Spring 2010 abstracts

Spring 2010 schedule

Date Speaker Title

January 26
Leland Schick
Rods and Balls: Liquid Crystals of Different Lengths

Rodlike polymers are frequently encountered in nature, and understanding their physical properties is key to understanding their biological
functions. This talk will examine a free energy functional for liquid crystals of different lengths, and some consequences of that functional.
A basic introduction to statistical mechanics will also be included.

February 2
Brendan Fry
Effects of Blood Flow Distribution on Oxygen Delivery in a Heterogeneous Microvascular Network

Due to its relatively low solubility in tissue, oxygen can only diffuse a short distance into oxygen-consuming tissue. Thus, oxygen levels in
the tissue are critically dependent on the arrangement of surrounding blood vessels. Irregular vascular geometries are not well-described by
simple models in which oxygen levels for an entire network are determined solely by oxygen transport in a small number of representative vessels.

Here, we use a model that can describe oxygen transport in a heterogenous blood vessel network to test what effect distribution of blood flow
has on tissue oxygenation. We will see that changing the distribution of flow can have a significant effect, which will demonstrate the importance
of local blood flow regulation. All necessary biology for this talk with be explained along the way.

February 9
Enrique Acosta

The 27 Lines on a Cubic Surface

Every smooth cubic surface in CP^3 contains exactly 27 lines. I will explain what these have to do with blowing up CP^2 at 6 sufficiently
general points, and will elaborate on the relation between the coordinates of the points and the equation of the cubic surface. Surprisingly, this
will involve an elliptic curve over a function field, and the minimal roots of a lattice.

February 16
2009 and 2010 GTEAMS Cohorts
GTEAMS and You

The G-TEAMS program provides an innovative and dynamic opportunity for graduate students and teacher partners to collaborate on the
development of novel, rigorous and relevant material for K-12 mathematics courses. Applications are now being accepted for the 2011-2012
academic year. Come hear more information about the program, the application process, and have your questions answered.

February 23
No speaker

March 2
John Gemmer
Introduction to non-Euclidean plates

The wave-like patterns observed in leaves, flowers, mushrooms, sea slugs, lichen, and even dresses is caused by simple lateral differential growth
of the tissue/cloth. A recent model of this phenomenon, based on the classic theory of large deflection in elastic plates, proposes that the equilibrium
configuration taken is one that minimizes a Föppl - von Kąrmąn type energy functional with the stretching energy measured by deviations away from
a fixed target Riemannian metric. In this talk we present a study of this functional in the small thickness regime when the Gaussian curvature is constant
and negative.

March 9
Scott Hottovy
What equation does a Brownian particle obey?

Using Newton's second law, a small particle in fluid is modeled by a second-order stochastic differential equation. For negligible mass, the
Smoluchowski-Kramers approximation is valid but may change depending on how the limit of the mass goes to zero. The limit is studied in
applications to homogenization and large deviations. We note that the fluctuation-dissipation theorem in thermodynamics complicates the limit.
However, the Smoluchowski-Kramers approximation can be derived using PDE theory to find the equation the Brownian particle obeys.

March 16
SPRING BREAK!!!

March 23
Michael Bishop
Weyl's Formula

Whether one can hear the shape of a drum is one of the more popular problems in modern mathematics. It was inspired by Mark Kac in his
famous 1966 paper where he pondered if the shape of a domain could be determined by the eigenvalues of the Dirichlet Lapacian. While it has
been disproven in many cases, there are many related problems still being investigated from various points of view. A central theme to all
such work is the formula of Weyl, which relates the number of eigenvalues of the Dirichlet Laplacian on a bounded domain to its volume. In
this talk, I will give an introduction to unbounded operators and Dirichlet-Neumann bracketing to setup a sketch of a proof of Weyl's formula.

March 30
Yuliya Gorlina
Weighted Delaunay Triangulations

In this talk, I will summarize my dissertation research, including some of my results. I will start with an accessible introduction of geometric
triangulations. Please come and give me feedback to help me write the talk for my defense.

April 6
Sarah Mann
Computability: Computer Science for the Mathematician

In this talk, I will discuss computability, the Church-Turing thesis, Turing machines, and other methods of computation such as tag systems,
glider systems, and cellular automata. This talk will be accessible to everyone; no background in computer science will be assumed.

April 13
Victor Piercey
Stock Options

A stock option is a contract to purchase or sell shares of stock for a fixed price K at some time N in the future. How much is the option worth
at time t=0? One way to price an option is through the celebrated Black-Scholes formula. I will present the discrete-time analogue of
Black-Scholes. Applications include how to simultaneously get unimaginably rich while everyone around you is put in the poor house, kind
of like the movie Trading Places.

April 20
Mandi Schaeffer Fry
Maximal Subgroups of the Finite Classical Groups and Representations

A theorem by Aschbacher provides a link between studying maximal subgroups of the finite classical groups and studying the behavior under
restriction of irreducible representations of its subgroups. The goal of my talk (aside from eating bagels) is to explain this statement and give
some insight into my current research on the subject.

April 27
Matt Johnson
Hecke Eigenforms and Products of Eigenforms

Certain modular forms have a Fourier expansion with very nice properties. These forms, call Hecke eigenforms, have multiplicative Fourier
coefficients. Occasionally the product of two eigenforms is again an eigenform. We will discuss this phenomenon and how often it occurs.
For the full group SL_2(Z) there are 16 (classically known) product identities, all forced by dimensional considerations. For modular forms
on Gamma_1(N) with certain weights we prove that there are only a finite number of such product identities. The Valenzetti equation will be
discussed if time permits.

May 4
Alex Tao
Galois Cohomology and Mordell-Weil Theorem

Given a set of rational points on an elliptic curve, one may employ the classical tangent-chord process to produce yet more rational points.
Question: How many rational points must one begin with in order to produce all of the rational points on a given elliptic curve? The Mordell-Weil
Theorem provides a partial answer:

$E(K)$ is finitely generated for any number field $K$.

The proof of the theorem involves two main parts: Descent Theorem on Heights and the Weak Mordell-Weil Theorem. In this talk, we will
concentrate on the latter, which states that $E(K)/mE(K)$ is finitely generated. In particular, I will introduce some basic group cohomology
and give an outline of the proof via Galois cohomology.

Date	Speaker	Title
January 26	Leland Schick	Rods and Balls: Liquid Crystals of Different Lengths
Rodlike polymers are frequently encountered in nature, and understanding their physical properties is key to understanding their biological functions. This talk will examine a free energy functional for liquid crystals of different lengths, and some consequences of that functional. A basic introduction to statistical mechanics will also be included.
February 2	Brendan Fry	Effects of Blood Flow Distribution on Oxygen Delivery in a Heterogeneous Microvascular Network
Due to its relatively low solubility in tissue, oxygen can only diffuse a short distance into oxygen-consuming tissue. Thus, oxygen levels in the tissue are critically dependent on the arrangement of surrounding blood vessels. Irregular vascular geometries are not well-described by simple models in which oxygen levels for an entire network are determined solely by oxygen transport in a small number of representative vessels. Here, we use a model that can describe oxygen transport in a heterogenous blood vessel network to test what effect distribution of blood flow has on tissue oxygenation. We will see that changing the distribution of flow can have a significant effect, which will demonstrate the importance of local blood flow regulation. All necessary biology for this talk with be explained along the way.
February 9	Enrique Acosta	The 27 Lines on a Cubic Surface
Every smooth cubic surface in CP^3 contains exactly 27 lines. I will explain what these have to do with blowing up CP^2 at 6 sufficiently general points, and will elaborate on the relation between the coordinates of the points and the equation of the cubic surface. Surprisingly, this will involve an elliptic curve over a function field, and the minimal roots of a lattice.
February 16	2009 and 2010 GTEAMS Cohorts	GTEAMS and You
The G-TEAMS program provides an innovative and dynamic opportunity for graduate students and teacher partners to collaborate on the development of novel, rigorous and relevant material for K-12 mathematics courses. Applications are now being accepted for the 2011-2012 academic year. Come hear more information about the program, the application process, and have your questions answered.
February 23	No speaker

March 2	John Gemmer	Introduction to non-Euclidean plates
The wave-like patterns observed in leaves, flowers, mushrooms, sea slugs, lichen, and even dresses is caused by simple lateral differential growth of the tissue/cloth. A recent model of this phenomenon, based on the classic theory of large deflection in elastic plates, proposes that the equilibrium configuration taken is one that minimizes a Föppl - von Kąrmąn type energy functional with the stretching energy measured by deviations away from a fixed target Riemannian metric. In this talk we present a study of this functional in the small thickness regime when the Gaussian curvature is constant and negative.
March 9	Scott Hottovy	What equation does a Brownian particle obey?
Using Newton's second law, a small particle in fluid is modeled by a second-order stochastic differential equation. For negligible mass, the Smoluchowski-Kramers approximation is valid but may change depending on how the limit of the mass goes to zero. The limit is studied in applications to homogenization and large deviations. We note that the fluctuation-dissipation theorem in thermodynamics complicates the limit. However, the Smoluchowski-Kramers approximation can be derived using PDE theory to find the equation the Brownian particle obeys.
March 16	SPRING BREAK!!!

March 23	Michael Bishop	Weyl's Formula
Whether one can hear the shape of a drum is one of the more popular problems in modern mathematics. It was inspired by Mark Kac in his famous 1966 paper where he pondered if the shape of a domain could be determined by the eigenvalues of the Dirichlet Lapacian. While it has been disproven in many cases, there are many related problems still being investigated from various points of view. A central theme to all such work is the formula of Weyl, which relates the number of eigenvalues of the Dirichlet Laplacian on a bounded domain to its volume. In this talk, I will give an introduction to unbounded operators and Dirichlet-Neumann bracketing to setup a sketch of a proof of Weyl's formula.
March 30	Yuliya Gorlina	Weighted Delaunay Triangulations
In this talk, I will summarize my dissertation research, including some of my results. I will start with an accessible introduction of geometric triangulations. Please come and give me feedback to help me write the talk for my defense.
April 6	Sarah Mann	Computability: Computer Science for the Mathematician
In this talk, I will discuss computability, the Church-Turing thesis, Turing machines, and other methods of computation such as tag systems, glider systems, and cellular automata. This talk will be accessible to everyone; no background in computer science will be assumed.
April 13	Victor Piercey	Stock Options
A stock option is a contract to purchase or sell shares of stock for a fixed price K at some time N in the future. How much is the option worth at time t=0? One way to price an option is through the celebrated Black-Scholes formula. I will present the discrete-time analogue of Black-Scholes. Applications include how to simultaneously get unimaginably rich while everyone around you is put in the poor house, kind of like the movie Trading Places.
April 20	Mandi Schaeffer Fry	Maximal Subgroups of the Finite Classical Groups and Representations
A theorem by Aschbacher provides a link between studying maximal subgroups of the finite classical groups and studying the behavior under restriction of irreducible representations of its subgroups. The goal of my talk (aside from eating bagels) is to explain this statement and give some insight into my current research on the subject.
April 27	Matt Johnson	Hecke Eigenforms and Products of Eigenforms
Certain modular forms have a Fourier expansion with very nice properties. These forms, call Hecke eigenforms, have multiplicative Fourier coefficients. Occasionally the product of two eigenforms is again an eigenform. We will discuss this phenomenon and how often it occurs. For the full group SL_2(Z) there are 16 (classically known) product identities, all forced by dimensional considerations. For modular forms on Gamma_1(N) with certain weights we prove that there are only a finite number of such product identities. The Valenzetti equation will be discussed if time permits.
May 4	Alex Tao	Galois Cohomology and Mordell-Weil Theorem
Given a set of rational points on an elliptic curve, one may employ the classical tangent-chord process to produce yet more rational points. Question: How many rational points must one begin with in order to produce all of the rational points on a given elliptic curve? The Mordell-Weil Theorem provides a partial answer: $E(K)$ is finitely generated for any number field $K$. The proof of the theorem involves two main parts: Descent Theorem on Heights and the Weak Mordell-Weil Theorem. In this talk, we will concentrate on the latter, which states that $E(K)/mE(K)$ is finitely generated. In particular, I will introduce some basic group cohomology and give an outline of the proof via Galois cohomology.

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