Summer 2016« Fall 2016 Lectures »Spring 2017

Date	Speaker	Title	Abstract
October 5	Raymond Cheng	Grothendieck Ring of Varieties	Since the time of Euler, we have known that the alternating sum of the number of vertices, edges and faces of a polytope is, somewhat remarkably, always 2. In this talk, I will discuss invariants of geometric objects called "motivic invariants". As a main example, I discuss motivic invariants in the context of algebraic geometry where the invariants will take values in the so-called Grothendieck ring of varieties, or more affectionately, the ring of "baby motives". Despite being a rather mysterious object, we will see that this ring will allow us to make many interesting calculations and find amazing ways to encode facts we already know.
October 12	Sam Mundy	Bernoulli Numbers and Arithmetic	I will discuss the role Bernoulli numbers play in modern Number Theory.
October 19	Shizhang Li	Poncelet's Closure Theorem	Let C and D be two plane conics intersecting transversely. For a point c on C, consider the following procedure: find a tangent line of D passing through c, which would cut C at another point, say, c'. Now consider the chain of points on C: c, c', c'',.... The theorem says: if you could find one point c on C such that after n times of procedure described above it coincides with the original point c, then for any point on C same thing would happen. I will introduce a little bit about elliptic curves and prove this (fantastic) theorem.
October 26	George Drimba	Eigenvalue Problems in Geometry	In this talk, we will explore Elliptic PDE theory and discuss tools from geometric analysis in the context of Eigenvalue problems.
November 2	Michael Thaddeus	Schubert Calculus	If four skew lines in three-dimensional space are chosen at random, how many lines pass through all four of them? The answer is two. Schubert calculus is a collection of theorems in enumerative geometry for answering such questions. To describe it, we will introduce and study the Grassmannian parametrizing all k-dimensional subspaces of a fixed n-dimensional space.
November 9		No meeting
November 16	Lauren Williams	Tableaux Combinatorics and Hopping Particles	The asymmetric exclusion process is a model of particles hopping on a 1d lattice. It has been cited as a model for traffic flow and for translation in protein synthesis. I'll explain how to use certain combinatorial tableaux to understand the probabilities in the ASEP. Note: The Fall Math Open House will take place before the talk.
November 23		No meeting - Thanksgiving Break
November 30	Mitchell Faulk	Vector Operator Algebras	Vertex operator algebras are algebraic objects where the multiplication structure consists of a Z-grading of multiplications. From their inception, they played an interesting role in pure math, exhibiting connections to the Monster group and modular functions. Later, it was shown by Huang that these algebraic structures arise precisely as the algebraic structures modeling interactions of strings in conformal field theory. In this talk, we outline this brief history of these interesting algebraic structures.
December 7	Adam Block	Fundamental Theorem of Algebra	As many proofs of the Fundamental Theorem of Algebra I can fit into one hour.
December 14	Daniel Litt	Graph Isomorphism and Representation Theory	Let g_{n,k} be the number of isomorphism classes of graphs with n vertices and k edges. I'll explain how to use the representation theory of the Lie algebra sl_2 to show that the sequence g_{n,0}, g_{n,1}, g_{n, 2}, ... is unimodal.
December 21		No meeting

Date

Speaker

Title

Abstract

October 5

Raymond Cheng

Grothendieck Ring of Varieties

Since the time of Euler, we have known that the alternating sum of the number of vertices, edges and faces of a polytope is, somewhat remarkably, always 2. In this talk, I will discuss invariants of geometric objects called "motivic invariants". As a main example, I discuss motivic invariants in the context of algebraic geometry where the invariants will take values in the so-called Grothendieck ring of varieties, or more affectionately, the ring of "baby motives". Despite being a rather mysterious object, we will see that this ring will allow us to make many interesting calculations and find amazing ways to encode facts we already know.

October 12

Sam Mundy

Bernoulli Numbers and Arithmetic

I will discuss the role Bernoulli numbers play in modern Number Theory.

October 19

Shizhang Li

Poncelet's Closure Theorem

Let C and D be two plane conics intersecting transversely. For a point c on C, consider the following procedure: find a tangent line of D passing through c, which would cut C at another point, say, c'. Now consider the chain of points on C: c, c', c'',.... The theorem says: if you could find one point c on C such that after n times of procedure described above it coincides with the original point c, then for any point on C same thing would happen. I will introduce a little bit about elliptic curves and prove this (fantastic) theorem.

October 26

George Drimba

Eigenvalue Problems in Geometry

In this talk, we will explore Elliptic PDE theory and discuss tools from geometric analysis in the context of Eigenvalue problems.

November 2

Michael Thaddeus

Schubert Calculus

If four skew lines in three-dimensional space are chosen at random, how many lines pass through all four of them? The answer is two. Schubert calculus is a collection of theorems in enumerative geometry for answering such questions. To describe it, we will introduce and study the Grassmannian parametrizing all k-dimensional subspaces of a fixed n-dimensional space.

November 9

No meeting

November 16

Lauren Williams

Tableaux Combinatorics and Hopping Particles

The asymmetric exclusion process is a model of particles hopping on a 1d lattice. It has been cited as a model for traffic flow and for translation in protein synthesis. I'll explain how to use certain combinatorial tableaux to understand the probabilities in the ASEP.

Note: The Fall Math Open House will take place before the talk.

November 23

No meeting - Thanksgiving Break

November 30

Mitchell Faulk

Vector Operator Algebras

Vertex operator algebras are algebraic objects where the multiplication structure consists of a Z-grading of multiplications. From their inception, they played an interesting role in pure math, exhibiting connections to the Monster group and modular functions. Later, it was shown by Huang that these algebraic structures arise precisely as the algebraic structures modeling interactions of strings in conformal field theory. In this talk, we outline this brief history of these interesting algebraic structures.

December 7

Adam Block

Fundamental Theorem of Algebra

As many proofs of the Fundamental Theorem of Algebra I can fit into one hour.

December 14

Daniel Litt

Graph Isomorphism and Representation Theory

Let g_{n,k} be the number of isomorphism classes of graphs with n vertices and k edges. I'll explain how to use the representation theory of the Lie algebra sl_2 to show that the sequence g_{n,0}, g_{n,1}, g_{n, 2}, ... is unimodal.

December 21

No meeting

Columbia Undergraduate Math Society

Summer 2016« Fall 2016 Lectures »Spring 2017