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
Manchester Building (Hall 2), Hebrew University Jerusalem
The rational solutions on an elliptic curve form a finitely generated abelian group, but the maximum number of generators needed is not known. Goldfeld conjectured that if one also fixes the j-invariant (i.e. the complex structure), then 50% of such curves should require 1 generator and 50% should have only the trivial solution. Smith has recently made substantial progress towards this conjecture in the special case of elliptic curves in Legendre form. I'll discuss recent work with Lemke Oliver, which bounds the average number of generators for general j-invariants.
For $\kappa < \lambda$ infinite cardinals let us consider the following generalization of the Lowenheim-Skolem theorem: "For every algebra with countably many operations over $\lambda^+$ there is a sub-algebra with order type exactly $\kappa^+$".
We will discuss the consistency and inconsistency of some global versions of this statement and present some open questions.
The goal of my talk will be to formulate local and global Langlands conjectures
Key words: smooth representations of p-adic groups, supercuspidal representations,
automorphic representations, cusp forms, Galois representations.
Abstract: I will discuss applications of algebraic results to combinatorics, focusing in particular on Lefschetz theorem, Decomposition theorem and Hodge Riemann relations. Secondly, I will discuss proving these results combinatorially, using a technique by McMullen and extended by de Cataldo and Migliorini. Finally, I will discuss Lefschetz type theorems beyond positivity.
Recommended prerequisites: basic commutative algebra