Columbia Undergraduate Math Society

Fall 2017« Spring 2018 Lectures »Summer 2018

Wednesdays, 7:30pm; Room 507, Mathematics
Contact UMS (Email Adam Block)
Sign up for weekly emails

Date Speaker Title Abstract
January 24
The Weil Conjectures
In this talk, we start with a well-known example of counting points on Grassmannians over finite fields. This will provide us motivation for introducing the deep and far-reaching connections between non-singular complex varieties and their realizations over finite fields, known as the Weil Conjectures. These conjectures concern properties of zeta functions, and we explain how some of these properties follow easily from the existence of an appropriate cohomology theory.
January 31
Symplectic Manifolds
and Embeddings
I will introduce and motivate the basic concepts in symplectic geometry and explain why they are important (in physics, for example).  One important problem in symplectic geometry is determining when one symplectic manifold embeds symplectically into another.  I will summarize some methods and results used to address this question.
February 7
How to Use Finite Fields
for Problems Concerning
Infinite Fields
Following J.P. Serre’s paper of the same title as this talk, I will give an algebraic proof of the Ax-Grothendieck Theorem - which appears to be a theorem of complex analysis - using finite fields. I will discuss what makes this argument work, and if time permits, I will prove another result that appears in Serre’s paper.
February 14

Topological Quantum
Field Theory and
Gauge Theories

The study of QFTs has inspired many modern mathematical constructions and results. QFTs which are unchanged by diffeomorphism are called topological; we will play around with the structure of such QFTs in (1+1) dimensions and prove a baby version of the celebrated Verlinde formula. If time permits, we’ll define gauge theories and their quantizations, and apply the baby Verlinde formula to them to get some interesting group/representation theoretic identities.
February 21
Definitely Maybe -
Probability and
Statistics in
Number Theory
There has been a series of profound advancements in number theory in the 20th century, thanks to the understanding of the anatomy of integers and the fruitful interactions between statistics, probability theory, analysis and number theory. This will be a light survey talk on the heuristics and results in this direction, with emphasis on the distributions of prime divisors and prime gaps.
February 28 Alex
From Morse to Floer:
Topological Invariants
from Functions
From generic nice functions, we could get information on the topological invariant of the manifold. Andreas Floer generalized this idea to functionals on loop space of a closed symplectic manifold to prove Arnold's conjecture under some further conditions. This construction has far richer structure than just the homology of the manifold and actually encodes all of its symplectic info. I will show some connection with the talks before on symplectic manifold and TQFT if time allows.
March 7 Semon
Feynman Diagram
(Canceled Due to Snow)
There is no reason why Feynman diagrams couldn't be a taught in an advanced calculus class. I will discuss something actually useful: how to compute asymptotic series for certain exponential integrals. We will start with one variable, where the asymptotic series will be sums over graphs, and try to get to matrix integrals (the Feynman diagrams for which involve surfaces with boundary, i.e. `interacting strings'.)
March 14
No meeting
March 21
Hypergeometric Series
and Igusa's Formula 
(Canceled Due to Snow)
Consider a 2nd order ODE: z(1-z)f'' + (1-2z)f' - (1/4) f = 0 known as hypergeometric differential equation. In the first part of my talk, I will briefly discuss it's solution found by Euler and studied systematically by Gauss known as Gauss hypergeometric series. Then, in the second part of my talk, I will discuss some seemingly completely unrelated formula (Igusa's formula) about counting points of Elliptic curves over characteristic $p$ (of Legendre form). For the rest of the talk, I will try to tell the audience why and how these two things are related.
March 28
A Categorification of
a Knot Polynomial
We will describe Khovanov's groundbreaking "categorification" of the Jones polynomial of a knot. Khovanov homology is a topological invariant of knots and links with an easy combinatorial definition. It has been used to answer some longstanding questions in three-dimensional topology such as the Milnor Conjecture. The objective of this talk is to define Khovanov homology, compute it for some examples, and state some of its properties.
April 4


Alexandrov Geometry
and its Applications
In this talk, I will motivate and explain various aspects and techniques of Alexandrov Geometry as well as applications to geometric questions.
April 11
Why Pants Are
The goal of this talk is to outline the proof that the moduli space of Riemann surfaces of genus $g$ is $3g-3$ complex dimensional. Along the way, we hope to illustrate how the combinatorics of simple closed curves on a surface give a great deal of information about the geometry of the moduli space of Riemann surfaces of genus $g$. Our proof of the main result heavily relies on understanding the hyperbolic geometry of pairs of pants, that is, closed disks with two boundary components. We will first study the hyperbolic geometry of a pair of pants.  After this, we will be able to introduce the Teichm\"{u}ller space of a surface and its associated Fenchel--Nielsen coordinates, which will enable us to prove the main result.
April 18
From Cutting Squares
to Combinatorics
and 2-adics
While it is easy to dissect a square into any even number of triangles of equal area, it turns out to be impossible to do the same with an odd number of triangles. The only proof of this result to date, due to Monsky (1970), requires input from both combinatorial topology (!) and the 2-adic numbers (!!), which is surprising given that the problem is geometric in nature. In this talk, we will introduce all the necessary concepts and explain Monsky's proof in detail.
April 25 Alex
Brauer Groups The Brauer group is an object with diverse applications in algebraic geometry. In this talk, I will focus on the Brauer group of a field, which classifies finite-dimensional division algebras over the field.
May 2
No meeting
designed by Nilay Kumar, maintained by Rex Lei and George Drimba