# 2017 - 2018

# Calendrier / Calendar

# QUÉBEC

**Date Heure/Time**: Le jeudi 1 mars 2018 - 15:30

**Lieu/Venue**: Université Laval, Pavillon Vachon, salle 3840

**Conférencier/Speaker**: Glenn Stevens, Boston University

**Titre/Title**: p-adic Variation in the Theory of Automorphic Forms

**Resume/Abstract**:

This will be an expository lecture intended to illustrate through examples the theme of p-adic variation in the classical theory of modular forms. Classically, modular forms are complex analytic objects, but because their fourier coefficients are typically integral, it is possible to do elementary arithmetic with them. Early examples arose already in the work of Ramanujan. Today one knows that modular forms encode deep arithmetic information about elliptic curves and galois representations. The main goal of the lecture will be to motivate a beautiful theorem of Robert Coleman and Barry Mazur, who constructed the so-called Eigenvariety, which leads to a geometric approach to varying modular forms, their associated galois representations, as well as their L-functions, in p-adic analytic families. We will briefly discuss important applications to Number Theory and Iwasawa Theory.

**Date Heure/Time**: Le jeudi 14 décembre 2017 - 15:30

**Lieu/Venue**: Université Laval, Pavillon Vachon, salle 2830

**Conférencier/Speaker**: Anthony Bonato, Ryerson University

**Titre/Title**: The new world of infinite random geometric graphs

**Resume/Abstract**:

The infinite random or Rado graph R has been of interest to graph theorists, probabilists, and logicians for the last half-century. The graph R has many peculiar properties, such as its categoricity: R is the unique countable graph satisfying certain adjacency properties. Erdös and Rényi proved in 1963 that a countably infinite binomial random graph is isomorphic to R. Random graph processes giving unique limits are, however, rare. Recent joint work with Jeannette Janssen proved the existence of a family of random geometric graphs with unique limits. These graphs arise in the normed space $\ell^n_\infty$ , which consists of $\mathbb{R}^n$ equipped with the $L_\infty$-norm. Balister, Bollobás, Gunderson, Leader, and Walters used tools from functional analysis to show that these unique limit graphs are deeply tied to the $L_\infty$-norm. Precisely, a random geometric graph on any normed, finite-dimensional space not isometric $\ell^n_\infty$ gives non-isomorphic limits with probability 1. With Janssen and Anthony Quas, we have discovered unique limits in infinite dimensional settings including sequences spaces and spaces of continuous functions. We survey these newly discovered infinite random geometric graphs and their properties.