Columbia Undergraduate Math Society

Summer 2016« Fall 2016 Lectures »Spring 2017

Wednesdays, 7:30pm; Room 507, Mathematics
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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. Put differently, if we cut up a sphere in any way we can imagine and then count the number of pieces we have, then if we count correctly, we will always get the same number, 2. In modern terms, this process of cutting up a sphere, for that matter, any topological space, into smaller bits and counting the resulting pieces in a careful way is encapsulated in an invariant known as the Euler characteristic. In this talk, I will discuss invariants of geometric objects called "motivic invariants". These are similar in spirit to the Euler characteristic in that they can be computed by cutting and pasting your space, but which sometimes result in objects much more interesting than just numbers. 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 TBD    
November 2 Michael Thaddeus Schubert Calculus  
November 9 TBD    
November 16   Math Department Open House  
November 23 TBD    
November 30 TBD    
December 7 TBD    
December 14 Daniel Litt    
December 21 TBD    
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