We usually meet on Wednesday in 508 Math at 7:00 PM and the talk takes place at 7:30 PM in 507 Math.
September 11 *Wednesday* Vivek Pal (Columbia University): Algebraic geometry in number theory.
Abstract: Algebraic geometry plays an important role in many fields, in particular it is really useful in number theory. In this talk we will explore some of
the ways algebraic geometry is used and how number theoretic properties can be expressed in terms of geometry.
September 18 *Wednesday* Peter Woit (Columbia University): Quantum mechanics and representation theory.
Abstract: The subjects of quantum mechanics and representation theory have been closely linked since their beginnings. I'll explain what a
"unitary representation of a Lie group or Lie algebra" is, give some examples, and show how these appear both in the fundamental formalism
of quantum mechanics and in some specific applications.
September 25 *Wednesday* Michael Harris (Institut de Mathématiques de Jussieu): Several ways to think about the Langlands program.
Abstract: The Langlands program is, among many other things, a way to understand solutions to polynomial equations with rational
coefficients in terms of the non-commutative version of Fourier analysis as well as a way to explain non-commutative Fourier analysis in
terms of solutions to polynomial equations. I will mainly take the latter approach, starting with the Galois theory of Q and showing how the
solution to the most extensively studied problem in Diophantine equations, the theory of cubic equations in two variables, is intimately tied to
the first non-trivial example of non-commutative Fourier analysis. Yet another way of thinking about the Langlands program is as a case study
in the attempt to explain disparate mathematical observations by a unified general program.
October 2 *Wednesday* Clément Hongler (Columbia University): Semi-circular law for random matrices.
October 9 *Wednesday* Joe Harris (Harvard University): Poncelet's theorem.
Abstract: Poncelet's theorem is an answer to the question, "Given two conic curves in the plane, when is there a polygon inscribed in the first and
circumscribed around the second?" Originally proved with some difficulty in the early 19th century, it turns out to be relatively transparent from the point of
view of algebraic geometry as it developed over the next century, illustrating some of the value of those developments. The talk is recorded by Sicong
Zhang: https://www.youtube.com/watch?
October 16 *Wednesday* Sicong Zhang (Columbia University): The game of Hex and Brouwer's fixed-point theorem - David Gale's proof.
Abstract: Hex is a strategy game where two players take turns to place stones on a hexagonal grid in order to connect opposite sides of the game
October 23 *Wednesday* No talk planned.
October 30 *Wednesday* (Department open house) John Yu (Columbia University): Finite dimensional algebras and quiver representations.
Abstract: I will talk about finite dimensional algebras, their relation to quiver representations, and some of the homological algebra behind these ideas. I will in particular talk about the case where the algebra in question has global dimension 1. By the end of the talk, I will hopefully have introduced a beautiful theorem due to Gabriel and have given a sketch of the proof.
November 6 *Wednesday* Anna Puskas (Columbia University): Burnside's theorem.
Abstract: Burnside's Theorem states that if the order of a finite group has only two different prime divisors, then the group is solvable. The original proof is an amusing application of results from the representation theory of finite groups. We shall discuss Maschke's theorem, the Schur orthogonality relations and some properties of algebraic integers, and then present a proof of Burnside's Theorem. Prerequisites will be kept to a minimum; some familiarity with modules might be useful.
December 4 *Wednesday* Nava Balsam (Columbia University): Elliptic curves in number theory.
board. Brouwer's fixed point theorem is a classical topology theorem which states that any continuous function from a topological closed disk to itself
always leave a point fixed. In 1979 David Gale gave a proof of the following: the fact that the game of Hex never ends in a draw is equivalent to Brouwer's
fixed-point theorem. In this talk I will present Gale's proof, emphasizing the beautiful interplay between combinatorics, analysis and topology.