# Recently Added

## IWASAWA: Lecture 2

## IWASAWA: Lecture 1

## Automorphy Lifting for Galois Representations With Small Residual Image

**GALOIS REPRESENTATIONS AND AUTOMORPHIC FORMS SEMINAR**

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

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.

## Eisenstein Congruences and Euler Systems

**GALOIS REPRESENTATIONS AND AUTOMORPHIC FORMS SEMINAR**

## On Automorphy of Certain Galois Representations of GO_4 Type

**GALOIS REPRESENTATIONS AND AUTOMORPHIC FORMS SEMINAR**

## Learning with Boolean Threshold Functions, a Statistical Physics Perspective

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].

## Efficiently Learning Mixtures of Gaussians

## An Elementary Proof of the Restricted Invertibility Theorem

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.