Department of Mathematics
Graduate Colloquium: Fall 2008

Graduate Colloquium
Fall 2008
Tuesdays, 4:35 - 5:35 PM, JWB 335
Math 6960-001
(credit hours available!)

GSAC Home | Past Graduate Colloquia

The goal of this Colloquium is to encourage interaction among graduate students, specifically between graduate students who are actively researching a problem and those who have not yet started their research. Speakers will discuss their research or a related introductory topic on a level which should be accessible to nonspecialists. The discussions will be geared toward graduate students in the beginning of their program, but all are invited to attend. This invitation explicitly includes undergraduate students.

September 9
Speaker: Andrejs Treibergs
Title: Can You Hear the Shape of a Manifold?
Closed, curved surfaces in three space and rectangles with periodic boundary conditions are examples of manifolds. If a manifold vibrates according to the wave equation, what frequencies occur as overtones, and how are the frequencies influenced by the geometry of the manifold? Knowing that some geometric quantities, such as the volume, can be determined from the frequencies, Mark Kac asked in 1966 whether the manifold itself (or a drum) can be completely determined knowing its frequencies. By the work of Milnor and Gordon-Webb-Wolpert the question has been resolved in the negative.

By separating variables, the frequencies turn out to be given by the eigenvalues of the Laplacian (the spectrum) of the manifold. I'll describe some results that relate the geometry to the spectrum including the variational formulation for eigenvalues and Weyl's asymptotic formula.

September 16
Speaker: Jay Newby
Title: Random Search Strategies for Delivering Cellular Resources to Active Synapses in Neurons
A synapse is a connection between two neurons that chemically propagates a nerve impulse from one neuron to another. It is thought that changing the strength of the synaptic connections between certain neurons is the mechanism underlying learning and memory. New studies have also implicated this mechanism in drug addiction. I will present recent work I've done with Paul Bressloff using random search theory to help understand part of this story. We model the transport of building materials necessary to strengthen a synaptic connection using a jump Markov process with both discrete and continuum states.

September 23
Speaker: Dylan Zwick
Title: A Number by Another Name
In this talk I'll explore some ideas relating to how we describe numbers. I'll first cover the notion of the set of describable numbers, and then touch upon some of the stranger, more subtle aspects of the real numbers, including the fact that "most" of the real numbers cannot be described individually. I'll then discuss the concept of Kolmogorov complexity, and ideas about how we measure how hard it is to describe a number.

September 30
Speaker: Brittany Bannish
Title: Path-Finding Slime Mold: A-MAZE-ing!
In this talk, I will discuss the paper by Atsushi Tero, et al. called "A mathematical model for adaptive transport network in path finding by true slime mold". When the true slime mold Physarum polycephalum is spread over a maze, with food placed at the maze entrance and exit, the slime mold rearranges itself until it connects the two food sources by the shortest path. This process of path-finding is attributed to an underlying physiological mechanism that we will explore in the talk. The main focus of the talk will be the mathematical model that the authors create to help explain how this amoeboid organism "solves" a maze.

October 7
Speaker: Julian Chan
Title: A Topic in Invariant Theory

October 14
Speaker: NONE
Title: Fall Break
Fall Break

October 21
Speaker: Zack Kilpatrick
Title: Symmetry on Acid
Symmetric causes in nature have long been known to produce symmetric effects (Curie, 1894). Group theory can be used to quantify the symmetry of equations governing a physical system and therefore predict the symmetry of its solutions. We review an example of such techniques in a model of neural activity in primary visual cortex (V1). Drug-induced geometric visual hallucinations are thought to arise spontaneously in V1 due to its functional architecture (Bressloff et al, 2001; Bressloff and Kilpatrick, 2008). Using symmetric bifurcation theory, perturbation methods, and Fourier analysis we can characterize steady-states of the model that match quite well with experimentally observed contoured hallucinations.

October 28
Speaker: Aaron Wood
Title: Partitions
A partition of a positive integer n is a non-increasing sequence a_1, ..., a_k of positive integers such that n = a_1 +...+ a_k, and the number of possible partitions of n is denoted p(n). For example, the partitions of 4 are 4, 3+1, 2+2, 2+1+1, 1+1+1+1, so p(4)=5. To study the partition function p(n), many tools and techniques have been developed using combinatorics, generating functions, and modular forms. Using combinatorics and generating functions, we will discuss various properties of p(n) like Euler's Theorem and the Ramanujan congruences, and we'll talk about the exact formula for p(n) due to Hardy and Ramanujan.

November 4
Speaker: Ben Murphey
Title: Foundations of Statistical Mechanics
In statistical physics one is faced with the problem of assigning probabilities to events based on a few significant bits of information. In practice this information is far from sufficient to obtain objective nor unique probabilities. It is common to use the concept of entropy in order to develop a theoretical description of the macroscopic properties of a system, based on its underlying microscopic properties, which are often not precisely known. I will discuss how the entropy maximization approach to statistical mechanics allows one to derive the system probability distribution and the first law of thermodynamics without any assumptions about the nature of the system or its evolutions, hence may be applied to many Hamiltonian systems.

November 18
Speaker: Erika Meucci
Title: Braid Groups
What is a braid in math? Braid groups were introduced explicitly by Emil Artin in 1925 although the idea was implicit in Adolf Hurwitz's work (1891). Artin found a beautiful presentation of the braid groups in terms of generators and relations. Moreover, he proposed to use braids to study knots and links. There are now applications involving braid groups in several mathematical fields such as topology (knots and links), geometry (mapping class group) and dynamical systems. In this talk I will define braid groups and I will present Artin's theorem and few applications of braid groups.

November 25
Speaker: Mike Purcell
Title: Dimension Estimation and Manifold Learning
In many modern applications, researchers are faced with very high-dimensional data which is not compatable with tradition methods of analysis. In many of these applications, however, the data can be thought of as living on a much lower dimensional manifold embedded in a high-dimensional Euclidean space. In this talk, I will discuss what one means by local Hausdorff dimension and how local dimension can be used to constuct an estimator for the intrinsic dimension of such a manifold without prior knowledge of the sampling method used to collect the available data.

December 2
Speaker: Christopher Hacon
Title: Birational Algebraic Geometry
In this talk, I will illustrate several ideas from birational algebraic geometry. I will then show how some of the geometric techniques can be used to study interesting problems in algebra.

December 9
Speaker: Tim Carstens
Title: Dangers in Key Reuse: WEP is old and broken
WEP is a common protocol used to encrypt communications on wireless networks. Due to a collection of small breaks, modern methods have completely eliminated the security once promised by WEP. In this talk we will explore the nature of these breaks and show how they can be used together to perform a key recovery attack with high probability in short time using common equipment.

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