Columbia Undergraduate Math Society

Fall 2017« Spring 2018 Lectures »Summer 2018

Wednesdays, 7:30pm; Room 507, Mathematics
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Date Speaker Title Abstract
January 24
Stanislav
Atanasov
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
Theo
Coyne
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
Noah
Olander
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
Henry
Liu

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
Kevin
Kwan
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
Zhang
 
 
March 7 Semon
Rezchikov
   
March 14
 
No meeting
 
March 21
Shizhang
Li
Igusa Formula and
Hypergeometric Series
 
March 28
Linh
Truong
 
 
April 4
Pak-Hin
Lee
 
 
April 11
Alex
Pieloch
 
 
April 18
George
Drimba 
 
 
April 25 Alex
Perry
   
May 2
 
No meeting
 
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