Recently Added

The Permanents of Gaussian Matrices

Scott Aaronson
Massachusetts Institute of Technology
November 29, 2010

In recent joint work with Alex Arkhipov, we proposed a quantum optics experiment, which would sample from a probability distribution that we believe cannot be sampled (even approximately) by any efficient classical algorithm, unless the polynomial hierarchy collapses. Several optics groups are already working toward doing our experiment.

A Classical Approximation Point of View on Some Results in the Spectral Theory of Jacobi Matrices

Mira Shamis
Institute for Advanced Study
December 10, 2010

Deift--Simon and Poltoratskii--Remling proved upper bounds on the measure of the absolutely continuous spectrum of Jacobi matrices. Using methods of classical approximation theory, we give a new proof of their results, and generalize them in several ways. First, we prove a sharper inequality taking the distribution of the values of the potential into account. Second, we prove a generalization of a "local" inequality of Deift--Simon to the non-ergodic setting. Based on joint work with Sasha Sodin

Univalent Foundations of Mathematics

Vladimir Voevodsky
Institute for Advanced Study
December 10, 2010

The correspondence between homotopy types and higher categorical analogs of groupoids which was first conjectured by Alexander Grothendieck naturally leads to a view of mathematics where sets are used to parametrize collections of objects without "internal structure" while collections of objects with "internal structure" are parametrized by more general homotopy types. Univalent Foundations are based on the combination of this view with the discovery that it is possible to directly formalize reasoning about homotopy types using Martin-Lof type theories.

Parahoric Subgroups and Supercuspidal Representations of p-Adic groups

Dick Gross
Harvard University
December 9, 2010

This is a report on some joint work with Mark Reeder and Jiu-Kang Yu. I will review the theory of parahoric subgroups and consider the induced representation of a one-dimensional character of the pro-unipotent radical. A surprising fact is that this induced representation can (in certain situations) have finite length. I will describe the parahorics and characters for which this occurs, and what the Langlands parameters of the corresponding irreducible summands must be.

On Families of Filtered phi Modules and Crystalline Representations

Eugen Hellmann
University of Bonn
December 8, 2010

We study families of filtered phi-modules associated to families of p-adic Galois representations as considered by Berger and Colmez. We show that the weakly admissible locus in a family of filtered phi-modules is open and that the groupoid of weakly admissible modules is in fact an Artin stack. Working in the category of adic spaces instead of the category of rigid analytic spaces one can show that there is an open substack of the weakly admissible locus over which the filtered phi-modules is induced from a family of crystalline representations.