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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...

Frustrated quantum magnets provide a rich playground for interplay of quantum fluctuations and competing interactions  and hence provide important theoretical and experimental settings to  explore unconventional phases of condensed matter including those with subtle signatures of many-body quantum entanglement. The developments in the field is a direct outcome of successful exchange of ideas between physics and material sciences and benefits spectacular progress in our understanding of controlled synthesis of materials with interesting properties many of which have promising technological applications.The program, second of its kind, aims to bring together physicists, chemists and material scientists working on various aspects of magnetism in the Asia-Pacific region in particular to exchange results on the recent developments in different areas of quantum magnetism as well as discuss new ideas at the frontiers of both theoretical as well as experimental research in the field. It also a...

Frustrated quantum magnets provide a rich playground for interplay of quantum fluctuations and competing interactions  and hence provide important theoretical and experimental settings to  explore unconventional phases of condensed matter including those with subtle signatures of many-body quantum entanglement. The developments in the field is a direct outcome of successful exchange of ideas between physics and material sciences and benefits spectacular progress in our understanding of controlled synthesis of materials with interesting properties many of which have promising technological applications.The program, second of its kind, aims to bring together physicists, chemists and material scientists working on various aspects of magnetism in the Asia-Pacific region in particular to exchange results on the recent developments in different areas of quantum magnetism as well as discuss new ideas at the frontiers of both theoretical as well as experimental research in the field. It also a...

Artificial intelligence (AI) is undergoing an explosive phase - machines can now accomplish complex specific tasks at a level that exceeds human skills. At the basis of this performance is the ability to understand sensory input from the external world and to associate it with effective strategies to achieve a desired goal.This advanced school aims to combine different yet strongly coupled perspectives: first, theoretical approaches, which focus on principles, algorithms, and their applications to computer science; second, the relationship with experimental neuroscience, which has inspired the latest generation developments in AI and has, in turn, benefited from the ability of AI to investigate the computations underlying complex cognitive processes; third, applications such as robotics, gaming, etc.The topics covered include:Deep learning and its relation to vision and languageReinforcement learning and decision makingSensorimotor learningThe ethics of artificial intelligence and its ...