8/22 
Ronen Mukamel 
Rice University 
Teichmuller curves in positive characteristic 

Abstract: We will investigate the arithmetic nature of Teichmuller curves in the spirit of the celebrated theory developed for the classical modular curves. For arithmetic Teichmuller curves in genus two, we introduce the Weierstrass polynomial, an analogue of the classical modular polynomial associated to modular curves. We find that Weierstrass polynomials often have integer coefficients and surprising congruences with modular polynomials in positive characteristic.

8/29 
School Closed 
9/5 
John Calabrese 
Rice University 
Quotient Categories 

Abstract: It is a well known fact that two varieties are birational if and only if they have the same function field. An a priori unrelated theorem is due to Gabriel: two varieties are the same if and only if their categories of coherent sheaves are the same. I'll explain how these two theorems are actually special cases of a broader one, involving quotients of Coh(X). This is joint work with Roberto Pirisi (UBC).

9/12 
Nathan Kaplan 
UC Irvine 
Counting Lattices by Cotype 

Abstract: The zeta function of Z^d is a generating function that encodes the number of its sublattices of each index. This function can be expressed as a product of Riemann zeta functions and analytic properties of the Riemann zeta function then lead to an asymptotic formula for the number of sublattices of Z^d of index at most X.
Nguyen and Shparlinski have investigated more refined counting questions, giving an asymptotic formula for the number of cocyclic sublattices L of Z^d, those for which Z^d/L is cyclic. Building on work of Petrogradsky, we generalize this result, counting sublattices for which Z^d/L has at most m invariant factors. We will see connections to cokernels of random integer matrices and the CohenLenstra heuristics. Joint work with Gautam Chinta (CCNY) and Shaked Koplewitz (Yale).

9/19 
Arindam Roy 
Rice University 
Unnormalized differences of the zeros of the derivative of the completed Lfunction 

Abstract: We study the distribution of unnormalized diferences between imaginary parts of the zeros of the derivative of the Riemann ξ function. Such distributions are capable of identifying the exact location of every zero of the Riemann zeta function. In particular, we prove that the Riemann hypothesis for the Riemann zeta function is encoded in the distribution of zeros of derivative of completed Dirichlet Lfunction. We also show that the differences tend
to avoid the imaginary part of low lying zeros of the Riemann zeta function.

9/26 
Zheng Zhang 
Texas A & M 
TBA 
10/3 
Izzet Coskun 
University of Illinois at Chicago 
Birational Geometry of moduli spaces of sheaves on surfaces and the BrillNoether Problem 

Abstract: In this talk, I will relate sharp Bogomolov inequalities on a surface to the ample cone of the moduli space of sheaves via Bridgeland stability. I will discuss weak BrillNoether theorems on rational surfaces. As a consequence, I will describe the classification of Chern characters on the plane and on Hirzebruch surfaces such that the general bundle of the moduli space is globally generated. This is joint work with Jack Huizenga.

10/10 
Midterm Recess 
10/17 
Beth Malmskog 
Villanova University 
TBA 
10/24 
Abbey Bourdon 
Wake Forest University 
TBA 
10/31 
Ari Shnidman 
Boston College 
TBA 
11/7 
Andrew Obus 
University of Virginia 
TBA 
11/14 
Renzo Cavalieri 
Colorado State University 
TBA 
11/21 
Nivedita Bhaskhar 
UCLA 
TBA 
11/28 
Leo Goldmakher 
Williams College 
TBA 