Columbia Undergraduate Math Society

Summer 2017« Fall 2017 Lectures »Spring 2018

Wednesdays, 7:30pm; Room 507, Mathematics
Contact UMS
Sign up for weekly emails

Date Speaker Title Abstract
September 13
Torelli Theorems
in Dimension One
Torelli Theorems, roughly speaking, classify geometric objects via linear algebraic invariants. In this talk, I will motivate and explain various Torelli Theorems for curves.
September 20
A Brief History of
Cyclotomic Fields
In this talk, I will develop chronologically the theory of cyclotomic fields, motivated by Fermat's Last Theorem and other interesting problems.
September 27
Rationality of
It was known to the ancients that the solutions to the equation $x^2+y^2=1$ can completely described by rational functions in a single parameter $t$: $x=2t/(t^2+1)$ and $y=(t^2-1)/(t^2+1)$. With some more theory, one can show that for equations of degree 3 or higher in two variables, such a parametrization usually doesn't exist. However, generalizations of this problem quickly become difficult: an important open problem in algebraic geometry asks whether the solutions of a cubic equation in 5 variables may be described using 4 parameters. In modern language, is a cubic hyperspace in 5-dimensional affine (or projective) space a rational variety? I will discuss examples of variants of these questions that are tractable, and indicate some areas of current research.
October 4
In this talk I will trace a series of classical theorems in Diophantine approximations, culminating in the celebrated Schmidt subspace theorem. Along the way, I will prove some finiteness results on the number of solutions of certain Diophantine equations as well as explore interesting distribution properties of the digits of algebraic numbers. The talk will end with a list of seemingly unrelated problems for which Schmidt subspace theorem has been the main tool for recent advances.
October 11
Geometry on
The classification of geometric structures has been used to understand the analytic properties of evolving flows on manifolds. In this talk, I will motivate and explain how one can begin to approach such a classification.
October 18
Ten Cool Deep
Learning Papers
in 2017
I'll give a summary of ten cool deep learning papers from 2017.
October 25
Quadratic Polynomials
and Modular Forms
We will describe a surprising phenomenon discovered by Zagier on values of quadratic polynomials. We will then provide an explanation using modular forms.
November 1
An Introduction
to [G_2] Manifolds
I will give an introduction to $G_2$ manifolds, setting up the relevant definitions, discussing their properties, and time-permitting outlining some of the classical constructions of these objects.
November 8
How to Count:
Your Daily Dose of
Intersection Theory
Given two curves in a plane, how many times do they intersect? How many tangents to a given curve pass through a given point? Given a set of $r$ polynomials in $r$ variables, how many points are fixed? I will introduce some basic algebraic geometry and intersection theory and answer all of these questions. 
November 15
Slice Knots and
Concordance Group
The central question of the manifold topology is "classification". Surprisingly, in some sense, this question is harder in dimension 4 compared to higher dimensions. Closely related to the classification question in dimension 4 is the study of knots in 3-manifolds. In this talk, we will discuss slice knots, concordance group and the slice-ribbon conjecture, one of the most important conjectures in low-dimensional topology. We will also discuss the role of invariants to study them.
November 22
No meeting
November 29
December 6
December 13
No meeting
designed by Nilay Kumar, maintained by Rex Lei and George Drimba