## Description

The probability seminar aims at exposing students to advanced probability theory and selected applications. A different topic is selected each year. The course for the academic year 2019-2020 will be focusing on asymptotic methods in probability.

The course material will be mainly based on the recent book by Joel Spencer and Laura Florescu entitled "Asymptopia".

The authors describes Asymptopia as an imaginary world populated by different diverging functions, and they claim that understanding "language and culture" of this imaginary world is fundamental for the education of a contemporary mathematician interested in probability theory and its applications.

To support this claim, after introducing basic concepts from asymptotic analysis, the book covers a number of probabilistic topics of apparently very different flavour from: discrete mathematics, combinatorics, geometry, and theory of stochastic processes. The common theme in all these topics is the implemented asymptotic method.

The idea of this course is to select parts of the material of the book aiming at introducing students to different problems of current research interest while learning a methodology which can serve as a set of asymptotic tool-boxes of general mathematical and probabilistic interest.

Topics that will be covered will include: estimates for sums and integrals, distribution of prime integers, counting subgraphs in network models via so-called "Erd"os magic", large deviations and concentration inequalities for random walks and other processes.

## Prerequisits

Familiarity with basic concepts in probability theory and calculus.

## Literature

The course material is based on the following reference, together with selected topical research papers that will be made available during the course:

- J. Spencer, with L. Florescu, Asymptopia, Student Mathematical Library 71, American Mathematical Society, 2014.

## Plan and exam

Three introductory lectures and two advanced lectures will be given by Avena and Verbitsky.

All the other lectures will be given by the participants.

Each participant gives 2 presentations:

one dedicated to some aspects of the general theory from the main book,

one chosen from a list of topics presented during the course.

For the second presentation, a short hand-out (to be distributed to all the participants) must be prepared

as well.

The final grade is based on active participation, the two presentations and the hand-out.