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Algebraic complexity aims at understanding the computational aspects of algebraic objects such as multivariate polynomials, tensors etc. The primary focus in this field has been the study of multivariate polynomials, and its hardness based on the number of addition/multiplication operations required to compute it (i.e. via `algebraic circuits'). This is a quest to answer the ``VP vs VNP'' question, an algebraic analogue of the classical ``P vs NP'' question and is a fundamental open problem in algebraic complexity.These questions about multivariate polynomial broadly fall in one of the following categories:Lower bounds: Can we find explicit polynomials that cannot be computed by small algebraic circuits?Polynomial Identity Testing: Given an algebraic circuit C, can we test efficiently if the circuit C is computing the zero polynomial? (Or equivalently, can we test if two circuits C_1 and C_2 are computing the same polynomial?)Reconstruction: Given blackbox access to a circuit C, can we...

Algebraic complexity aims at understanding the computational aspects of algebraic objects such as multivariate polynomials, tensors etc. The primary focus in this field has been the study of multivariate polynomials, and its hardness based on the number of addition/multiplication operations required to compute it (i.e. via `algebraic circuits'). This is a quest to answer the ``VP vs VNP'' question, an algebraic analogue of the classical ``P vs NP'' question and is a fundamental open problem in algebraic complexity.These questions about multivariate polynomial broadly fall in one of the following categories:Lower bounds: Can we find explicit polynomials that cannot be computed by small algebraic circuits?Polynomial Identity Testing: Given an algebraic circuit C, can we test efficiently if the circuit C is computing the zero polynomial? (Or equivalently, can we test if two circuits C_1 and C_2 are computing the same polynomial?)Reconstruction: Given blackbox access to a circuit C, can we...

The focal area of the program lies at the juncture of three areas: Probability theory of random walks,Ergodic theory of flows on negatively curved spaces,Gromov hyperbolic groups.Random walks on finite and infinite, finitely generated groups is a topic of considerable vintage. It is well-known, starting with Kesten's characterization of amenability, that asymptotic properties of such random walks are intimately connected to the large scale geometry of the underlying group. Vershik and Kaimanovich (1983) introduced entropic techniques to study the Poisson boundary of random walks on countable discrete groups, which is a natural measure theoretic space "at infinity" associated with the random walk. In a seminal paper in 2000, Kaimanovich gave a very general sufficient condition on the one step distribution of the walk on a hyperbolic group for the Poisson boundary to equal the (geometric) Gromov boundary. Further probabilistic methods have started being applied to hyperbolic groups relat...

The focal area of the program lies at the juncture of three areas: Probability theory of random walks,Ergodic theory of flows on negatively curved spaces,Gromov hyperbolic groups.Random walks on finite and infinite, finitely generated groups is a topic of considerable vintage. It is well-known, starting with Kesten's characterization of amenability, that asymptotic properties of such random walks are intimately connected to the large scale geometry of the underlying group. Vershik and Kaimanovich (1983) introduced entropic techniques to study the Poisson boundary of random walks on countable discrete groups, which is a natural measure theoretic space "at infinity" associated with the random walk. In a seminal paper in 2000, Kaimanovich gave a very general sufficient condition on the one step distribution of the walk on a hyperbolic group for the Poisson boundary to equal the (geometric) Gromov boundary. Further probabilistic methods have started being applied to hyperbolic groups relat...