School of Mathematics

Decoupling in harmonic analysis and the Vinogradov mean value theorem

Jean Bourgain

IBM von Neumann Professor; School of Mathematics

December 17, 2015

Based on a new decoupling inequality for curves in $\mathbb R^d$, we obtain the essentially optimal form of Vinogradov's mean value theorem in all dimensions (the case $d = 3$ is due to T. Wooley). Various consequences will be mentioned and we will also indicate the main elements in the proof (joint work with C. Demeter and L. Guth).

Modularity and potential modularity theorems in the function field setting

Michael Harris

Columbia University

December 17, 2015

Let $G$ be a reductive group over a global field of positive characteristic. In a major breakthrough, Vincent Lafforgue has recently shown how to assign a Langlands parameter to a cuspidal automorphic representation of $G$. The parameter is a homomorphism of the global Galois group into the Langlands $L$-group $^LG$ of $G$. I will report on my joint work in progress with Böckle, Khare, and Thorne on the Taylor-Wiles-Kisin method in the setting of Lafforgue's correspondence.

Toward the KRW conjecture: cubic lower bounds via communication complexity

Or Meir

University of Haifa

December 14, 2015

One of the major challenges of the research in circuit complexity is proving super-polynomial lower bounds for de-Morgan formulas. Karchmer, Raz, and Wigderson suggested to approach this problem by proving that formula complexity behaves "as expected" with respect to the composition of functions. They showed that this conjecture, if proved, would imply super-polynomial formula lower bounds.

Locally symmetric spaces and torsion classes

Ana Cariani

Princeton University; Veblen Research Instructor, School of Mathematics

December 14, 2015

The Langlands program is an intricate network of conjectures, which are meant to connect different areas of mathematics, such as number theory, harmonic analysis and representation theory. One striking consequence of the Langlands program is the Ramanujan conjecture, which is a statement purely within harmonic analysis, about the growth rate of Fourier coefficients of modular forms. It turns out to be intimately connected to the Weil conjectures, a statement about the cohomology of projective, smooth varieties defined over finite fields.