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The goal of the program is to survey the progress in the theory of arithmetic and Zariski-dense subgroups achieved in the last 10-15 years, including a variety of applications to algebraic and differential geometry, combinatorics and other areas. Special attention will be given to open problems and directions of further research. The program will showcase an array of techniques employed to investigate Zariski-dense subgroups but the focus will be on the use of methods from algebraic and analytic number theory and arithmetic theory of algebraic groups. These have been successfully used to tackle long-standing problems such as fake projective planes, isospectral and length-commensurable locally symmetric spaces, expanding graphs and multi-dimensional expanders, and many others, with new applications to geometry, topology and mathematical physics likely to emerge in the near future.Also, recent results on bounded/non-bounded generation (``diophantine techniques in linear groups''), bounde...

The goal of the program is to survey the progress in the theory of arithmetic and Zariski-dense subgroups achieved in the last 10-15 years, including a variety of applications to algebraic and differential geometry, combinatorics and other areas. Special attention will be given to open problems and directions of further research. The program will showcase an array of techniques employed to investigate Zariski-dense subgroups but the focus will be on the use of methods from algebraic and analytic number theory and arithmetic theory of algebraic groups. These have been successfully used to tackle long-standing problems such as fake projective planes, isospectral and length-commensurable locally symmetric spaces, expanding graphs and multi-dimensional expanders, and many others, with new applications to geometry, topology and mathematical physics likely to emerge in the near future.Also, recent results on bounded/non-bounded generation (``diophantine techniques in linear groups''), bounde...

Turbulence is an important problem of science and engineering. The flow arising from Navier-Stokes equations at large Reynolds number is always turbulent. Physicists, mathematicians, and engineers have been engaged in modelling turbulence for quite some time, but the problem is unfinished despite considerable progress. Buoyed by the success of Quantum Field Theory (QFT), which is one of the most successful theories in high energy and condensed matter physics, many researchers have applied it to model turbulence. In addition, turbulent flows exhibit universal power laws, similar to those in critical phenomena. There have been serious efforts to develop a theory for these properties faithfully (i.e. without approximations) from the Navier-Stokes equations. Researchers have also attempted to relate turbulence to conformal field theory. The slowly varying black-hole solutions of Einstein equations with negative cosmological constant have been mapped to solutions of relativistic fluid equat...

Turbulence is an important problem of science and engineering. The flow arising from Navier-Stokes equations at large Reynolds number is always turbulent. Physicists, mathematicians, and engineers have been engaged in modelling turbulence for quite some time, but the problem is unfinished despite considerable progress. Buoyed by the success of Quantum Field Theory (QFT), which is one of the most successful theories in high energy and condensed matter physics, many researchers have applied it to model turbulence. In addition, turbulent flows exhibit universal power laws, similar to those in critical phenomena. There have been serious efforts to develop a theory for these properties faithfully (i.e. without approximations) from the Navier-Stokes equations. Researchers have also attempted to relate turbulence to conformal field theory. The slowly varying black-hole solutions of Einstein equations with negative cosmological constant have been mapped to solutions of relativistic fluid equat...

Cells are the basic units of all living matter. They harness energy flows to drive biological functions. Most bio-chemical networks involved in energy transduction are nowadays well-characterized. However, the energetic costs and constraints for specific cellular processes remain largely unknown. In particular, what are the energy budgets of cells? What are the constraints and limits energy flows impose on cellular processes? Do cells operate near these limits, and, if so, how do energetic constraints impact cellular functions? Physics has provided poweerful tools to study nonequilibrium systems and to define these physical limits, but applying these tools to cell biology remains a challenge. Physical bioenergetics resides at the interface of nonequilibrium physics, energy metabolism, and cell biology. It seeks to understand how much energy cells use, how they partition this energy between different cellular processes, and the associated energetic constraints. In this School, we shall ...

Cells are the basic units of all living matter. They harness energy flows to drive biological functions. Most bio-chemical networks involved in energy transduction are nowadays well-characterized. However, the energetic costs and constraints for specific cellular processes remain largely unknown. In particular, what are the energy budgets of cells? What are the constraints and limits energy flows impose on cellular processes? Do cells operate near these limits, and, if so, how do energetic constraints impact cellular functions? Physics has provided poweerful tools to study nonequilibrium systems and to define these physical limits, but applying these tools to cell biology remains a challenge. Physical bioenergetics resides at the interface of nonequilibrium physics, energy metabolism, and cell biology. It seeks to understand how much energy cells use, how they partition this energy between different cellular processes, and the associated energetic constraints. In this School, we shall ...

India has a rich and long heritage in probability theory and stochastic Processes. Indian researchers, working both within India and abroad, are making quality contribution to the subject individually as well as collaboratively. To sustain this effort it is imperative to expose the young research scholars of India to the latest techniques and advances in the subject as well as increase the interaction between researchers in India. In order to achieve this and create collaborative probability groups within India, we believe that there is a need to have annually organized workshops.Towards this, Lectures on Probability and Stochastic Processes, an annual workshop was started in 2006. It is designed to have two mini-courses on topics of recent interest given by experts in the respective areas. These courses are expected to expose the participants to the important problems being considered in the current literature, as well as the available techniques to work with them. A large audience, w...

India has a rich and long heritage in probability theory and stochastic Processes. Indian researchers, working both within India and abroad, are making quality contribution to the subject individually as well as collaboratively. To sustain this effort it is imperative to expose the young research scholars of India to the latest techniques and advances in the subject as well as increase the interaction between researchers in India. In order to achieve this and create collaborative probability groups within India, we believe that there is a need to have annually organized workshops.Towards this, Lectures on Probability and Stochastic Processes, an annual workshop was started in 2006. It is designed to have two mini-courses on topics of recent interest given by experts in the respective areas. These courses are expected to expose the participants to the important problems being considered in the current literature, as well as the available techniques to work with them. A large audience, w...

The representation theory of infinite-dimensional Lie (super)algebras, Quantum groups, and Vertex algebras is an active area of research with deep connections to other areas of mathematics and to physics. The first week of this program will be a workshop consisting of four mini-courses (6 lectures each) on various themes in modern representation theory.  Vertex algebras and associated varieties (Anne Moreau, Université Paris-Saclay, France)  Representations of quantum affine algebras and yangians (Arun Ram, University of Melbourne, Australia)  Representations of quantum superalgebras and their crystal bases (Jae-Hoon Kwon, Seoul National University, Republic of Korea)  The eigenvalue problem for reductive groups (Shrawan Kumar, University of North Carolina at Chapel Hill, USA) These lectures will be aimed at graduate students, postdoctoral scholars, and researchers who wish to enter the field. The second week of this program will be on “Algebraic and Combinatorial Methods in Representa...

The representation theory of infinite-dimensional Lie (super)algebras, Quantum groups, and Vertex algebras is an active area of research with deep connections to other areas of mathematics and to physics. The first week of this program will be a workshop consisting of four mini-courses (6 lectures each) on various themes in modern representation theory.  Vertex algebras and associated varieties (Anne Moreau, Université Paris-Saclay, France)  Representations of quantum affine algebras and yangians (Arun Ram, University of Melbourne, Australia)  Representations of quantum superalgebras and their crystal bases (Jae-Hoon Kwon, Seoul National University, Republic of Korea)  The eigenvalue problem for reductive groups (Shrawan Kumar, University of North Carolina at Chapel Hill, USA) These lectures will be aimed at graduate students, postdoctoral scholars, and researchers who wish to enter the field. The second week of this program will be on “Algebraic and Combinatorial Methods in Representa...

Active matter is a novel class of driven nonequilibrium systems wherein a many-body system with an energy throughput consumes and dissipates energy at the level of the individual units. This field has grown in prominence in the last couple of decades with immense applicability in diverse areas such as granular and glassy systems, control theory, and biological systems. Many biologically crucial processes, such as the dynamics in the cellular cytoskeleton, the collective motion of cells, the progression of cancer, and the large-scale migratory behavior of bird flocks, mammal herds, ants, and fish schools have been modeled within the active matter framework.The meeting will feature research talks that will discuss recent progress in the field. Additionally, several pedagogical talks will provide a broad overview for young researchers in the country and help develop an appreciation for the significant new developments. The meeting will provide ample opportunities for young researchers to ...

Active matter is a novel class of driven nonequilibrium systems wherein a many-body system with an energy throughput consumes and dissipates energy at the level of the individual units. This field has grown in prominence in the last couple of decades with immense applicability in diverse areas such as granular and glassy systems, control theory, and biological systems. Many biologically crucial processes, such as the dynamics in the cellular cytoskeleton, the collective motion of cells, the progression of cancer, and the large-scale migratory behavior of bird flocks, mammal herds, ants, and fish schools have been modeled within the active matter framework.The meeting will feature research talks that will discuss recent progress in the field. Additionally, several pedagogical talks will provide a broad overview for young researchers in the country and help develop an appreciation for the significant new developments. The meeting will provide ample opportunities for young researchers to ...