Title: Combinatorics and geometry in non-commutative algebraic geometry

Abstract:

Algebraic geometry studies the structure of sets of solutions (varieties) over fields and commutative rings. Starting in the early 1960's ring theorists (Cohn, Bergman and others) have tried to study the structure of varieties over non-commutative rings (notably free associative algebras). The lack of unique factorization that they tackled and studied in detail, and the pathologies that they were aware of, prevented any attempt to prove or even speculate what can be the properties of such varieties.

Using combinatorial and geometric techniques that are partly based on concepts and results from geometric group theory and from low dimensional topology, we study the structure of these varieties.

We manage to formulate concrete conjectures on this structure, and slowly prove preliminary results in the direction of these conjectures.

No prior knowledge will be assumed.

Abstract:

Algebraic geometry studies the structure of sets of solutions (varieties) over fields and commutative rings. Starting in the early 1960's ring theorists (Cohn, Bergman and others) have tried to study the structure of varieties over non-commutative rings (notably free associative algebras). The lack of unique factorization that they tackled and studied in detail, and the pathologies that they were aware of, prevented any attempt to prove or even speculate what can be the properties of such varieties.

Using combinatorial and geometric techniques that are partly based on concepts and results from geometric group theory and from low dimensional topology, we study the structure of these varieties.

We manage to formulate concrete conjectures on this structure, and slowly prove preliminary results in the direction of these conjectures.

No prior knowledge will be assumed.

## Date:

Mon, 04/11/2019 - 10:00 to 12:00

## Location:

CS building, C-400