Department of Mathematics
Graduate Colloquium: Fall 2007

Graduate Colloquium
Fall 2007
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.

August 28
Speaker: Mike Purcell
Title: Sounds Like Math
When first faced with the complexities of the western musical tradition, many people are understandably frustrated by the prevalence of (seemingly) arbitrarily chosen structures. It is tempting to ask, for example, why is the piano keyboard designed like it is? Why are there only 12 notes? Why do we spend so much time working with major scales and chords, and why do they sound "good"?

The answers to all of these questions (and many more) have their basis in the same physical phenomenon, the overtone series. In this talk we will see that the overtone series appears as a consequence of the general solution of the one-dimensional scalar wave equation and explore some of the acoustic consequences of these solutions (including a magic trick!).

We'll also take a look at some of the problems inherent in attempting to view all of music theory through the lens of the overtone series, and how modern music theorists are using topological techniques to bring a new perspective to their field of study.

This talk will NOT require any prior knowledge of music theory, and we'll take the time to develop any mathematical ideas that we'll need along the way. Therefore, the talk will be largely self contained.

September 4
Speaker: Amber Smith
Title: Presentations Using LaTEX: The Beamer Class
LaTeX is one of the most powerful tools a scientist can use. Not only can you write articles that correctly typeset mathematics, but you can use it for slide presentations. Yes, your days of using powerpoint are over (HALLELUJAH!)!! Using the Beamer class in LaTeX allows for production of great beautiful presentations that will WOW your audience. It is easy to use, automatically makes a PDF document, has sweet features for everyone including using movies, makes navigating around your talk extremely smooth (NO MORE flipping through every slide when someone asks a question!), and gives a final product that your audience will remember.

September 11
Speaker: Zack Kilpatrick
Title: Traveling pulses and wave propagation failure in inhomogenous neural media.
We use averaging and homogenization theory to study the propagation of traveling pulses in an inhomogeneous excitable neural network. The network is modeled in terms of a nonlocal integro-differential equation, in which the integral kernel represents the spatial distribution of synaptic weights. We show how a spatially periodic modulation of homogeneous synaptic connections leads to an effective reduction in the excitability of the network and a corresponding reduction in the speed of a traveling pulse. In the case of large amplitude modulations, the traveling pulse represents the envelope of a multibump solution, in which individual bumps are non-propagating and transient. The appearance (disappearance) of bumps at the leading (trailing) edge of the pulse generates the coherent propagation of the pulse. Wave propagation failure occurs when activity is insufficient to maintain bumps at the leading edge.

September 18
Speaker: Andrew Nelson
Title: Sparse Approximations and High Dimensional Geometry
By a lucky coincidence, most digital camera sensors are able to have millions of sensors on a single silicon chip. Unfortunately, many of these measurements are thrown away, as the image is then compressed to a smaller size. It is then natural to ask whether or not we can take fewer measurements and get a compressed image directly. The surprising answer is yes, and that this sampling procedure is non-adaptive. The seminar today will discuss this phenomenon and give some geometric arguments about why it is true.

September 25
Speaker: Jason Preszler
Title: Leopoldt's Conjecture
The idea of viewing a number field K as a subfield of the real or complex numbers has resulted in some of the most fundamental results of algebraic number theory. Embedding K into it's other completions should yield results of the same, if not greater, importance; but in these p-adic situations there is a plethora of unsolved problems. Leopoldt's conjecture is the result of trying to transport the Dirichlet Unit Theorem into this p-adic realm. After a discussion of these ideas and a description of Leopoldt's conjecture we will conclude with a recent algorithm to verify the conjecture.

October 2
Speaker: Julian Chan
Title: Sicherman Dice
The regular dice are labeling one through 6. When you play with two dice there is a certain probability of landing a number. How would one construct die with different numerical values on their faces that gives the same probability of landing a certain number as the regular dice.

October 16
Speaker: Bill Casselman
Title: Dirichlet Does Distributions
In the {\sl Disquisiiones Arithmeticae}, Gauss mentioned that quadratic reciprocity followed from an evaluation of the sign of certain quadratic Gauss sums. At that time he had found a great deal of empirical evidence for such an evaluation, but was able only a few years later to prove his conjecture. His proof was purely algebraic, but somewhat indirect, and dealt separately with different cases. The second evaluation of these signs was by Dirichlet, many years later. His very beautiful proof is much more direct than Gauss', albeit much less elementary. It relies on the theory of Fourier series, which as rigorous mathematics was due entirely to him, and Fresnel integrals, which had been introduced only recently in the theory of diffraction. I shall present a modern version of Dirichlet's proof, which seems to be new.

October 23
Speaker: Damon Toth
Title:Dynamics of Age-Structured Populations in a Chemostat
A chemostat is a laboratory apparatus that provides a controlled, resource-limited environment for microorganisms. Chemostat models are of considerable interest to theoretical ecologists because they can provide predictions with broad ecological ramifications that are testable in a laboratory. I will present the research I did at the University of Washington on chemostat models: a single-species chemostat model that couples a PDE describing an age-structured population with a system of ODEs describing a chemostat, and a periodically forced version of the model that displays strongly resonant and chaotic behavior.

October 30
Speaker: Dylan Zwick
Title: The Principle of "Least" Action
One of the most amazing results in physics is the principle of "least" (actually stationary) action and its application to mechanics. In this talk I will discuss some classical problems, including refraction and the Brachistochrone, and their history and how the method for solving these problems led to a subject in analysis called the calculus of variations. I will then go over some of the basic results in the calculus of variations, and discuss their application to mechanics. Time permitting I will even construct a reformulation of Newtonian mechanics based on this principle of "least" action. Nothing too advanced, and the talk should be comprehensible to any graduate student, but if you've never seen this before it's definitely worth knowing.

November 6
Speaker:Peter Trapa
Title: Flags Fixed by Matrices
This talk is about an elementary construction from linear algebra which, perhaps surprisingly, turns out be very important in several areas of mathematics. Start by considering an $n$-by-$n$ complex matrix, say $N$. Consider ``flags'' of nested subspace $V_1 \subsetneq V_2 \subsetneq \cdots \subsetneq V_n$ inside $\mathbb{C}^n$ with the property that if $v_i$ is a vector in $V_i$, then $Nv_i$ is also in $V_i$. (In this sense the flag is ``fixed'' by $N$.) It turns out that space of such fixed flags is incredibly interesting for all sorts of reasons. We'll uncover a few of them.

November 13
Speaker: Bobby Hanson
Title:A Survey of Geometric Constructions, and the Resulting Subfields of $\mathbb{C}$
We are all familar with so-called Euclidean constructions (also called Straightedge-and-Compass constructions). Recall that some constructions are possible with a straightedge and compass, while others are impossible. In particular, it is always possible to bisect a given angle; while some angles are impossible to trisect. It is possible to find the square-root of any previously constructed length, but it is usually impossible to find a cube-root.

Recall, also, that the lengths that are constructible in this fashion (called the Euclidean numbers) form a subfield of $\mathbb{C}$. In fact, it is the smallest subfield of $\mathbb{C}$ which is closed under taking square-roots. For this survey, we will look at the set of axioms that give us Euclidean constructions, and see that subsets of these axioms lead to other (smaller) subfields of $\bbC$. We will also see that all of the axioms of Euclidean constructions can be realized by merely making folds in our paper version of $\mathbb{C}$, rather than using the clumsy straightedge and compass. Moreover, an addional axiom can be added, also realizable through folding, which allows such amazing things as trisecting angles or computing cube-roots!

November 20

November 27
Speaker: Liz Copene
Title: Potassium Coupled Oscillations
Cardiac cells are electrically coupled through gap junction channels, which allow ionic current to spread intercellularly from one cell to the next. However, because the extracellular junctional cleft space between neighboring cells is so narrow and tortuous, it might act as a microdomain for ionic concentrations. In this microdomain, ionic concentrations (potassium in particular) might vary drastically and rapidly enough to conduct an electrical signal from one cell to the next. I will present a model of two cells coupled through junctional potassium and discuss the resulting dynamics of the coupled cells. Even if your not into the physiology, the dynamical system itself is pretty cool!

December 4
Speaker: Nessy Tania
Title: Stochastic Simulation of Chemical Reactions
In living cells, biochemical reactions often involve only a small number of molecules. As a result, a continuous deterministic approximation is often inadequate and a full discrete stochastic formulation is required to study the system. The Gillespie algorithm commonly used to numerically simulate stochastic chemical reactions will be introduced. Although the method is exact, it can be very expensive because it simulates every reaction event. This is especially true when fast and slow reactions are occurring simultaneously. The Gillespie algorithm will spend the majority of the time simulating the fast events even though the slow reactions often have a greater impact on the behavior of the system. Two approaches to accelerate the simulation will be discussed. The first approach involves approximating the number of reactions by a Poisson process. The second approach involves taking quasi-steady-state approximations of the fast reactions. The strengths and weaknesses of each method will be discussed. To elucidate the concepts introduced in this talk, a simple reversible isomerization reaction will be used as an example, and the basic result will be discussed.

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