Columbia Undergraduate Math Society

Fall 2016« Spring 2017 Lectures »Summer 2017

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 d-bar 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 d-bar equation for a domain in the complex plane.
February 8
Willie
Dong
Spectral Theorem
for Compact
Self-Adjoint
Operators
In linear algebra, one is tasked with the problem of diagonalizing an n x n matrix, or a linear transformation from a finite-dimensional vector space to itself. Diagonalizing a linear transformation on an infinite-dimensional 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, self-adjoint operators on a Hilbert space. Familiarity with linear algebra and modern analysis is assumed.
February 15
Nathan
Dowlin
Invariants and
Smooth Structures
in Low-Dimensional
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 if and only if n = m.
March 1
Luis
Diogo
Symplectic
geometry and
Fibonacci
numbers
Since Gromov's groundbreaking non-squeezing 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 4-dimensional ellipsoids into 4-dimensional 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
 
 
March 15
 
No meeting
 
March 22
Gerhardt
Hinkle
 
 
March 29
 
 
 
April 5
Anton
Zeitlin
 
 
April 12
 
 
 
April 19
 
 
 
April 26
Remy Van
Dobben
de Bruyn
 
 
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