Growth, dynamics, and approximations of infinite-dimensional algebras

The growth of an infinite-dimensional algebra is a fundamental tool to 'measure its infinitude'. Growth of algebras plays an important role in noncommutative geometry, representation theory, differential algebraic geometry, symbolic dynamics and various homological stability results in number theory and arithmetic geometry.
We analyze the space of growth functions of algebras, answering a question of Zelmanov on the existence of certain holes in this space. We then prove a strong quantitative version of the Kurosh Problem on algebraic algebras.
An important property implied by subexponential growth (both for groups and for 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 (naturally extending a wide open conjecture of Vershik on amenable group rings).
Finally, we discuss sofic algebras, that is, algebras which can be approximated by almost-representations. We answer a question of Arzhantseva and Paunescu on soficity and stable finiteness, and discuss the connection with the soficity of groups.
This talk is partially based on joint works with J. Bell and with E. Zelmanov.

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https://us02web.zoom.us/j/87856132062

Meeting ID: 878 5613 2062