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. 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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