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 socalled 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 threedimensional 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 kdimensional subspaces of a fixed ndimensional 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 Zgrading 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 