The Graduate Student Seminar is put on by the Mathematics Graduate Student Association . GSS meets approximately every other Monday.
For the Fall 2014 semester, the GSS meets at 4:00p in 125 HH.
To volunteer to give a talk, or for anything else regarding the seminar, contact Dominic Culver.
|Monday, September 8||Katrina Barron||An Introduction to the Algebraic Aspects of Conformal Field Theory and Applications|
|Monday, September 22||Augusto Stoffel||2-Dimensional Topological Field Theories|
|Monday, October 6||Edward Burkard||A Primer on Pseudoholomorpic Curves|
|Monday, November 3||Leandro Lichtenfelz||Singularity Theorems in General Relativity|
|Monday, November 17||Weijia Wang||A Finite State Automaton for Coxeter Groups|
|Monday, December 1||Mike Haskel||Git: LaTeX's Best Friend|
|Monday, December 15||Philip Jedlovec||Arrow's Impossibility Theorem|
|Monday, April 13, 2015||Eric Wawerczyk||Fourier Analysis and Zeta Functions|
Conformal Field Theory (CFT) is an attempt to unify all fundamental forces, including gravity, by modeling particles as vibrating strings. In the genus-zero, two-dimensional setting, vertex operator algebras (VOAs) describe the particle interactions. Independently from physics, VOAs were discovered in mathematics in the study of representations of infinite-dimensional Lie algebras and the Monster finite simple group. This study led to the surprising connection between CFT, VOAs and number theory. I will briefly give a flavor of some of the mathematics involved.
We will show that the notion of a topological field theory, in one of its simplest variations (namely, 2-dimensional and oriented), recovers a classical notion from algebra, that of a Frobenius algebra. This fact is a well-known “folk theorem” but, as we will see, it has a nontrivial proof relying on Morse theory.
I will begin with an overview of almost complex manifolds and complex manifolds. Next, I will define pseudoholomorphic curves (or (J-)holomorpic curves) and I will try to convince you that these are just the usual notion of a holomorphic function in a special case. After that, I will give some basic definitions in symplectic geometry and relations with almost complex structures. Finally, I will state some results in symplectic geometry which are applications of holomorphic curves, one of which can be thought of as a generalization of the Riemann Mapping Theorem.
I will begin with a brief introduction to the language of General Relativity, including some examples of spacetimes. After that, I will move on to discuss causality on such spacetimes, and finally try to give a general idea of how Hawking’s and Penrose’s Singularity Theorems work.
The problems of determining when two words represent the same element in the group and finding a finite state automaton that recognizes the language of reduced words are fundamental ones in the combinatorial group theory. Using the standard geometric representation and the theory of root system, I will show that both problems can be solved for Coxeter groups. In particular I will show how to construct such an automaton using a partial order structure on the set of roots.
Writing up a large project is complex business. Multiple reorganizations can either lead to a mess of files with names like "thesis_rewrite2_september_with_changes.tex", or to being afraid to make big structural changes for fear of losing work. And if you fix a typo in one version, it takes a lot of focus to always remember to fix it in every version you might use. This whole situation is much worse if you have to deal with multiple authors, or keep track of one version for submission to journals and another as a dissertation. Fortunately, there's a better way! Git is free software written by Linus Torvalds, the creator of Linux, to solve the very similar problems faced by a complicated software project with many contributors. I'll show you how Git works and how to use it on your projects.
Social choice theory, an area of study closely tied to fields as diverse as mathematics, economics, political theory, and philosophy, seeks to aggregate individual preferences into overall societal preferences in order to aid the collective decision-making process within a society. Perhaps the most famous and influential theorem in modern social choice theory is Arrow's Impossibility Theorem, or Arrow's Paradox, which states that no social choice function which combines individual rank-order preferences into a societal rank-order preference can simultaneously satisfy the following three very minimal fairness and consistency criteria: the Weak Pareto Criterion, Independence of Irrelevant Alternatives, and Nondictatorship. In this talk, I will discuss the mathematics underlying this famous theorem, outline some of its economic and philosophical implications, and finally give a sketch of its proof.
We will be discussing the contents of the 1950 dissertation by John Tate entitled "Fourier analysis in number fields, and Hecke's zeta-functions". In this cult-classic paper Tate develops Fourier Analysis on the Real, Complex, and p-adic numbers simultaneously in the "local theory" to define local zeta functions. By carrying over the classical results like the Fourier Inversion Theorem and the Poisson Summation formula, one can derive a functional equation and analytic continuation for these local zeta functions. Afterwards the Adéles are introduced, a restricted direct product of all of the completions of the rationals, and the Fourier Analysis on this space of Adéles leads to a generalized proof of the functional equation and analytic continuation of the Riemann Zeta function as well as the Hecke zeta functions.