School of Mathematics

Time quasi-periodic gravity water waves in finite depth

Massimiliano Berti
International School for Advanced Studies
November 8, 2017
We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water waves solutions, namely periodic and even in the space variable $x$, of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a set of asymptotically full measure. This is a small divisor problem.

Topological and arithmetic intersection numbers attached to real quadratic cycles

Henri Darmon
McGill University
November 8, 2017

Abstract:  I will discuss a recent conjecture formulated in an ongoing project with Jan Vonk relating the intersection numbers of one-dimensional topological cycles on certain Shimura curves to the arithmetic intersections of associated real multiplication points on the Drinfeld p-adic upper half-plane.  Numerical experiments carried out with Vonk and James Rickards supporting the conjecture will be described.

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.