Speaker: Zlil Sela
Title: Basic conjectures and preliminary results in non-commutative algebraic geometry

 Abstract: Algebraic geometry studies the structure of varieties over
 fields and commutative rings. Starting in the 1960's ring theorists
 (Cohn, Bergman and others) have tried to study the structure of varieties
 over some 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 techniques and concepts from geometric group theory and from low
 dimensional topology, we formulate concrete conjectures about the
 structure of these varieties, and prove preliminary results in the
 direction of these conjectures. Further (possible) applications to the
 study of representations into free associative algebras will also be
 discussed.

 We intend to survey the relevant background from geometric group theory,
 and some from low dimensional topology, including the structure of
 varieties over free groups and semigroups that motivate the conjectures.