Notre Dame Math Graduate Student Seminar, 2010-2011

The Graduate Student Seminar is put on by the Mathematics Graduate Student Association . GSS meets approximately every other Monday.

All talks are at 4:15 in HH231 (Fall)/ HH229 (Spring) unless otherwise noted.

Previous Semesters

To volunteer to give a talk, or for anything else regarding the seminar, contact Megan Patnott.

Schedule

Abstracts

September 6, 2010

Speaker

Prof. Stephan Stolz

Title

The Index Theorem and K-theory

Abstract

The Atiyah-Singer Index Theorem is a celebrated result relating differential operators, topology and geometry. If D is a differential operator on a compact manifold, which is elliptic (a condition easy to check), then the kernel and cokernel of D are finite dimensional so that its index

index(D)=dim ker D - dim coker D

can be defined. The Atiyah-Singer Index Theorem gives a recipe for how to calculate index(D) in terms of algebraic topology, specifically in terms of a generalized cohomology theory called K-theory. Applied to a specific differential operator known as the Dirac operator, this can be used to show that some manifolds do not admit Riemannian metrics of positive scalar curvature.

September 20, 2010

Speaker

Jeff Madsen

Title

Elliptic Curves and Cryptography

Abstract

Elliptic curves are important objects of study in number theory. One of their most important properties is that the points on the curve may be made into a group. After defining the curve and the group law, I will show how elliptic curves can be applied to public key cryptography.

October 4, 2010

Speaker

Chris Porter

Title

Extracting Randomness from the Tosses of a Biased Coin

Abstract

Given a biased coin (one for which the probability of heads is strictly between 0 and 1/2), can we use it to simulate a fair coin? In this talk, I will consider von Neumann's simple yet elegant solution to this problem, as well as generalizations of this solution. I'll also look at generalizations of the problem, such as the case in which we try to simulate a fair coin using a sequence of biased coins. As I'll show, we can solve these generalized versions of the problem using techniques from computability theory and effective randomness. No background in these areas will be presupposed.

October 25, 2010

Speaker

Ryan Grady

Title

Curvature and Physics

Abstract

In a similar spirit to Prof. Stolz' talk, we discuss the existence of positive scalar curvature metrics on manifolds. We will then discuss the situation for Ricci curvature and the relationship with ideas coming from physics. All relevant notions will be introduced (or recalled) and the goal is to paint a picture of the landscape that lies on the boundary of physics and geometry/topology.

November 8, 2010

Speaker

Xiaoyang Chen

Title

Foliation structure on vector bundles

Abstract

In this talk I will give an introduction to foliation structure on vector bundles. Foliation of vector bundles reflects the twisting of vector bundles in some sense. I will discuss some obstructions to the existence of foliation on vector bundles. Some natural constructions will be also covered.

November 22, 2010

Speaker

David Karapetyan

Title

Triple Your Latexing Speed

Abstract

In this talk I will discuss a highly efficient workflow for editing latex documents. Specifically, I will be discussing the text editor Vim, and a variety of add-ons and customizations I have introduced.

More information, including installation materials and instructions for OS X users, can be found at http://davidkarapetyan.com/computing.php.

Though geared towards Mac users, much of the material in this talk is applicable to Linux and Windows users as well.

February 7, 2011

Speaker

Curtis Holliman

Title

Solitons

Abstract

Solitons, or self-reinforcing solitary waves, arise in the theory of partial differential equations and dynamical systems including several well-known instances such as in fiber optics, magnets, and DNA. We will begin with a basic introduction to the theory of such waves and more generalized traveling waves and conclude by presenting an application of this theory to new results regarding the Degasperis-Procesi equation.

March 21, 2011

Speaker

Sarah Cotter

Title

An Overview of the Surreal Numbers

Abstract

The surreal numbers, a John Conway creation, provide a way of starting from nothing and constructing the real numbers. Along the way, though, they grow to include less-standard numbers, too - infinite numbers and infinitesimals, as well as things like the square root of infinity. We'll look at two different constructions of the surreal numbers, examine some structural results, and see how several more familiar number systems fit into the surreals.

April 18, 2011

Speaker

Brandon Rowekamp

Title

The Geometry of Planar Pixelations

Abstract

A shape in the plane can be approximated by a set of square pixels. This pixelation is a poor approximation in that it does not capture important information about the set, such as length or Euler characteristic. However, given certain niceness conditions on the original set, it is possible to generate a piecewise linear set which approximates which recovers these invariants of the original set using only information from the pixelation.

May 2, 2011

Speaker

Steven VanDenDriessche

Title

Games, Determinacy, and Descriptive Set Theory

Abstract

We will discuss a formalization of two-person games of perfect information. Chess, Go, and checkers are a few examples of these games. Given the formalization, we see that these games are determined. Namely, one of the players must have a winning strategy! We will then see how this formalization naturally extends to infinite games, and discuss determinacy in this context. We will finish with an application of a determinacy result.

Previous Years

• 2009-2010
• 2008-2009
• 2007-2008
• 2006-2007
• 2005-2006
• 2004-2005
• 2003-2004
• 2002-2003