India

Geometric group theory is a relatively new line of research on its own, inspired by pioneering works of M. Dehn, G.D. Mostow and M. Gromov. It is mainly devoted to the study of countable groups by exploring connections between algebraic properties of such groups and geometric properties of spaces on which these groups act, such as the deck transformation group of a Riemannian manifold. Geometric group theory is a very broad area, and this program aims at introducing young students to different aspects of the theory.

The theory of surfaces of finite or infinite types, with their geometric structures, with moduli spaces of geometric structures and with some related dynamical systems on these moduli spaces, constitutes some of the most important aspects of low-dimensional topology, geometry and dynamics. In this school, we propose a set of coordinated courses that will concentrate on several aspects of this field and which will give the students the opportunity, at the same time, to learn the basic aspects of these topics, and to have access to important research problems.

Quantum computers threaten to break most of the cryptography we currently use to protect our information security systems. In a quantum computer, performing operations comes from a quan- tum physical notion that works differently from a classical computer setting, and it gives an expo- nential speed-up for certain computations. To construct quantum-resistant cryptographic systems, we need a new class of hard mathematical problems. Many of them are currently competing in the National Institute of Standards and Technology (NIST) Post Quantum Cryptography Standardizati- on.

The study of error-correcting codes has posed a large number of intriguing and important questions in several areas of mathematics, such as algebra, number theory, combinatorics, algebraic geometry etc. In addition it is vital tool in making transmission of data robust against noise. In this school, we are going to present several lectures on the usage of techniques in finite geometry in the study of error-correcting codes.

An $L$-function is a function defined additively by a Dirichlet series with a multiplicative Euler product.
The initial work of Dirichlet was generalized to number fields by Hecke and given an adelic interpretation by Tate
which paved the way to move from $GL(1)$ to higher degree $L$ functions associated to automorphic forms of $GL(n)$ for a general n .

In this project our focus will be on the study of Differential and Algebraic Geometry from a
Complex Analytic perspective. Certain Riemannian and semi-Riemannian objects, as well as
various properties of them in a complex context will be analysed.

The main aim of this school is to introduce young researchers and graduate students to fundamental and advanced tools of homological algebra as applied to non-commutative algebra and number theory, with an emphasis on algebraic, geometric, topological, and numerical methods.