Recently Added

Completed Cohomology of Shimura Curves and a p-Adic Jacquet-Langlands Correspondence

James Newton
Member, School of Mathematics
February 9, 2011

In this talk, I will describe a construction of a geometric realisation of a p-adic Jacquet-Langlands correspondence for certain forms of GL(2) over a totally real field. The construction makes use of the completed cohomology of Shimura curves, and a study of the bad reduction of Shimura curves due to Rajaei (generalising work of Ribet for GL(2) over the rational numbers). Along the way I will also describe a p-adic analogue of Mazur's principle in this setting.

Learning with Boolean Threshold Functions, a Statistical Physics Perspective

Rémi Monasson
Ecole Normale Superieure; Simons Center for Systems Biology, IAS
January 25, 2011

Boolean Threshold Functions (BTF) arise in many contexts, ranging from computer science and learning theory to theoretical neurobiology. In this talk, I will present non-rigorous approaches developed in the statistical physics of disordered systems to characterize BTF in a quantitative way [1], with an emphasis on computational and geometrical aspects. These techniques will be illustrated on two particular cases: the celebrated perceptron (Linear Threshold Function) [2], and the more realistic tempotron model of a neuron [3,4].

An Elementary Proof of the Restricted Invertibility Theorem

Nikhil Srivastava
Institute for Advanced Study
November 9, 2010

We give an elementary proof of a generalization of Bourgain and Tzafriri's Restricted Invertibility Theorem, which says roughly that any matrix with columns of unit length and bounded operator norm has a large coordinate subspace on which it is well-invertible. Our proof gives the tightest known form of this result, is constructive, and provides a deterministic polynomial time algorithm for finding the desired subspace.