Spring 2018 Seminar Schedule

1/9 No Seminar
1/16 No Seminar (Colloquium)
1/23 No Seminar
1/30 David Zureick-Brown Emory University Canonical rings of stacky curves
  • Abstract: We give a generalization to stacks of the classical (1920's) theorem of Petri -- we give a presentation for the canonical ring of a stacky curve. This is motivated by the following application: we give an explicit presentation for the ring of modular forms for a Fuchsian group with cofinite area, which depends on the signature of the group. This is joint work with John Voight.
2/6 Tom Sutherland University of Mainz Stability conditions and rational elliptic surfaces
  • Abstract: I will describe the space of Bridgeland stability conditions of the derived category of some CY3 algebras of quivers coming from dimer models on the Riemann sphere. We give a biholomorphic map from the upper-half plane to the space of stability conditions lifting the period map of a meromorphic differential on a 1-dimensional family of elliptic curves, the complement of certain singular fibres in a rational elliptic surface. The map is equivariant with respect to the actions of a congruence subgroup of PSL(2, Z), on the left by monodromy of the rational elliptic surface and on the right by autoequivalences of the derived category.
2/13 Benjamin Antieau University of Illinois at Chicago The topological period-index conjecture
  • Abstract: I will survey recent progress on the topological period-index problem for Brauer groups and its connection to the algebraic period-index problem.
2/20 Dan Erman University of Wisconsin - Madison Big polynomial rings and Stillman's Conjecture
  • Abstract: Ananyan--Hochster's recent proof of Stillman's conjecture is based on a key principle: if $f_1, \ldots, f_r$ are sufficiently general forms in a polynomial ring, then as the number of variables tends to infinity, they will behave increasingly like independent variables. We show that this principle becomes a theorem if ones passes to a limit of polynomial rings, using either the inverse limit or the ultraproduct. This yields the surprising fact that these limiting rings are themselves polynomial rings (in uncountably many variables). It also yields two new proofs of Stillman's conjecture. This is joint work with Steven Sam and Andrew Snowden.
2/27 Ronen Mukamel Rice University
3/6 Gordon Heier University of Houston A generalized Schmidt subspace theorem for closed subschemes
  • Abstract: I will discuss our recent generalized version of Schmidt's subspace theorem for closed subschemes in general position in terms of suitably defined Seshadri constants with respect to a fixed ample divisor. I will also explain how our theorem allows to recover a higher-dimensional Diophantine approximation theorem of K.F. Roth-type due to D. McKinnon and M. Roth with a significantly shortened proof. This is joint work with A. Levin.
3/13 Spring Break
3/20 Anastassia Etropolski Rice University Explicit rational point calculations for certain hyperelliptic curves
  • Abstract: Given a curve of genus at least 2, it was proven in 1983 by Faltings that it has only finitely many rational points. Unfortunately, this result is ineffective, in that it gives no bound on the number of rational points. 40 years earlier, Chabauty proved the same result under the condition that the rank of the Jacobian of the curve is strictly smaller than the genus. While this is obviously a weaker result, the methods behind that proof could be made effective, and this was done by Coleman in 1985. Coleman's work led to a procedure known as the Chabauty-Coleman method, which has shown to be extremely effective at determining the set of rational points exactly, particularly in the case of hyperelliptic curves. In this talk I will discuss how we implement this method using Magma and Sage to provably determine the set of rational points on a large set of genus 3, rank 1 hyperelliptic curves, and how these calculations fit into the context of the state of the art conjectures in the field. The subject of this talk is joint work with Jennifer Balakrishnan, Francesca Bianchi, Victoria Cantoral-Farfan, and Mirela Ciperiani.
3/27 No Seminar
4/3 No Seminar
4/10 No Seminar
4/17 Jen Berg Rice University
Fall 2017 Schedule