School of Mathematics

Profinite rigidity and flexibility for compact 3-manifold groups

Alan Reid
University of Texas, Austin; Member, School of Mathematics
February 2, 2016
This talk will discuss the question: To what extent are the fundamental groups of compact 3-manifolds determined (amongst the fundamental groups of compact 3-manifolds) by their finite quotients. We will discuss work that provides a positive answer for fundamental groups of hyperbolic 1-punctured torus bundles.

The space of surface shapes, and some applications to biology

Joel Hass
University of California, Davis; Member, School of Mathematics
February 1, 2016
The problem of comparing the shapes of different surfaces turns up in different guises in numerous fields. I will discuss a way to put a metric on the space of smooth Riemannian 2-spheres (i.e. shapes) that allows for comparing their geometric similarity. The metric is based on a distortion energy defined on the space of conformal mappings between a pair of spheres. I'll also discuss a related idea based on hyperbolic orbifold metrics. I will present results of experiments on applying these techniques to biological data.

Constant-round interactive-proofs for delegating computations

Ron Rothblum
Massachusetts Institute of Technology
February 1, 2016
Interactive proofs have had a dramatic impact on Complexity Theory and Cryptography. In particular, the celebrated IP=PSPACE Theorem [LFKN92,Shamir92] allows an all-powerful but untrusted prover to convince a polynomial-time verifier of the validity of extremely complicated statements (as long as they can be evaluated using polynomial space). The interactive proof system designed for this purpose requires a polynomial number of communication rounds.

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.

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.

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.