This is an archived page, from a previous year. Links may be broken.
The current schedule can be found here.
The 2008-2009 schedule can be found here.

All talks are at 4:15 in HH229 HH-231 unless otherwise noted. (Please note the time change from previous semesters!)

To volunteer to give a talk, or for anything else regarding the seminar contact Don Brower.

## Schedule

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Date Speaker Title
Monday, September 10 Prof. Jeff Diller Newton's Method and complex dynamical systems
Monday, September 24 Stacy Hoehn The Method of Infinite Repetition in Topology
Monday, October 8
Monday, October 15
Tom Edgar Wyld Weyl and Twisted Bruhat
Monday, October 29 Jon Hauenstein Homotopy continuation and intersecting algebraic sets without defining equations
Monday, November 12 Sara Quinn A tour of computable structure theory
Monday, November 26 Tanya Salyers Coupled Oscillators and Biological Synchronization
Monday, December 10 Fernando Galaz-Garcia Riemannian geometry via symmetries
Monday, January 21 Daniel Cibotaru Cohomology of the unitary groups and Morse Theory
Monday, February 4 Logan Axon A Short Tour of Algorithmic Randomness
Monday, February 18 Stuart Ambler A Brief Look at the Dirac Equation of Physics
Monday, March 10 Steven Broad Umbilics and Loewner's Conjecture
Monday, March 24
Monday, March 31
Chrissy Maher Math Games People Play
Monday, April 7
Monday, April 14
Angie Kohlhaas Pictures of Commutative Algebra
Monday, April 21
Monday, April 28
Sarah Cotter Goodstein's Theorem and other abstract nonsense
Goodstein’s Theorem and Incompleteness

## Abstracts

### September 10, 2007

Speaker
Jeffery Diller
Title
Newton's Method and complex dynamical systems
Abstract
If you think there's not much left to be said about rational functions of a single complex variable, then try iterating one and analyzing the results. This has, at any rate, kept a number of mathematicians busy for the past few decades. In this talk, I'll make a particular example of Newton's method applied to find the roots of a polynomial in one variable in order to try and give you some idea of what the subject of complex dynamics is about.

### September 24, 2007

Speaker
Stacy Hoehn
Title
The Method of Infinite Repetition in Topology
Abstract
A cobordism is a compact manifold whose boundary is made up of two disjoint pieces. In general, cobordisms can be quite complicated, but somewhat surprisingly, once you remove one of the pieces of the boundary from an invertible cobordism, the result always has a simple form. The proof of this uses a technique of infinite repetition (sometimes called an Eilenberg swindle) which has also been used for many other applications in topology, including the Generalized Schoenflies Theorem. We will discuss a few of these applications and how they are related to the notion of groups (or groupoids) with infinite products. There will be lots of pictures!

### October 15, 2007

Speaker
Tom Edgar
Title
Wyld Weyl and Twisted Bruhat
Abstract
The Bruhat order is a partial order on Coxeter groups with many remarkable properties. It plays an interesting role in many different areas of mathematics including algebraic geometry, representation theory, and combinatorics. We will begin by introducing Coxeter groups (in particular dihedral groups and finite Weyl groups) and the definition of Bruhat order. After that, we will discuss how to create Bruhat-like orders which we call "twisted Bruhat orders." We will give a result that classifies all of the possible twisted Bruhat orders on a Coxeter group that arise in a manner similar to the Bruhat order. Throughout the talk, we will use the simple example of S3 (the symmetric group on three letters) to illustrate the ideas presented.

### October 29, 2007

Speaker
Jon Hauenstein
Title
Homotopy continuation and intersecting algebraic sets without defining equations
Abstract
Homotopies are a continuous deformation of one object to another, e.g. coffee cup to a doughnut. In algebraic geometry, homotopies provide a (numerical) method for solving systems of polynomial equations. We will begin by introducing homotopies of polynomial systems and look at some of their basic properties. After looking at some example problems, we will use homotopies to intersect algebraic sets where the defining equations are not known explicitly.

### November 12, 2007

Speaker
Sara Quinn
Title
A tour of computable structure theory
Abstract
This talk will be a tour of computable structure theory, using equivalence structures as our tour guide. I will introduce equivalence structures and define some of the themes in computable structure theory, such as computable spectrum, index set complexity, and computable embeddings. I will also give some results in these areas as they relate to equivalence structures. No logic background will be assumed.

### November 26, 2007

Speaker
Tanya Salyers
Title
Coupled Oscillators and Biological Synchronization
Abstract

At night, in certain parts of southeast Asia, thousands of male fireflies gather in trees at night and flash on and off in unison in an attempt to attract the females that cruise overhead. When the males arrive at dusk, their flickerings are uncoordinated. As the night deepens, pockets of synchrony begin to emerge and grow. Eventually whole trees pulsate in a silent, hypnotic concert that continues for hours.

Fireflies are the epitome of a "pulse-coupled" oscillator system: they interact only when one sees the sudden flash of another and shifts its rhythm accordingly. Pulse coupling is awkward to handle mathematically because it introduces discontinuous behavior into an otherwise continuous model. In this talk, I will discuss the emergence of synchrony in a population of pulse-coupled oscillators and explain what it tells us about fireflies.

### December 10, 2007

Speaker
Fernando Galaz-Garcia
Title
Riemannian geometry via symmetries
Abstract
The geometry of isometry groups provides a useful tool in the study of Riemannian manifolds. We will survey the main ideas and some important results in this area, focusing on the nonnegative and positive curvature setting.

### January 21, 2008

Speaker
Daniel Cibotaru
Title
Cohomology of the unitary groups and Morse Theory
Abstract
The unitary groups have a pervasive presence in geometry and far beyond, and you shouldn't be amazed if a Google search for these two words alone returns more than a hundred thousand results. If popularity is not something that you might consider worthy of esteem then one should say that their rich structure is probably equalled only by the simplicity of their definition. This talk aims to explain the words in the title and then the title as a whole and give also a little, cute corollary of what will have been said up to that point. They say that after the cute corollary there's free pizza. Does it get any better?:)

### February 4, 2008

Speaker
Logan Axon
Title
A Short Tour of Algorithmic Randomness
Abstract

Combining computablility theoretic ideas and measure allows us to identify a collection of random'' binary sequences.Random'' because they are unpredictable, incompressible, and effectively average. The digits of any one random sequence behave pretty much as a sequence of independent identically distributed random variables (taking values 0 or 1).

Effective randomness has been extended to the space of continuous functions on the real interval, a probability space under the Wiener measure. A sequence of random variables taking values 0 or 1 determines a random walk. Taking a random sequence (random as above) determines a sequence of random walks. Limits of random walks have a well established connection with the Wiener measure so it is no surprise that random sequences correspond with random'' functions.

Recently some effort has been made to extend algorithmic randomness to the space of closed subsets of $2^{\mathbb{N}}$. Early efforts seemed unnatural and unsatisfying to some. Recently I have been trying to fit that work into the framework of probability theory for this space.

### February 18, 2008

Speaker
Stuart Ambler
Title
A Brief Look at the Dirac Equation of Physics
Abstract

Quoting Wikipedia, "In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928" that "provides a description of elementary spin-½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. The equation demands the existence of antiparticles and actually predated their experimental discovery, making the discovery of the positron, the antiparticle of the electron, one of the greatest triumphs of modern theoretical physics."

This talk will explain, using math mostly understandable by engineering undergraduates who learned what they should have in multivariable calculus and linear algebra and differential equations, and with only a few references to physics, how the use of things called spinors allowed Dirac to take the square root of a second order differential operator, obtaining a first order operator, and his equation.

### March 10, 2008

Speaker
Title
Umbilics and Loewner's Conjecture
Abstract
We will discuss the connection between Carath\'eodory's conjecture on the minimum number of umbilical points present in deformations of a sphere and Loewner's Conjecture about the index of vector fields of the form $\partial_{\bar{z}}^n f$ for functions $f:\mathbb {C}\to\mathbb{R}$. A recent result of F. Xavier allows the index of such vector fields to be computed in terms of the set of eigenvalues of the Hessian of $f$ in the case $n=2$. A further result allows us to compute the index of such vector fields for $n \geq2$.

### March 31, 2008

Speaker
Chrissy Maher
Title
Math Games People Play
Abstract
I'll give an overview of some math games, both in the sense of mathematical content of games which many people have played and in the sense of a mathematical technique to demonstrate certain properties. In the first part I will talk about a few of games I liked to play as a kid or still like to play now, such as 24, Set, Tic Tac Toe, and Dots and Boxes, all of which either have mathematical content, or promote mathematicial/logical reasoning. Tic Tac Toe and Dots and Boxes will serve as the transition into the second part of the talk, as they are games between two people in which each is trying to play in order to maximize their chance of winning. A game between two opponents, possibly with one having a winning strategy is a method of proof in mathematical logic. For example, an Ehrenfeucht-Fraisse game on two structures is a way of showing isomorphism or non-isomorphism, and a strong Choquet game can be played on a metric space with a winning strategy for player II if and only if the metric space is completely metrizable.

### April 14, 2008

Speaker
Angie Kohlhaas
Title
Pictures of Commutative Algebra
Abstract
Commutative algebra is often thought of as a technical, non-pictorial version of algebraic geometry. However, when we consider the case of monomial ideals in polynomial rings, many algebraic properties can be illustrated with lattice diagrams and graphs. In this talk, we will first state some of the basic questions in commutative algebra. Using the combinatorial properties of monomial ideals, we can then create pictures to answers these questions in the monomial case. We will talk briefly about how monomial ideals provide information about general ideals, and hopefully end by introducing the core of an ideal.

### April 28, 2008

Speaker
Sarah Cotter
Title
Goodstein’s Theorem and Incompleteness
Abstract
Gödel’s First Incompleteness Theorem guarantees the existence of statements which are true, but not provable, in Peano Arithmetic. Goodstein’s Theorem was one of the first examples of an unprovable statement in number theory. In this talk, we will prove Goodstein’s 1944 theorem, and outline a proof of Kirby and Paris’s 1982 result that Goodstein’s Theorem is unprovable from PA. In the process, we will cover some necessary background in the ordinal numbers; discuss a related result by Kirby and Paris concerning the killing of hydra; and attempt to set the 2007-2008 record for greatest integer used in a GSS talk.

## Previous Years

Don Brower - Math Department - University of Notre Dame