School of Mathematics

An Euler system for genus 2 Siegel modular forms

David Loeffler
University of Warwick
November 8, 2017

Abstract:  Euler systems are compatible families of cohomology classes for a global Galois represenation, which plan an important role in studying Selmer groups.  I will outline the construction of a new Euler system, for the Galois representation associated to a cohomological cuspidal automorphic representation on the symplectic group GSp(4).  This is joint work with Chris Skinner and Sarah Zerbes. 

A derived Hecke algebra in the context of the mod $p$ Langlands program

Rachel Ollivier
University of British Columbia
November 8, 2017

Abstract: Given a p-adic reductive group G and its (pro-p) Iwahori-Hecke algebra H, we are interested in the link between the category of smooth representations of G and the category of H-modules. When the field of coefficients has characteristic zero this link is well understood by work of Bernstein and Borel.

Modularity lifting theorems for non-regular symplectic representations

George Boxer
University of Chicago
November 7, 2017

Abstract:  We prove an ordinary modularity lifting theorem for certain non-regular 4-dimensional symplectic representations over totally real fields.  The argument uses both higher Hida theory and the Calegari-Geraghty version of the Taylor-Wiles method.  We also present some applications of these theorems to abelian surfaces.  (Joint work with F. Calegari, T. Gee, and V. Pilloni.) 

Language edit distance, $(\min,+)$-matrix multiplication & beyond

Barna Saha
University of Massachusetts, Amherst
November 6, 2017

The language edit distance is a significant generalization of two basic problems in computer science: parsing and string edit distance computation. Given any context free grammar, it computes the minimum number of insertions, deletions and substitutions required to convert a given input string into a valid member of the language. In 1972, Aho and Peterson gave a dynamic programming algorithm that solves this problem in time cubic in the string length. Despite its vast number of applications, in forty years there has been no improvement over this running time.

Morse-Bott cohomology from homological perturbation

Zhengyi Zhou
University of California, Berkeley
November 6, 2017
Abstract: In this talk, I will give a new construction of the Morse-Bott cochain complex, where the underlying vector space is generated by the cohomology of the critical manifolds. This new construction has two nice features: (1) It requires the minimum amount of transversality. (2) The choices made in the construction do not depend on the moduli spaces. I will explain its relation to three other constructions in literature, namely Austin-Braam's push-pull construction, Fukaya's push-pull construction and the cascades construction.