Wednesdays, 7:30pm; Room 407, Mathematics
~
ums [at] math.columbia.edu
Date  Speaker  Title  Abstract 
January 25 
Ivan
Corwin

Universal
phenomena
in random
systems

More than 200 years ago, the Gaussian distribution was discovered by De Moivre (and again by Laplace) from analyzing the binomial distribution arising from coin flips. This distribution is the basis for classical statistics and arises is many different types of physical and mathematical systems. There are, however, many systems for which classical statistics fails and new distributions arise. These distributions are also universal within certain classes of systems. In this talk we will begin to describe where these universal probability distributions come from, what they describe and why they are important. We will touch on many applications such as big data, random growth, traffic flow and more.

February 1 
Sebastien
Picard

Hormander's
L2 Esitmates

The technique of solving the dbar equation and using Hormander's L2 estimates has been used to obtain many interesting results in complex and algebraic geometry. In this talk, we will solve the one dimensional dbar equation for a domain in the complex plane.

February 8 
Willie
Dong

Spectral Theorem
for Compact
SelfAdjoint
Operators

In linear algebra, one is tasked with the problem of diagonalizing an n x n matrix, or a linear transformation from a finitedimensional vector space to itself. Diagonalizing a linear transformation on an infinitedimensional vector space is far more involved, and a spectral theorem is a result about when can a linear transformation be diagonalized. In this talk, I will state and prove the spectral theorem for compact, selfadjoint operators on a Hilbert space. Familiarity with linear algebra and modern analysis is assumed.

February 15 
Nathan
Dowlin

Invariants and
Smooth Structures
in LowDimensional
Topology

Most of the major open problems in topology involve understanding how many smooth manifolds there are of a given topological type. This problem is particularly difficult in dimension 4, the only dimension in which the smooth Poincare conjecture remains unsolved. I will discuss several of these problems, as well as the role of invariants in shedding light on them.

February 22 
Roxane
Sayde

Topological
dimension
theory

In the absence of coordinates, how does one define the dimension of a topological space? In this talk, we'll give a leisurely introduction to dimension theory, which attempts to answer this question. After motivating the field with several examples, we'll give three possible definitions of topological definition. We'll then use one of these to prove that R^n is homeomorphic to R^m iff n = m.

March 1 
Luis
Diogo

Symplectic
geometry and
Fibonacci
numbers

Since Gromov's groundbreaking nonsqueezing theorem, a central question in symplectic geometry is when can a subset of even dimensional Euclidean space be embedded into another symplectically. I will explain what this means and talk about work by McDuff and Schlenk, who showed a remarkable relation between embeddings of 4dimensional ellipsoids into 4dimensional balls and Fibonacci numbers. I will not assume any knowledge of symplectic geometry, but familiarity with Fibonacci numbers will be useful.

March 8 
Daniel
Gulotta

Tilings and
Groups

You may have seen this problem before: can a checkerboard with two opposite corners removed be tiled with dominoes? The answer is no, and one can prove that no tiling exists by counting the number of light and dark squares on the board. I will explain how some similar tiling problems can be solved by "counting" with nonabelian groups instead of the integers.

March 15 

No meeting 

March 22 
Irit Huq
Kuruvilla

The Combinatorial
Nullstellensatz

I will state and prove the Combinatorial Nullstellensatz, and show some of the very diverse areas in which it can be applied, including number theory, graph theory, as well as contest problems.

March 29 



April 5 
Anton
Zeitlin



April 12 



April 19 



April 26 
Remy Van
Dobben
de Bruyn


