In a famous paper published in Annals of Mathematics (2003), Killip and Simon gave a Szegö like theorem for the circular law (sum rule). The proof of this sum rule relies on mathematical analysis. This sum rule relates the Kullback-Leibler information with respect to the semi-circular law to coefficients involved in the recursive construction of the orthogonal polynomials. Using the theory of large deviations for random matrices we recover easily this sum rules and provide a full machinery to establish and prove new ones.

I will describe the role played by an Apollonian circle packing in the scaling limit of the abelian sandpile on the square grid Z^2. The sandpile solves a certain integer optimization problem. Associated to each circle in the packing is a locally optimal solution to that problem. Each locally optimal solution can be described by an infinite periodic pattern of sand, and the patterns associated to any four mutually tangent circles obey an analogue of the Descartes Circle Theorem. Joint work with Wesley Pegden and Charles Smart.

: I will give an introductory course of 3-4 lectures on the topics mentioned in the title to an audience without any prior knowledge of the subject which is a meeting ground for Differential geometry, Algebraic geometry, and Number theory.

This series of talks will cover various notions of zeta functions associated to graphs

See attachment.

We will explain constructions of Robba rings and $(\varphi, \Gamma)-modules of p-adic Hodge theory. We will describe new proofs of some results on finiteness of cohomology of these modules, and indicate their applications to the theory of $p$-adic families of automorphic forms. This is part of ongoing work with Eugen Hellmann and Ruochuan Liu.

I will give an introductory course of 3-4 lectures on the topics mentioned in the title to an audience without any prior knowledge of the subject which is a meeting ground for Differential geometry, Algebraic geometry, and Number theory.