India

Fourier theory has been a useful analytic tool in studying discrete structures. Some of the areas where this theory has been particulaly fruitful are additive combinatorics, eigen values of graphs and random walks on finite groups or in the study of Boolean functions used in computer sciences.

The development of Computational methods for partial differential equations (PDEs) is a key tool for the development of science and technology. For its development, it is important to have a deep understanding of the classical and new methodologies used in numerical methods. The summer school will provide an overview of some techniques that allow one to address the computational challenges encountered in different applications.

The scope of problems accessible for a numerical treatment has been constantly broadened over the last fifty years. In particular, there has been a lot of research activity in the recent decades aimed at the problems with multi-scale and multi-physics features. The dedicated numerical methods (model reduction, micro-macro models and model coupling, non standard FEM) stem from diverse techniques and ideas such as homogenization, asymptotic analysis, statistical physics, domain decomposition methods, etc.

Recent developments in non-commutative algebra include parallel theories of graph C*-algebras and Leavitt path algebras. In this research school, we mainly focus on theories of Leavitt path algebras and their connections with other areas of mathematics.

Geometric flows are burning topic now a days which involve the evolution of Riemannian metric along with some geometric concepts.

Numerical analysis, simulation and scientific computing for the solution of partial differential equations (PDEs) using FEM and related methods have been cornerstones of applied mathematics over the last fifty years. In fact, it is well known that the study of computational methods for PDEs is extremely crucial for the development of science and technology.

Soft computing is a consortium of methodologies which work synergetically and provides in one form or another flexible information processing capabilities for handling real life ambiguous situations. Its aim is to exploit the tolerance for imprecision, uncertainty, approximate reasoning, and partial truth in order to achieve tractability, robustness, low cost solution and close resemblance with human like decision making. The relevance of soft computing for pattern recognition and image processing is already established during the last few years.

The objective of the school is to expose recent results in the domain of probability theory and stochastic analysis on groups and other geometrical or algebraic structures such as symmetric spaces, symmetric cones and semigroups. And to present recent methods, very diversified and often not easily available, used in the probability theory on groups, in particular methods of the Lie-group theory, harmonic analysis, ergodic theory and the theory of Jordan algebras.

Since a couple of decades, Finsler geometry has been a very active field of research, with a particular stress on the use of purely metric methods in the investigation of various Finsler metrics that appear naturally in geometry, topology and convexity theory.

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.