Department of Mathematics
Graduate Colloquium: Fall 2006

Graduate Colloquium
Fall 2006
Tuesdays, 4:35 - 5:35 PM, JWB 335
Math 6960-003
(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 29
No Talk - Introductory Meeting

September 5
Speaker: Scott Crofts
Title:Theorem of the Highest Weight
The "Theorem of the Highest Weight" gives a remarkably explicit parameterization of irreducible, finite-dimensional representations of a complex semisimple Lie algebra (up to some natural notion of equivalence). In addition to explaining what some of these words mean, the goal of this talk is to explain this important theorem primarily by considering the special case of sl(2,C). The only prerequisites should be a good background in linear algebra.

September 12
Speaker: Karim Khader
Title: An Introduction to Brownian Motion
This talk will mainly focus on the history of Brownian Motion. I will briefly discuss it's origin as well as the contributions of Einstein, Bachelier, Wiener and Donsker towards a mathematical model and understanding of Brownian Motion. I will discuss some of the interesting properties of Brownian Motion and if there is time, I will motivate some applications using Brownian Motion. I will not assume prior knowledge of measure theory and any concepts needed from probability will be provided during the talk.

September 19
Speaker: Russ Richins
Title: A Brief Intro to Direct Methods in the Calculus of Variations
Most natural systems seek to minimize their free energy, and it quite often happens that that energy can be expressed in terms of an integral functional. This suggests that functions which describe physical systems should be minimizers of the energy functional over an admissible class of functions. The direct method in the Calculus of Variations gives us tools to determine when such a minimizer exists. We will discuss the general framework of the Calculus of variations and discuss how the minimization process relates to PDE.

September 26
Speaker: Erin Chamberlain
Title: The geometry of commutative algebra
Commutative algebra is closely linked with algebraic geometry. One area studies rings and the other studies spaces, but we will discuss how they are related by studying something as elementary as polynomials. We will see that certain properties of rings correspond to certain geometric properties of spaces. During the talk we will also introduce a conjectures which, though easy to understand, has been open since 1939.

October 3
Speaker: Meagan McNulty
Title: A Model of Respiratory Inflammation
Chronic inflammation occurs when the immune system is unable to return to a neutral state. This inflammation can be devastating to the health of the individual. Mathematical models can provide insight to controlling unwanted inflammation and even lend some ideas about the underlying causes. I will introduce a model of respiratory inflammation and give a quick overview of the immune system. The analysis of this model identifies some conditions commonly seen in diseases which are associated with chronic inflammation.

October 10
Speaker: Josh Thompson
Title: Fractal Dust and Schottky Dancing
The Book Indra's Pearls is a visually stunning story of a computer aided exploration of unusually symmetrical shapes which arise when two special spiral motions interact. In other words, the book has lots of pictures of 'fractals' in it - and it describes these images in an incredibly accessible way. I will shamelessly use slides created by the book's author (with permission) to convey some of the mathematics behind the striking images. Beautiful pictures are guaranteed, a good description of them is likely. This talk should be accessible to mathematical people of all varieties.

October 17
Speaker: Berton Earnshaw
Title: Biophysical model of AMPA Receptor Trafficking and its Regulation during LTP/LTD
AMPA receptors mediate the majority of fast excitatory synaptic transmission in the central nervous system, and evidence suggests that AMPA receptor trafficking regulates synaptic strength, a phenomenon implicated in learning and memory. There are two major mechanisms of AMPA receptor trafficking: exo/endocytic exchange of surface receptors with intracellular receptor pools, and the lateral diffusion or hopping of surface receptors between the postsynaptic density and the surrounding extrasynaptic membrane. I will present a biophysical model of these trafficking mechanisms under basal conditions and during the expression of long term potentiation (LTP) and depression (LTD). I will show that the model reproduces a wide range of physiological data, and how the model reveals features that a synapse must have to be "physiological."

October 24
Speaker: Jason Preszler
Title: Linear and Arboreal Galois Representations
This talk will introduce the idea of a Galois representation and explain why these objects are of central importance in algebraic number theory. The connections between Galois representations and other objects, such as elliptic curves and modular forms, will then be discussed as further motivation for studying these objects. The last part of the talk will focus on recently developed arboreal Galois representations that can be understood by constructing trees with a natural Galois action.

October 31
Speaker: Dan Margalit
Title: Moduli Space
If you take the unit square and glue parallel sides, you get a surface known as the torus. What if you start with a parallelogram that is not a square? You get a different kind of torus. In fact, there is an entire parameter space of these tori. In this talk, we will start to get a basic understanding of what this space of tori looks like. For example, we will determine its dimension and decide if it is a connected space.

November 7
Speaker: Lindsay Crowl
Title: How to Stream and Collide Gracefully
The Lattice Boltzmann Method (LBM) is a relatively new computational approach to solving fluid flow problems. Instead of assuming that the fluid is a continuum, as in the Navier Stokes equations, the Lattice Boltzmann Method uses a mesoscopic particle based technique. One major advantage of this is that it allows for more complex boundary conditions.

In this talk I briefly discuss the derivation of the LBM from kinetic theory and describe its numerical ``stream and collide'' algorithm. I then show how to apply this method to biofluid flow problems, such as blood moving through an artery. Since blood is comprised of macroscopic particles such as red blood cells and platelets moving within plasma, it is a non-Newtonian fluid. This property makes the computational flow problem difficult to solve numerically.

November 14
Speaker: Dave Novom
Title: The Witt Ring
With a little bit of linear algebra we shall capture an overview of E. Witt's written paper of 1936, discussing the construction of the Witt Ring via the Grothendiek Ring. In a nutshell of unintelligible words: the Witt ring is a ring of classes of non-degenerate quadratic forms on finite-dimensional vector spaces over a field, modulo hyperbolic planes. From here, I would like to consider two cases: the Witt Ring over the real numbers and over finite fields.

November 21
Talk cancelled this week

November 28
Speaker: Sarah Kitchen
Title: Generalizing Resultants
The inspiration for the construction of the classical resultant is the following simple observation: A polynomial is completely determined by its coefficients, and the coefficients of a polynomial have a very special relationship with the roots of that polynomial. Consequently, if two polynomials have a common root, it is not unreasonable to expect that we should be able to detect this by some relationship among the coefficients of those polynomials. The resultant precisely describes this relationship. In this talk, we will define the classical resultant, explain the generalization to n+1 polynomials in n variables, as well as a formalization of the construction, and state a geometric application of the formalism.

December 5
Speaker: Mike Purcell
Title: An Examination of the Limitations of Classical Probability Theory
Classical probability theory was developed largely during the latter part of the 19th century. This theoretical framework provides many tools that allow us to analyze both discrete and absolutely continuous random variables. These tools, however, are insufficient for even the most rudimentary analysis of continuous random variables that are not absolutely continuous. In this talk I'll briefly review the "tools" under consideration, explore the nature of these limitations by way of a standard example, and introduce the alternative framework on which we build "Modern" probability theory. Some familiarity with undergraduate probability and/or basic measure theory will be helpful but not required as the talk is (should be!) self contained.

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