##### Department of Mathematics,

University of California San Diego

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### Zoom for Thought

## Cameron Cinel - Ph.D. student

#### UC San Diego

## Polynomial Identity Algebras and the Kurosh Problem

##### Abstract:

The Kurosh problem can be seen as an analogue of the Burnside problem for algebras. It asks whether or not a finitely generated algebra over a field has finite dimension. While the answer is negative in general, you don't have to go all the way to \textbf{Chinatown} to find a class of algebras for which the answer is affirmative. \\ \\ In this talk, we will show that for algebras satisfying a polynomial identity (PI algebras), the Kurosh problem is true. Along the way, we will have a \textbf{(The) Conversation} about the basics of the theory of PI algebras, discussing their properties, constructing specific identities for classes of algebras, and looking at their structure. Time permitting, before we say our \textbf{(Long) Goodbyes} we will also look at another type of \textbf{Nice (Guys)} identities: central polynomials. Additionally, we will use them to prove a not so \textbf{Small (Town Crime)} result, Rowan's theorem. \\ \\ Hopefully, after this talk, as \textbf{Twilight} turns into \textbf{Night (Moves)}, you will go to \textbf{(The Big) Sleep} dreaming about PI algebras.

### January 19, 2021

### 1:00 PM

### Please see email with subject ``Zoom for Thought Information."

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