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The primary focus of this program will be on the theme of universality in the following three different classes of discrete random structures. All three are active areas of ongoing research.(1) Randomly growing interfaces and (1+1) dimensional polymer models: A large class of models in this area are believed to exhibit the so-called KPZ universality. Despite intense activity in the last decade, which saw immense progress in the study of exactly solvable models, the understanding of universality beyond integrable models remains rather limited. (2) Eigenvalues of random matrices and other point processes: Random matrix theory is an area where universality has been shown in non-integrable settings. This owes to fundamental progress in techniques in the last 10 years. Relationships between different aspects of random matrix theory and other branches of probability, or even mathematics at large, continue to be actively explored and developed.(3) Sandpile models and other Laplacian growth mo...

The primary focus of this program will be on the theme of universality in the following three different classes of discrete random structures. All three are active areas of ongoing research.(1) Randomly growing interfaces and (1+1) dimensional polymer models: A large class of models in this area are believed to exhibit the so-called KPZ universality. Despite intense activity in the last decade, which saw immense progress in the study of exactly solvable models, the understanding of universality beyond integrable models remains rather limited. (2) Eigenvalues of random matrices and other point processes: Random matrix theory is an area where universality has been shown in non-integrable settings. This owes to fundamental progress in techniques in the last 10 years. Relationships between different aspects of random matrix theory and other branches of probability, or even mathematics at large, continue to be actively explored and developed.(3) Sandpile models and other Laplacian growth mo...

The great observational progress in cosmology has revealed some very intriguing puzzles, the most important of which are the existence of new mysterious components of the Universe: the dark matter and the dark energy. While a standard model of cosmology — the Lambda cold dark matter (LCDM) paradigm — has emerged, many puzzles are as yet unsolved. Is dark energy really just a cosmological constant (Lambda), or is it something dynamical, perhaps even a clue that Einstein’s general relativity needs modifications at cosmological scales? Is dark matter a new particle beyond the Standard Model, and do its microscopic properties leave any imprint at cosmological or galactic scales (e.g., deep inside galaxy clusters, or in dwarf galaxies)? The nature of dark matter would also have very interesting consequences for the reionization history of the Universe, which is already being constrained by observations of high redshift quasars and galaxies and is expected to be determined in considerable de...

The great observational progress in cosmology has revealed some very intriguing puzzles, the most important of which are the existence of new mysterious components of the Universe: the dark matter and the dark energy. While a standard model of cosmology — the Lambda cold dark matter (LCDM) paradigm — has emerged, many puzzles are as yet unsolved. Is dark energy really just a cosmological constant (Lambda), or is it something dynamical, perhaps even a clue that Einstein’s general relativity needs modifications at cosmological scales? Is dark matter a new particle beyond the Standard Model, and do its microscopic properties leave any imprint at cosmological or galactic scales (e.g., deep inside galaxy clusters, or in dwarf galaxies)? The nature of dark matter would also have very interesting consequences for the reionization history of the Universe, which is already being constrained by observations of high redshift quasars and galaxies and is expected to be determined in considerable de...

ML (Machine Learning) has enjoyed tremendous practical success in the last decade with applications ranging from e-commerce to self-driving cars. The success of deep networks in vision and speech recognition are particularly notable examples. However, the theoretical understanding and characterization of these techniques has not kept pace with the real-world achievements of ML. Traditional approaches such as generalization bounds, stability-based justifications, capacity arguments, regularization etc., have only gone part way towards rationalizing the uncanny success of modern ML methods. In addition there are a number of empirical phenomena such as adversarial examples, effectiveness of gradient descent, failure of NLP GANS, etc., deserving of deeper observational scrutiny and rigorous theoretical treatment. Lastly, there are a number of desirable properties such as explainability, debuggability, ability to selectively ignore discriminatory biases in data, verifiability, etc., that st...

ML (Machine Learning) has enjoyed tremendous practical success in the last decade with applications ranging from e-commerce to self-driving cars. The success of deep networks in vision and speech recognition are particularly notable examples. However, the theoretical understanding and characterization of these techniques has not kept pace with the real-world achievements of ML. Traditional approaches such as generalization bounds, stability-based justifications, capacity arguments, regularization etc., have only gone part way towards rationalizing the uncanny success of modern ML methods. In addition there are a number of empirical phenomena such as adversarial examples, effectiveness of gradient descent, failure of NLP GANS, etc., deserving of deeper observational scrutiny and rigorous theoretical treatment. Lastly, there are a number of desirable properties such as explainability, debuggability, ability to selectively ignore discriminatory biases in data, verifiability, etc., that st...

This discussion meeting is aimed at promoting fruitful interactions between theoretical physicists and mathematicians in areas that could be of common interest. In the past few decades, the interaction between algebraic geometry and quantum field theory has contributed substantial insights to both areas. Now seems to be an opportune moment to extend the areas of  cross-fertilisation to include arithmetic geometry, comprising the study of arithmetic schemes and their Diophantine geometry, the theory of Galois representations, and the arithmetic Langlands programme.The ICTS discussion meeting will focus on some ideas that Minhyong Kim and others have been pursuing (see Link). There are analogies between the moduli spaces of arithmetic principal bundles and constructions in quantum field theory, in particular, Chern-Simons theory. The idea of the meeting is to have about a dozen number theorists and string theorists (and some other local participants) to explore this common ground. There ...

This discussion meeting is aimed at promoting fruitful interactions between theoretical physicists and mathematicians in areas that could be of common interest. In the past few decades, the interaction between algebraic geometry and quantum field theory has contributed substantial insights to both areas. Now seems to be an opportune moment to extend the areas of  cross-fertilisation to include arithmetic geometry, comprising the study of arithmetic schemes and their Diophantine geometry, the theory of Galois representations, and the arithmetic Langlands programme.The ICTS discussion meeting will focus on some ideas that Minhyong Kim and others have been pursuing (see Link). There are analogies between the moduli spaces of arithmetic principal bundles and constructions in quantum field theory, in particular, Chern-Simons theory. The idea of the meeting is to have about a dozen number theorists and string theorists (and some other local participants) to explore this common ground. There ...

The study of spaces of complex structures on a Riemann surface, the so-called Moduli space and Teichmüller space is a classical and well-studied area of mathematics, with relations and interconnections with  different areas of mathematics and also  theoretical physics.  In the case of surfaces with genus at least two, complex structures can be uniformized to hyperbolic structures, which are discrete, faithful representations of surface groups in the group of  isometries of the hyperbolic plane. A natural generalization is to consider surface group representations in other semisimple Lie groups.  In the last few years, spectacular advances have been made towards generalizing existing tools and techniques to the study of these representations, and their moduli spaces. Remarkably, in many cases there is a natural generalization of discrete, faithful representations which provides an analogue of Teichmüller space. Our program shall focus on two perspectives:The dynamic point of view, leadi...

The study of spaces of complex structures on a Riemann surface, the so-called Moduli space and Teichmüller space is a classical and well-studied area of mathematics, with relations and interconnections with  different areas of mathematics and also  theoretical physics.  In the case of surfaces with genus at least two, complex structures can be uniformized to hyperbolic structures, which are discrete, faithful representations of surface groups in the group of  isometries of the hyperbolic plane. A natural generalization is to consider surface group representations in other semisimple Lie groups.  In the last few years, spectacular advances have been made towards generalizing existing tools and techniques to the study of these representations, and their moduli spaces. Remarkably, in many cases there is a natural generalization of discrete, faithful representations which provides an analogue of Teichmüller space. Our program shall focus on two perspectives:The dynamic point of view, leadi...

In 2000, S. Fomin and A. Zelevinsky introduced Cluster Algebras as abstractions of a combinatoro-algebraic phenomenon that is found in many different areas throughout mathematics. Defined as a sub-algebra of the ambient field of rational functions in finitely many variables, it is generated by union of cluster variables. The cluster variables are distributed across clusters. The clusters arise from an original "seed'' by a process known as mutation. For example, given a regular polygon with n sides the triangulations of this polygon with non-crossing diagonals can be obtained from a given triangulation of this type by a sequence of diagonal flips. Combinatorial data at a given cluster may be defined in terms of a quiver or alternatively a skew-symmetric matrix and using this quiver or matrix, the other clusters may be obtained by mutations. The clusters can be visualized as a graph with vertices being the clusters and edges being the mutations.As it turns out the coordinate rings of Gr...

In 2000, S. Fomin and A. Zelevinsky introduced Cluster Algebras as abstractions of a combinatoro-algebraic phenomenon that is found in many different areas throughout mathematics. Defined as a sub-algebra of the ambient field of rational functions in finitely many variables, it is generated by union of cluster variables. The cluster variables are distributed across clusters. The clusters arise from an original "seed'' by a process known as mutation. For example, given a regular polygon with n sides the triangulations of this polygon with non-crossing diagonals can be obtained from a given triangulation of this type by a sequence of diagonal flips. Combinatorial data at a given cluster may be defined in terms of a quiver or alternatively a skew-symmetric matrix and using this quiver or matrix, the other clusters may be obtained by mutations. The clusters can be visualized as a graph with vertices being the clusters and edges being the mutations.As it turns out the coordinate rings of Gr...