Seminars and Colloquia by Series

Wednesday, March 11, 2015 - 15:05 , Location: Skiles 006 , Aaron Bertram , University of Utah , Organizer: Matt Baker
Jacobians aren't particularly interesting from the point of view of the minimal model program, and neither are the moduli spaces of vector bundles on curves. But once we pass to vector bundles of higher rank (or torsion-free sheaves) on surfaces, then the birational geometry becomes very interesting. In this talk, I want to describe some recent results that rely on "tilting" the category of coherent sheaves on a surface to produce birational models of moduli that are themselves moduli spaces that come up naturally in the minimal model program.
Monday, March 9, 2015 - 15:00 , Location: Skiles 006 , Igor Klep , University of Auckland , Organizer: Greg Blekherman
Given  a tuple A=(A_1,...,A_g) of symmetric matrices of the same size, the affine linear matrix polynomial L(x):=I-\sum A_j x_j is a monic linear pencil. The solution set S_L of the corresponding linear matrix inequality, consisting of those x in R^g for which L(x) is positive semidefinite (PsD), is called a spectrahedron. It is a convex basic closed semialgebraic subset of R^g. Given a spectrahedron S_L, the matrix cube problem of Nemirovskii asks for the biggest cube [-r,r]^g included in S_L. We solve a relaxation of this problem based on``matricial’’ spectrahedra and estimate the error inherent in this relaxation. The talk is based on joint work with B. Helton, S. McCullough and M. Schweighofer.
Wednesday, February 18, 2015 - 15:05 , Location: Skiles 006 , Eric Katz , University of Waterloo , Organizer: Joseph Rabinoff
In this talk, I discuss our recent proof that there is a uniform bound forthe number of rational points on genus g curves of Mordell-Weill rank atmost g-3, extending a result of Stoll on hyperelliptic curves.  I outlinethe Chabauty-Coleman for bounding the number of rational points on a curveof low Mordell-Weil rank and discuss the challenges to making the bounduniform.  These challenges involving p-adic integration and Newton polygonestimates, and are answered by employing techniques in Berkovich spaces,tropical geometry, and the Baker-Norine theory of linear systems on graphs.
Monday, January 26, 2015 - 15:05 , Location: Skiles 006 , Kenny Jacobs , University of Georgia , Organizer: Matt Baker
Let K be a complete, algebraically closed, non-Archimedean field, and let $\phi$ be a rational function defined over K with degree at least 2.  Recently, Robert Rumely introduced two objects that carry information about the arithmetic and the dynamics of $\phi$. The first is a function $\ord\Res_\phi$, which describes the behavior of the resultant of $\phi$ under coordinate changes on the projective line. The second is a discrete probability measure $\nu_\phi$ supported on the Berkovich half space that carries arithmetic information about $\phi$ and its action on the Berkovich line. In this talk, we will show that the functions $\ord\Res_\phi(x)$ converge locally uniformly to the Arakelov-Green's function attached to $\phi$, and that the family of measures $\nu_{\phi^n}$ attached to the iterates of $\phi$ converge to the equilibrium measure of $\phi$.​
Friday, November 21, 2014 - 15:00 , Location: Skiles 006 , Jennifer Park , McGill University , Organizer: Joseph Rabinoff
Using the ideas of Poonen and Stoll, we develop a modified version of Chabauty's method, which shows that a positive proportion of hyperelliptic curves have as few quadratic points as possible.
Wednesday, November 19, 2014 - 15:00 , Location: Skiles 006 , Jennifer Park , McGill University , Organizer: Joseph Rabinoff
Faltings' theorem states that curves of genus g> 1 have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the number of rational points, but this bound is too large to be used in any reasonable sense. In 1985, Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is smaller than g, can be used to give a good effective bound on the number of rational points of curves of genus g > 1. We draw ideas from nonarchimedean geometry and tropical geometry to show that we can also give an effective bound on the number of rational points outside of the special set of the d-th symmetric power of X, where X is a curve of genus g > d, when the Mordell-Weil rank of the Jacobian of the curve is at most g-d.
Friday, November 14, 2014 - 14:00 , Location: Skiles 202 , Ralph Morrison , Berkeley , Organizer: Joseph Rabinoff
Smooth curves in the tropical plane correspond to unimodulartriangulations of lattice polygons. The skeleton of such a curve is ametric graph whose genus is the number of lattice points in the interior ofthe polygon. In this talk we report on work concerning the followingrealizability problem: Characterize all metric graphs that admit a planarrepresentation as a smooth tropical curve. For instance, about 29.5 percentof metric graphs of genus 3 have this property. (Joint work with SarahBrodsky, Michael Joswig, and Bernd Sturmfels.)
Tuesday, November 4, 2014 - 16:00 , Location: Skiles 005 , Arul Shankar and Wei Zhang , Harvard University and Columbia University , Organizer:
The Joint Athens-Atlanta Number Theory Seminar meets once a semester, usually on a Tuesday, with two talks back to back, at 4:00 and at 5:15. Participants then go to dinner together.
Monday, November 3, 2014 - 15:00 , Location: Skiles 005 , François Charles , MIT and Paris-Sud , Organizer: Kirsten Wickelgren
We will discuss a proof of the following result: if E and E' are two elliptic curves over a number field, there exist infinitely many places p of k such that the reduction of E and E' modulo p are isogenous. We will explain the relationship with the dynamics of Hecke correspondences on modular curves and the heuristics behind such results.
Monday, October 20, 2014 - 15:05 , Location: Skiles 005 , Karl Schwede , University of Utah , Organizer: Anton Leykin
I will discuss recent work of Bhargav Bhatt, myself and Shunsuke Takagi relating several open problems and generalizing work of Mustata and Srinivas. First: whether a smooth complex variety is ordinary after reduction to characteristic $p > 0$ for infinitely many $p$. Second: that multiplier ideals reduce to test ideals for infinitely many $p$ (regardless of coefficients). Finally, whether complex varieties with Du Bois singularities have $F$-injective singularities after reduction to infinitely many $p > 0$.

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