Abstract: What are the noncommutative rings that are most analogous to polynomial rings? One class of such rings are the regular algebras first defined by Artin and Schelter in 1987. Since then such algebras have been extensively studied. We give a survey of these interesting examples and their associated projective geometry.

  Abstract: Given a topological group G, one considers the space of its closed subgroups, called the Chabauty space. I will talk about the structure and features of this space, and show how various algebraic and topological properties of a group are expressed there.

  Abstract: Basics of random-walks in a finite group, super-approximation, and recent developments in this subject will be discussed. (More recent results are parts of my joint works with Srivatsa Srinivas.)

  Abstract: Tensor categories have played an important role in areas as diverse as topology, mathematical physics, operator algebras and representation theory. This is an introductory talk. I will mostly talk about classification of tensor categories with given tensor product rules and module categories for certain important examples.

  Abstract: The growth of an infinite-dimensional algebra is a fundamental tool to measure its infinitude. Growth of noncommutative algebras plays an important role in noncommutative geometry, representation theory, differential algebraic geometry, symbolic dynamics and various recent homological stability results in number theory and arithmetic geometry. We analyze the space of growth functions of algebras, answering a question of Zelmanov (2017) on the existence of certain 'holes' in this space, and prove a strong quantitative version of the Kurosh Problem on algebraic algebras. We use minimal subshifts with highly correlated oscillating complexities to resolve a question posed by Krempa-Okninski (1987) and Krause-Lenagan (2000) on the GK-dimension of tensor products. An important property implied by subexponential growth (for both groups and algebras) is amenability. We show that minimal subshifts of positive entropy give rise to amenable graded algebras of exponential growth, answering a conjecture of Bartholdi (2007; naturally extending a wide open conjecture of Vershik on amenable group rings). This talk is partially based on joint works with J. Bell and with E. Zelmanov.

  Abstract: The Balmer spectrum of a tensor triangulated category is a topological tool analogous to the usual spectrum of a commutative ring. It provides a universal theory of support, giving a categorical framework to (among others) the support varieties that have been used to great effect in modular representation theory. In this talk I will present an approach to the Balmer spectrum of the stable module category of a Hopf algebra using twisted tensor products and emphasizing examples. This will include an unpretentious introduction to twisted tensor products, the Balmer spectrum, and the relevance of both in representation theory.

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