We usually meet on Wednesday in 508 Math at 7:00 PM and the talk takes place at 7:30 PM in 507 Math.
May 8 *Wednesday*
Summer Initial Meeting.
We are still looking for potential topics. Please email ums@columbia.edu if you have any suggestion (either for a topic, or a "host" who will be responsible for planning talks on that topic over the summer). So far we have
1) Comparison Theorems between Complex Differential and Algebraic Geometry. (Host: Yifei Zhao)
2) Modular Forms and Algebraic Number Theory. (Host: Joseph Eddy)
3) Commutative Algebra. (Host to be found)
May 1 *Wednesday*
Jonathan Hanselman (Columbia Grad Student)
Topic and abstract: to come!
April 24 *Wednesday*
Corrin Clarkson (Columbia Grad Student): Strange Properties of the Curve Complex.
The curve complex is a geometric space that encodes relationships between curves on a surface. It is relatively easy to define, but the resulting geometry is highly pathological. We will discuss the definition of this complex, some of its strange properties and if time permits some of the ways it is used in the study of low dimensional topology.
April 17 *Wednesday*
Tristan Collins (Columbia Grad Student): The Newton Polygon in Analysis and Algebraic Geometry.
Let F(x,y) be a bivariate polynomial. In 1676 Newton observed that, by associating to F(x,y) a certain convex body called the Newton Polygon, he could recursively construct solutions to the equation F(x,y)=0. It turns out that the Newton Polygon encodes a great deal of algebraic and analytic data about the function F. In this talk I will discuss the relationship between the Newton Polygon and the so called log canonical threshold of F, which is an important quantity in birational geometry. The relationship is obtained by finding sharp estimates for a class of singular integrals. This talk will only require some knowledge of multivariable calculus.
April 10 *Wednesday*
Karsten Gimre (Columbia Grad Student): The Poincaré Conjecture
The Poincaré conjecture states that a simply-connected compact manifold is diffeomorphic to a sphere. In the 1960s the conjecture was proved in dimension greater than four, and in the 1980s a slightly weaker version was proved in dimension four. In the 1980s and 1990s Richard Hamilton developed a framework for proving the conjecture in dimension three via the Ricci flow, which is a means of evolving a Riemannian metric in time. In 2002 and 2003 Grigori Perelman resolved all of the loose ends in Hamilton's program, proving the conjecture. We will outline Hamilton's program, as well as some of Perelman's resolutions of its difficulties. The talk is designed to be understood by those with a small amount of familiarity with smooth manifolds.
April 3 *Wednesday*
What is Mathematics?
This week we will have a special meeting (suggested by Nava Balsam) on "What is Mathematics? 5 Columbia graduate students will be our guest speakers, and each of them will discuss one major area of mathematics. The topics and speakers are:
Rahul Krishna: What is Number Theory?
Andrew Kiluk: What is Algebraic Geometry?
Stephane Benoist: What is Probability?
Karsten Gimre: What is Differential Geometry?
Rob Castellano: What is Topology?
March 27 *Wednesday*
Connor Mooney (Columbia Grad Student): Regularity Theory for Elliptic PDE.
You may know that the first partial derivatives of the components of a holomorphic function satisfy a seemingly unremarkable relationship, the Cauchy-Riemann equations. However, these simple equations tell us something very remarkable: the functions are smooth, and in fact real-analytic (their power series converge)! This phenomenon holds for a large class of partial differential equations known as "elliptic PDE," and the study of the local geometry and smoothness of solutions is called "regularity theory." In this talk I will motivate the main ideas of regularity theory by discussing two extremely important examples: Laplace's equation and the Monge-Ampère equation.
March 13 *Wednesday*
Mu-Tao Wang (Columbia Professor): The Role Mathematics Plays in General Relativity.
I would like to discuss the role mathematics plays in the historical development and frontier research of general relativity.
March 6 *Wednesday*
Department Open House.
All Columbia and Barnard prospective and current mathematics majors, joint majors, and concentrators are invited to meet faculty and other students who can answer questions about the Mathematics Department, the courses it offers and the major.
February 20 *Wednesday*
Sicong Zhang (Columbia Undergrad): Cluster Algebras.
Cluster algebras are a class of constructively defined commutative rings with a set of distinguished generators (cluster variables) grouped into overlapping subsets (clusters) of a fixed size (the rank). Since their introduction in 2002 in the context of Lie theory, cluster algebras have provided a unifying structure for phenomena in a variety of algebraic and geometric contexts. In this talk I will construct cluster algebras in terms of two elementary but perhaps puzzling operations (a combinatorial "quiver mutation" and an algebraic "exchange relation"), and give motivating examples such as Somos sequences, triangulation of regular polygons and coordinate rings of Grassmannians. If time permits, I will discuss some relations between cluster variables and perfect matchings of graphs. No prior knowledge is required, although familiarity with ring theory would be helpful.