The Summer School in Mathematics 2022 will take place from June 13^th to June 30^th, 2022 and will be dedicated to Cohomology, Geometry and Explicit number theory (COGENT).

The goal of this school is to introduce young mathematicians to some recent developments of the field of  cohomology of groups, with a focus on arithmetic groups, from geometrical, topological and computational aspects with applications to  number theory. We also want to promote greater interaction among researchers and PhD students, and  strenghten the COGENT network through new collaborations.

The first two weeks are dedicated to mini-courses and mini-workshops around some computational tools and new computational techniques, as well as short term team projects.

There will be also an "Illustrating mathematics event" with an art exhibitions showcasing artistic views related to the geometries of arithmetic groups.

The third week will be devoted to a workshop on the COGENT topics by young researchers and leading specialists as well.

Notice that all courses and talks will be in hybrid mode.

Registration for on-site participation is now closed.

Registration for online participation is now closed.

In mathematics, the theory of groups attacks problems by studying the algebraic structure of inherent symmetries. Groups are vital to modern algebra and numerous physical systems such as crystals, and atomic structures are modeled by symmetry groups. They have applications in physics, chemistry, materials science, and computer science. Group theory is also central to modern cryptography. The Cohomology of Groups attempts to understand groups of symmetries through coarse topological properties, expressed in terms of elementary algebra, possessed by geometric spaces associated to the groups. The Cohomology of Arithmetic Groups focuses on problems in number theory, and provides a means of translating these problems into elementary algebra. Explicit calculations play an important role in the development of the cohomology of groups and its applications, not only in mathematics but also in  theoretical physics, crystallography and condensed matter physics.

As the scale of these computations increased, it became more common for them to be performed with the aid of a computer, and currently, we face numerous challenges that are close to computational limits.

The cohomology of arithmetic groups, and more generally matrix groups, is a rich subject with links to geometry, topology, ring theory, number theory and theoretical physics. It is also deeply connected to the famous Langlands program

Explicit computation of the cohomology, and especially the action of the Hecke operators on cohomology, can be used not only to test various number theoretic conjectures, but also to build new, and large,  datasets to spur further investigations. Moreover, with the introduction of new computational techniques in number theory and geometry such as machine learning we expect to reveal new facets of old problems as seen in recent work.

The main geometric objects involved in such computations are based on the geometry of numbers. These objects are called lattices and they allow to build more sophisticated geometric models through what is known as reduction theory.  Lattices and reduction theory have played an important  role in mathematics and their applications since the XIXth century. On one hand, there has been a growing interest in  explicit calculations involving Euclidean lattices and their generalizations from the mathematical community to test various number theoretic conjectures.

The goal of this summer school is to introduce a wide audience to this topics, from computational group theory to theoretical and practical aspects of the  cohomology of arithmetic groups, throught geometrical and topological tools and applications to number theory. To this purpose, we will also introduce the audience to software like GAP and PARI/GP.