Conjectures" (Rated R)

The Stark Conjectures predict that the leading term of Taylor series of the L-functions for an abelian extension K/k can be evaluated using certain algebraic units in K. These units should also produce abelian extensions of k, which allows one to do explicit class field theory. It is possible to numerically verify the Stark Conjectures by evaluating the L-function to high precision, then recognizing the coefficients of the defining polynomial of the units. After giving a gentle introduction to evaluating L-functions, we give an example of how the Stark Conjectures can be computationally verified.

**Katherine Ott** is a Ph.D. candidate at the University of Virginia. She will be giving two talks during her visit; the first is aimed particularly at students.** Thursday, January 31st at 2:30 p.m., in Tutt Science Center 122**"Buffon's Needle Problem" (Rated PG), by Katherine Ott, University of Virginia.

In this talk Katy will introduce a classic problem in geometric probability known as Buffon’s Needle Problem. Using geometry and calculus techniques Katy will solve the problem two ways and discuss its very surprising answer. Finally, we’ll see how this simply-stated problem applies to Monte Carlo methods and some homeless ants.

*Friday, February 1st at 2:30 p.m., in Tutt Science Center 229*

"Boundary Integral Equations in Non-Smooth Domains" (Rated R), by Katherine Ott, University of Virginia.

A method for solving boundary value problems, such as the Dirichlet problem, on non-smooth domains is the method of layer potentials. In this talk Katy will discuss how she employs the techniques of layer potential operators and the tools of Mellin transform analysis to study the well-posedness of several other boundary integral equations related to mathematical physics, engineering and computer science.

## BLOCK 4, 2007 (November-December)

**Friday, November 30th at 2:30 p.m. in TSC229:**

"Heegaard Decompositions and the Combinatorics of 3-Manifolds", by Bob Pelayo (Rated PG13)

A Heegaard structure on a 3-manifold M is a natural decomposition of M into two particular subsets (called handle-bodies), which are glued together via a self-homeomorphism of a 2-dimensional surface. Since every 3-manifold has a Heegaard decomposition, understanding the various Heegaard structures on M would allow us to combinatorially classify all 3-manifolds. Furthermore, since these Heegaard decompositions are given in terms of a 2-dimensional surface S, we will explore a simplicial complex called the complex of curves, which categorifies the space of simple closed curves on S. Using these various notions, we will explore the combinatorics, dynamics, geometry, and algebra associated to Heegaard decomposition and rephrase important topological theorems (e.g., the Poincare Theorem) in terms of Heegaard structures. This talk will begin with an introduction to 2- and 3-manifold topology and culminate in the unification of various seemingly disparate subfields of mathematics involved in 3-manifold topology.

**Friday, December 7th, at noon in TSC122:**

"Inferring Gene Activity Dynamics from Reporter Protein Data, or You Might Already Know Enough Math to Do Research",

by David Brown (Rated PG) **PIZZA will be served**

**View seminar slides **from David's talk (pdf)

We all know that genes encode proteins. What most of us don't think about is that they do so in a highly regulated, dynamic manner. Genes can be "turned on", "turned off", or somewhere in between, depending on the type of cell and the cues that it receives from other cells and the environment. Measuring the activity level of a gene directly is difficult, so biologists measure the accumulation of a "reporter" protein that is easy to detect. This type of data reveals average gene activity over a period of time, but the underlying dynamic patterns are not immediately obvious. This summer Phoebe Lostroh and I realized that the dynamic patterns are in fact encoded in the reporter protein data, and that they can be extracted using some basic calculus. I will discuss the method and its biological uses, and reflect on how I stumbled across this problem and what it implies about doing applied mathematical research.

## BLOCK 3, 2007 (November)

**Friday, November 2nd at 2:30 p.m. in TSC 229**

"Robust multivariate statistics and data depth", byRYAN ELMORE, Colorado State University (Rated PG-13)

In this talk, Ryan will discuss multivariate analogues to the univariate “sign”-based suite of procedures. Specifically, we will introduce a multivariate median and hypothesis test for estimating and testing central location of a multivariate distribution. The methods are developed using the concept of spherical and elliptical data depth. The talk will focus on the geometry and computational details related to Ryan’s proposed methods. Basic issues pertaining to multivariate statistics will be introduced as part of the talk; however, a rudimentary understanding of beginning statistical concepts is assumed.

*Friday, November 9th at noon in TSC122 (Kresge Lecture Hall)*

"The Archimedes Palimpsest", by MARLOW ANDERSON, Colorado College (Rated PG)

In 1906 J. L. Heiberg discovered that a medieval prayer book contained previously unknown works by Archimedes, as an erased text on the parchment used to construct the prayer book. His discovery revolutionized the study of Archimedes – one of these works (The Method) contained valuable insights into how Archimedes approached problems we would today solve by calculus. The manuscript was in private hands for much of the twentieth century – during its disappearance it suffered considerable damage. In 1998 Christie’s sold it at public auction for $2 million to an anonymous bidder – who has made it available for study by scholars. In this talk Marlow will relate a bit more of this romantic tale, and in addition explain how Archimedes approached what we would call integration.

*Friday, November 16th at 2:30 p.m. in TSC229*

"Applications of Mathematical Computing to Probability" by FRED TINSLEY, Colorado College (Rated PG 13)

The Math Department bestows upon the calculus-based probability and statistics sequence a role central to the statistics track of our major. One objective for this talk is to institute a dialogue about the reasons for placing this subject matter at the heart of the curriculum. Fortunately, liberal arts students crave to understand the whys and wherefores of conventional practice, and the probability-statistics sequence continues to satisfy this desire. However, incorporating concrete applications to understanding actual phenomena would yield a significant benefit. Fred will combine mathematical probability with modern mathematical computing to highlight useful ways of analyzing data. Fred will argue that in a rush to purge mathematics from their courses, contemporary statistics educators miss a valuable opportunity to support their students in discovering the wonderful connections among all these subjects.

At the conclusion Fred will suggest ways for students interested in the statistics track of the major to engage in research incorporating the mathematical side of these subjects.

## BLOCK 2, 2007 (October)

**Friday, October 12th, at noon, in TSC 122**

JONATHAN BREDIN of Colorado College (Rated PG)*A Coming to Terms:Bodies Trump Brains,Elephants Can’t Play Chess,Boys Blow Things Up,and We All Hate Complexity(Managing Complexity with a Subsumptive Architecture)*

Controlling and planning actions for a mobile robot is a classic challenge in artificial intelligence (AI). The traditional approach leverages a sense-model-plan-act loop and found success in formulating and solving complex goals in laboratory settings, but also serves as one of the most visible research directions to succumb to the AI winter—a collapse in AI research and funding lasting into the mid 1980s. Several obstacles stymied the traditional approach. Researchers had trouble modeling real-world settings and logical planning proved fragile in dynamic environments. Furthermore, few projects could scale to utilize multiple processors, accommodate multiple robot goals, or handle conflicting sensor measurements.

The introduction of subsumption architecture is frequently credited as the paradigm that thawed the AI winter for robotics. The idea coordinates independent modules, each competing to represent a simple low-level behavior, and serves as a useful paradigm to tackle complexity.

We’ll look back at a critical moment in AI history, introduce subsumption-style organization, and apply the paradigm through designing a subsumption architecture to control virtual spaceships inside a multi-player programming game. The bulk of the material presented will be accessible to a general audience. No programming experience will be assumed.

## BLOCK 1, 2007 (September)

**Friday, September 7th at Noon in TSC 122:**

Mathematics and the Chessboard by JOHN WATKINS (Rated PG)

The simple and familiar 8×8 grid of squares we call the chessboard has been the inspiration for countless puzzles and mathematical problems over the years. The Knight's Tour Problem — in which a knight visits every square of the board — is perhaps the most famous of these puzzles and was recently featured by magician Ricky Jay in his hit Broadway show, even though Euler found a solution to this problem 250 years ago! The 8-Queens Problem — place eight queens on the board so that none of them attacks any of the others — was solved by Gauss. This talk, in which John will discuss these problems as well as many of the variations that continue to challenge mathematicians to this day, is based on his book Across the Board: The Mathematics of Chessboard Problems, which has just been reissued in paperback. FOLLOWED IMMEDIATELY BY THE FALL MATH PICNIC, FEATURING FUN & GAMES AND MEXICAN FOOD!

**Friday, September 14th at noon, in TSC 122:**

Frodo Meets Neo: Studying rings using matrices by AMELIA TAYLOR (Rated PG-13)

Frodo Meets Neo: Studying rings using matrices by AMELIA TAYLOR (Rated PG-13)

Recently, quotients of polynomial rings have been popping up in unexpected places, such as the study of phylogenetic trees in evolutionary biology and the game SET. We will start the talk with an introduction to quotients of polynomial rings and motivate our study of rings by introducing a few of these recent new uses for rings. A powerful tool for studying such rings (and modules) is the minimal free resolution. Minimal free resolutions in this context consist of a sequence of maps given by matrices and we can give many examples. In fact, there are algorithms for computing minimal free resolutions, but describing the structure based only on data from the ideal can be difficult and many of the results combine algebra, topology and combinatorics. We conclude with recent results and open questions on the structure of the minimal free resolution of certain quotient rings, including results by undergraduates. FREE PIZZA AT THIS TALK!

**Friday, September 22nd at 2:30pm, in TSC 223:**

Isomorphisms between matrix rings over Levitt algebras: some elementary number theory yields a not-so-elementary result by GENE ABRAMS of UCCS (Rated PG-13)

For a field K, the Levitt K-algebra R=L_K(1,n) was defined and investigated in the early 1960s, and has enjoyed an intense revival of interest over the past few years. Although this algebra might look somewhat strange at first glance, it is pretty easy to describe in terms of generators and relations.

Consider the ring M(t, R) of t by t matrices with entries from the Levitt K-algebra R. In this talk Gene will prove that R and M(t, R) are actually isomorphic as rings if and only if gcd(t,n-1)=1. (Really! A ring isomorphic to a matrix ring over itself!) The proof of this result involves some straightforward number theory (which Gene will describe in detail).

This result is joint work with E. Pardo and P. N. Anh, and was obtained during the fall of 2006, when Professor Anh was a visitor at Colorado College.

If there’s time, Gene will also talk about a connection between the Levitt algebra and Cuntz C* algebras from analysis.

**Friday, May 12th, at 2:30 in TSC 229:**

*PRIMES OF THE FORM x^2+ny^2 (Rated PG13)**By Stefan Erickson of Colorado College*A fundamental question in number theory is: "Which primes can be written as x^2 + n*y^2?" Pierre Fermat was the first to give complete answers for n = 1, 2, and 3. As n grows larger, the solution becomes more difficult to obtain. The search for an answer for arbitrary n leads first to the study of quadratic forms, which produces partial success. Class field theory provides a straightforward answer for all n (certain polynomials must have a solution modulo p). However, finding these polynomials remains a challenge. Complex multiplication finally provides an algorithm for explicitly constructing the desired polynomials. Along this mathematical journey, we will encounter the likes of Euler, Lagrange, Legendre, Gauss, Dirichlet, Hilbert, and others.

**Friday, May 5th, at noon in TSC122:**

**PROBLEMS FROM THE PUTNAM AND RAWLES EXAM (Rated PG)**

Presentations by CC students who took these exams, plus Wojciech Kosek and Rahbar Virk

Each year CC fields a team of students who take the national Putnam Exam, and we also sponsor a local competition called the Rawles Exam. Come for pizza and the opportunity to hear about solutions to selected problems from these two exams.

**Friday, April 28th at 2:30 p.m. in TSC 229:Dynamical methods in combinatorial number theory: history and some new results about prime numbers**

*By Wojciech Kosek of Colorado College (Rated PG13)*

A deep connection between Ramsey theory and the measurable and topological dynamics was first discovered in 1977, when Furstenberg proved the Multiple Recurrence Theorem and showed its combinatorial consequences. This led to new proofs of known theorems in combinatorics such as theorems of Hindman, van der Waerden and Szemeredi. Dynamical perspective on combinatorics stimulated a very fruitful interplay between the two disciplines, leading to new combinatorial results as well as to the development of new tools in dynamics.

Prime numbers have fascinated people since the dawn of time. There are statements and conjectures about them, which are easy to state, yet whose proof is still evading us. In 2004, Ben Green and Terrence Tao announced their latest result: the primes contain arithmetic progressions of arbitrary length. To prove this result they adapted some methods from ergodic theory, harmonic analysis and number theory. Time permitting, some ideas used in the proof of this result will be discussed. Also, we will look at a short an elegant proof of the Hindman’s theorem which uses ultrafilters.

Except for a few brief comments, technical details will be avoided and the talk will be accessible to students majoring in mathematics, who are encouraged to come. Hope you will join us.

## BLOCK 7, 2006

**Friday, April 14: 2:30, TSC 122**

**Representations of groups and other algebraic objects in mathematics**Fred Tinsley

**PG-13 (linear algebra is a help but not a necessity)**

In algebra, groups are sets with a single (binary) operation imposed that satisfies certain properties. A favorite example is the group of symmetries of the equilateral triangle. For example, we can let j be the group element that reflects the triangle through one of its medians and r the group element that rotates the triangle counterclockwise by 120°:

These two elements can be used to generate the entire group of symmetries of the equilateral triangle that turns out to have 6 elements. The complete multiplication table is given below where i is the identity (i.e., leaves the triangle alone):

Classical representation theory desires to realize this group in terms of matrices and matrix multiplication. Computations become mechanical and do not require a table. Students of linear algebra can check that the assignments,

precisely realize these symmetries.

We will look at representations of this type and also as objects other than matrices. We hope to appreciate the utility of viewing algebraic objects in several different settings.

**Thursday , April 6th: (TSC 122)**

**Student Distinction Talks 2:30-4pm **

**Friday, March 31st: Noon (TSC 122) How To Solve Sudoku**Robin Wilson, Open University

Rated G

This talk is rated G, and should be of wide interest to those of you who are addicts, and those of you who are not! The sudoku epidemic has swept the world faster than you can count to nine. But what is sudoku? How did it arise? How many possible sudoku puzzles are there? And how do you solve the wretched things? Robin Wilson is the author of two sudoku books and he will reveal all! Enjoy the talk, and the pizza provided.

## BLOCK 6, 2006

**Tuesday, March 14th, **

**at 3:30 p.m. in TSC229: Sobolev on ends: a story of growth and decay - (Rated R)Stephen Buckley, National University of Maynooth, Ireland**

In 2001, Li and Wang showed that if the Dirichlet Problem for the Laplacian on an end of a complete manifold has a positive leasteigenvalue, then that end must have at least a certain exponential rate of growth or decay, and these rates are sharp. In this talk we discuss similar sharp results for (q,p)-Sobolev inequalities, the case p=q=2 of which implies the Li-Wang theorem. This is joint work with Pekka Koskela.

**Friday, March 10thTSC229 , at 2:30 p.m. **

*Revisiting Calculus, and a few Function Spaces*- (Rated PG13)J

**ane McDougall**

Over the past 100 years, updated versions of integration and differentiation have become mainstream. Although calculus is useful for solving so many problems, its processes have some limitations in terms of what converges where and when. In this talk, I will present some of the ideas that allow us to extend our notion of what can be integrated and/or differentiated. We will also consider some particular sets of functions where convergence works better (be prepared for the infamous D). One of the themes will be that life is easier when we have dense subsets and decent limit theorems.

**Friday, March 3rd, **

**TSC122 , Noon The Banach-Tarski Paradox - (Rated PG)by John Watkins**

The worlds of mathematics and physics have provided us with many instances of things that are almost impossible to believe and yet are true. Quite naturally, some of the more delightful of these “impossible, yet true” statements have been given names: Zeno's Paradox, Russell's Paradox, Einstein's Twin Paradox, Braess's Traffic Flow Paradox, and so on. But, surely, the finest paradox of all is the Banach-Tarski Paradox.

The Banach-Tarski Paradox says that it is possible to dissect a ball - that is, a solid sphere - into a finite number of pieces and then rearrange these pieces to form two balls exactly the same size as the original ball. Yet, nothing has been created or stretched in the process! All that has happened is that pieces of the original ball have been moved around in space like the pieces of a jigsaw puzzle and yet somehow you manage to end up with twice as much stuff as you started with.

Impossible, you say. If that’s what you think, then come and hear this talk. I will even show you a remarkable relationship between this paradox and the famous woodcut of Escher, Circle Limit IV, that is hanging in our hallway.

## BLOCK 5, 2006

**Friday, January 27thTSC 223, 2:30pm**

*Divisibility in the Fibonacci Numbers*-(Rated PG)by Stefan Erickson

The Fibonacci numbers arose from a famous problem posed 800 years ago about reproducing rabbits. There are many remarkable properties of this sequence. We will dive into two questions. Which Fibonacci numbers divide other Fibonacci numbers? Which primes divide some Fibonacci number? The incredible answers will be revealed, along with their analogues for a similar sequence called the Lucas numbers. In the process, we will uncover some of the marvelous interconnections between the Fibonacci numbers, the Lucas numbers, and the Golden Ratio.

## BLOCK 4, 2005

**Friday, December 16th, at 2:30 p.m., TSC 223 The dynamics of multiplication by 2 and 3 mod 1By Dan Rudolph, Colorado State University, Rated R**

The dynamics of the single endomorphism x maps to nx mod 1, with n > 1, is extremely chaotic but not difficult to understand. It is modeled by the one-sided shift map on infinite sequences of digits from {0,1, ..., n-1}. In this form it is seen to have many periodic points, many closed invariant subsets and many invariant measures.

> When one extends this to the dynamics of a commuting pair of such maps, the simplest being x maps to 2x and x maps to 3x mod 1, the situation changes dramatically. Dr. Rudolph will offer a proof of the celebrated theorem of Furstenberg, that any closed invariant set for both these maps is either finite or all of [0,1). He will discuss what is known about the space of invariant measures and what remains unknown.

**Friday, December 9th, **

**Noon, TSC 122 Penrose tilesBy Kathy Merrill, Rated PG**

Most tilings of the plane that we are familiar with, such as those that appear in M.C.Escher prints or on floors or wallpapers, are periodic. That is, most tilings have a pattern that repeats in two independent directions. Roger Penrose discovered a pair of tiles called kites and darts that fit together to tile the plane, but only in ways that have no repeating pattern. This talk will introduce you to Penrose tiles, their relationship to the Golden ratio, and their applications to understanding quasi-crystals of aluminum and manganese and even to making toilet paper. As a part of the talk, you will have a chance to build some Penrose tilings yourself.

**Friday, December 2nd, 2005**

**2:30 p.m., TSC 122 The Indian Wars, Eugenics, and Statistics: A Broader View of Scientific Racism Before the Outbreak of World War IIBy Mike Siddoway, Rated G**

Racial theories figured to devastating effect in the "Indian Wars" of the American West, and a half-century later in the "Final Solution" in Nazi Germany. Arguments surfaced from the across the political spectrum in late 19th century America in support of eradicating the native population. When the Minneconjou followers of Chief Big Foot were shot down at Wounded Knee Creek in 1890, marking the end of the war on the plains, US laws were already excluding immigrants on the basis of race and ethnicity. The British anthropologist Francis Galton coined the term "eugenics" in the 1880's. The US eugenics movement was very strong through the early part of the 20th century and was followed closely by adherents in Europe. Sterilization and miscegenation laws appeared in numerous states. Two of the most prominent figures in the history of statistics, Karl Pearson and Ronald Fisher both held positions in England as Professors of Eugenics in the decades before the start of WWII. By the mid 30's, when National Socialism was tightening its grip on Germany, eugenics had become a fixture in the intellectual landscape in the US and Europe. The rigor that mathematics brought to the eugenics movement through the development and study of statistics made it all the more accepted as a serious branch of science. In this talk I will look a little wider than Stefan

Kühl did in his excellent book on the links between American eugenics and National Socialism by including discussion of American racial attitudes during the "Indian Wars" and the possible role that mathematics played in legitimizing the broader eugenics movement before the outbreak of WWII.

## BLOCK 3, 2005

**Establishing Multi-Path Connectivity in Sensor NetworksJonathan Bredin, Colorado CollegeRated PGNovember 18th at 2:30 (TSC 223)**

A sensor network is a collection of small devices that use radios and sensors, such as photometers, microphones, and accelerometers, to collect information to monitor an environment and distributively process their data. We are interested in methods to distributively detect and repair multi-path connectivity (k-vertex connectivity) of the radio network imposed by a sensor-network deployment. Guaranteeing k-vertex connectivity simultaneously provides fault tolerance against node failures and high capacity through multi-path routing.

Jonathan will present the first approximation algorithms that compute the placement of a minimum number of additional sensors to augment an existing network into a k-connected network, for any desired parameter k. He will also present a distributed augmentation algorithm, and its implementation using Mica2 Motes.

This talk will cover a wide range of work from theory to application and will be accessible to students with little college math or computing experience.

**Predicting Retirements of CC Faculty Fred Tinsley, The Colorado CollegeRated PGNovember 11th at noon (TSC 122)**

Fred and John Stinespring (Economics) have been working on a model to predict major trends in the cost of paying full-time faculty at Colorado College. Given the rather careful adherence by the Administration to the clearly delineated Faculty Salary Model, much of the work is entirely straightforward. However, the most unpredictable and, unfortunately, the most significant element is the distribution of retirements. The replacement of highly paid full professors by lower paid beginning assistant professors mitigates the factors that drive up costs.

Fred will discuss their approach, which is a blend of two major methodologies in statistics: Bayesian and frequentist. Don’t let these terms scare you off. If you have taken the equivalent of either Calculus I or Intro to Stats, then you will see an intriguing application of what you have learned.

Of course, an extensive literature on predicting retirements exists. However, Fred will argue the importance of first investigating the problem on your own before diving into the literature. This has always been for Fred one of the most valuable approaches to learning.

**The mathematics of representing human population density on a mapMichael Gastner, The Sante Fe InstituteRating: PG13November 4th at 2:30 (TSC 223)**

Dr. Gastner will present a method to construct "cartograms," maps in which the sizes of geographic regions such as countries or provinces appear in proportion to their population. Such maps, although unconventional, are often the best way to represent human data. The construction of cartograms turns out to be a challenging mathematical problem. A variety of methods have been put forward, but none is entirely satisfactory. In particular, they often produce highly distorted maps that are difficult to read or projections that are badly behaved under some circumstances. In many cases the methods proposed are also computationally demanding, sometimes taking hours to produce a single map.

The method discussed, based on the mathematics of diffusing gas particles, makes low demands on our computational resources, while also producing elegant, well-behaved, and useful cartograms. The speaker will illustrate the technique with applications to election results, disease incidence, and the optimum distribution of service facilities.

## BLOCK 2, 2005

**Friday, October 21st, 2:30 in TSC 229 Luis Melara Title: "A Homotopy Method in Regularization of Total Variation Image Denoising Problems"**

**Rating: R ***(Some graduate mathematics assumed)*

Homotopy methods can often be used to make mathematical programming problems easier to solve. One source of notoriously difficult problems is total variation image

denoising. In this talk, a direct relationship is established between the radius of the Kantorovich ball guaranteeing the convergence of Newton's method and the regularization parameter. This radius of convergence increases with the choice of the regularization parameter. A homotopy method in the regularization parameter is employed to improve numerical performance of Newton's method applied to these problems. The talk will summarize a convergence analysis and present numerical results.

**Friday, October 7th, 2:30 p.m. in TSC 229: **

**Dr. Craig Guilbault of the University of Wisconsin-Milwaukee**

Title: "Exotic Behavior near infinity on Manifolds"

*Rating: R (Some graduate mathematics assumed)*

Most of Dr. Guilbault's research involves the topology of manifolds. Currently, he studies "open" or "non-compact" manifolds. In simple terms, these are surfaces that are (in some sense) infinite in size. For example, the Euclidean plane? that extends infinitely outward? is non-compact; whereas the 2-sphere is compact. The extra flexibility permitted by non-compactness opens the door to exotic behavior "near infinity". There are many examples of non-compact spaces that decompose into simple finite pieces but have quite complicated structure at infinity. More technically, Professor Guilbault's research is motivated by current work in geometric group theory and by new approaches to the well known Borel and Novikov Conjectures in manifold topology. He is especially interested in Z-compactifications of open manifolds and complexes, and structure theorems for open manifolds with non-stable fundamental groups at infinity. Of special interest are universal covers of aspherical manifolds, and manifolds which admit CAT(0) metrics (a generalized version of non-positive curvature). Professor Fred Tinsley of Colorado College has collaborated with Professor Guilbault in studying non-stable fundamental groups at infinity.

**Friday, October 14th, noon in TSC 227**

*Steven JankeTitle: "Making Rotations Easier: The Quaternion Method"Rating: PG13 (A lot of undergraduate mathematics assumed)*

Rotating objects in three dimensions is a problem key to many fields: computer graphics, aeronautics, physics. The traditional way to find the coordinates of points on a rotated object is to use trigonometry to describe rotations around each of the standard axes. However, in the early nineteenth century, William Hamilton discovered another method that falls in the intersection of abstract algebra and geometry. Although his method of quaternions fell into disfavor at the end of the nineteenth century, it has been revived recently by those interested in computer graphics and offers an often-useful perspective on rotations.

## BLOCK 2, 2005

October 7th at 2:30: Craig Guilbault with a research talk

October 14th at noon: The second Friday student talk, by Steven Janke

October 21st at 2:30: Luis Melara

## BLOCK 1, 2005

**Friday, September 23rd, noon, TSC 122: Mike Siddoway**

**Title: "Closed Chains with Tetrahedral Links?"**

**Rating: PG.**

Abstract: Is it possible to make a complete chain using tetrahedrons that share a

complete face with nearest neighbors? We can make a chain in two-space using

equilateral triangles or squares that share edges. In three-space there are

infinitely many complete chains with cubes as links that share sides. What if

we use tetrahedrons in three-space? Josh Laison and CC student Luc Wilson

posed this problem last spring during the distinction talks and Mike will give

a solution that uses linear algebra and some polynomial theory. It turns out

that the deciding factor is whether 2/3 can be the root of certain polynomials

with integer coefficients.

**Friday, September 16th, 2:30 p.m., TSC 229: **

**Jane McDougall, on “More about star domains”Rated R.**

Jane gives a follow-up talk to her seminar last spring on her work with Beth

Schaubroeck on Harmonic Analysis on Star Domains, including some really recent

developments.

**Friday Sept. 9, 2:30, TSC 229**

*Jonathan BredinTitle: Models for Truthful Online Double AuctionsRating: G*

Abstract:

An online double auction (DA) is a market where traders arrive and depart

a market at arbitrary times to exchange a commodity. Each trader wishes

to maximize its economic surplus and possess private information regarding

its values, costs, and lifetimes. We will present an auction primer and

move on to present a general method to design truthful online DAs, such

that

no agent can benefit from misreporting its arrival time, duration, or

valuation. The family of DAs is parameterized by a pricing rule, and

includes a generalization of McAfee's truthful DA to this dynamic setting.

We present an empirical study, in which we study the allocative-surplus and

agent surplus for a number of different DAs. Our results illustrate that

dynamic pricing rules are important to provide good market efficiency for

markets with high volatility or low volume.

No mathematical, economic, or computational background is assumed. Our work

should be consumable for all interested parties (rated G). The work

presented is a joint effort with David Parkes.

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