VIGRE 2 Summer School Program

2008 Summer Graduate Student Seminar

Title: World's Shortest proof of Birkhoff's ergodic theorem
Speaker:  Justin Manning
Date: July 29, 2008
Time: 4:30-5:30 pm
Abstract:  We will discuss a short proof of Birkhoff's pointwise ergodictheorem on a probability space as developed by B. Bassler and A. Fieldsteel.
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Title: Trees With Unique Minimum Locating-Dominating Sets
Speaker: Steve Lane
Date: July 22, 2008
Time: 4:30-5:45* pm
Abstract:  In graph theory, given a graph G, a locating-dominating set is a subset of vertices L for which every vertex of the graph not in L has a nonempty set of neighbors in L that is different from such a set of neighbors in L for every other vertex not in L.  That is, every vertex not in L is both dominated and located.  We seek to characterize trees which have a unique minimum locating-dominating set.  This talk requires only knowledge of what a (simple) graph is.
* Lirong Yu will finish her talk from July 15.
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Date: July 15, 2008 4:30pm
Location: Boyd 304
*Speaker 1* (30 min): Lirong Yu
Title:  The Merton Portfolio Selection Problem
Abstract:  Suppose one wants to invest money to have capital gain. He can either play it safe and put it in the bank, or he can set up a money market acount for stocks(mutual fund, SP500 and such). The optimal plan is to pull money back to the bank when the stock market is not doing well, and to put it back when the stock market is producing more profit. The proportion is the issue. We are going to use Stochastic differential equation to model the stock price and use a continuous compounding ODE to describe the bank acount value.
*Speaker 2* (30 min): Ben Connell
Title: A discussion of character theory
Abstract: Character theory is an algebraic topic which relates to finite groups. This will be a self contained talk where I will give some basic definitions and concepts, and talk about how the theory relates to algebraic theorems more generally. I will also discuss classification of characters. The talk should be accessible to anyone who has taken 8000 Algebra.
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Title: Aggregation Model of Marine Particles by Moment Method of Multi Modes
Speaker: Yang Liu
Date: July 11, 2008, 4:30 pm
Location: Boyd 304
Abstract: I'll talk about the moments for log-normal distribution, its application on size distribution of marine particles, the multi-modes model, and some results from numerical experiments. This is a part of joint work with Adrian Burd.
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Speaker: Nathan Walters, UGA
Date: July 8, 2008, 4:30 pm
Room: Boyd 304
Abstract: The well-known Weierstrass approximation theorem states that a function f(x) can be approximated by polynomials on the closed interval [a,b] given only that f(x) is continuous.  In this talk I will discuss requirements for approximating functions with polynomials whose coefficients are integers; these requirements will be both on the function and the set on which the function is to be approximated.  This will lead naturally to the concept of the transfinite diameter of a set, and a basic application of this concept.  The talk should be accessible to anyone with a semester of undergraduate-level analysis.
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Title: "The Special Linear Group acting on the Projective Line" apres K.Conrad
Speaker: Jim Stankewicz Date: *July 1, 2008, 4:30 pm *
Room: Boyd 304
Abstract: "We look at the Special Linear Group acting on the Projective Line over a number field K by Moebius Transformations and a curious relation between the orbits of the action and the ideal class group of the ring of integers of K"
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Title: Construction of Ito Integral, properties and application
Speaker: Tin Kong, Jie Yu
Date: *June 27, 2008, 4:30 pm *
Room: Boyd 304
Abstract: Tin will give a 30min talk about construction of Ito- Integral which is different from Riemann integral in Calculus or Lebesgue Integral in Real Analysis. It is the fundamental of Stochastic Analysis. During the next 30 minutes I will talk about some interesting properties of Ito- Integral. Then the proof of Ito Isometry and some applications, examples will be provided.
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Title:  Some applications of linear systems of plane cubics.
Speaker: Maxim Arap, UGA
Date: June 24, 2008, 4:30 pm
Room: Boyd 304
Abstract: Abstract: This presentation is going to concern the applications of families of plane cubic curves. After proving that the dimension ofthe space of cubics passing through 8 general points in the plane has dimension 2, I will give applications. The first one is the classicaltheorem of Pascal about the "mystic hexagon" and the second is the associativity of the group law on the smooth plane cubic. If time permits, I'll state a more technical application, which is the fact that the projective plane blown up in 6 general points is isomorphic to a smooth cubic surface in projective 3-space.
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Title: Weierstrass Data of Minimal Surfaces
Speaker: Whitney Montgomery, UGA
Date: June 17, 2008, 4:30 pm
Room: Boyd 304
Abstract:  Given a minimal immersion, /X/, we will investigate the corresponding "conjugate" minimal immersion, /X*/, and how these relate to complex analysis. We will then use this to understand how the Gauss map, /g/, and height differential, /dh/, represent the minimal  immersion for /X/.  Together, (/g, dh/) is called the Weierstrass Data for/ X./
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Title: Introduction to Minimal Surfaces
Speaker: Matt Mastin, UGA
Date*: June 13, 2008, 4:30 pm
Room: Boyd 304
Abstract: We will begin by discussing the various equivalent definitions of minimal surfaces.  There are several famous minimal surfaces that will be presented, and as time allows we will prove a few nice results including the strong half space theorem.

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