Columbia Undergraduate Math Society

Summer 2016« Fall 2016 Lectures »Spring 2017

Wednesdays, 7:30pm; Room 507, Mathematics
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ums [at] math.columbia.edu

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    
December 21   No meeting  
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