STARS: Superalgebra Theory and Representations Seminar

A quasi-morphism on a group G​​ is a real-valued function which satisfies the homomorphism equation up to a bounded error. Let N_g​​ be a non-orientable surface of genus g≥ 3​​ and let Homeo₀(N_g,μ)​​ be the identity component of the group of measure-preserving homeomorphisms of N_g​​.
We prove that the space of homogeneous quasi-morphisms on the group Homeo₀(N_g,μ)​​ is infinite-dimensional. This project is part of an M.Sc. thesis under the supervision of Dr. Brandenbursky. 

This is the first lecture of STARS (Superalgebra Theory and Representations Seminar) series. 

Meeting ID: 919 8455 8186
Password: 663945
https://weizmann.zoom.us/j/91984558186?pwd=dzl2Z2hPRWtzaDF6R1pySXd4OGFIQT09  

Consider the function field F of a smooth curve over F_q, with q > 2.    

L-functions of automorphic representations of GL(2) over F are important objects for studying the arithmetic properties of the field F. Unfortunately, they can be defined in two different ways: one by Godement-Jacquet, and one by Jacquet-Langlands. Classically, one shows that the resulting L-functions coincide using a complicated computation.

I will present a conceptual proof that the two families coincide, by categorifying the question. This correspondence will necessitate comparing two very different sets of data, which will have significant implications for the representation theory of GL(2). In particular, we will obtain an exotic symmetric monoidal structure on the category of representations of GL(2).

No prior knowledge of automorphic forms is assumed. This work is a part of my PhD thesis under the supervision of J. Bernstein.

Root-reductive Lie algebras are relatively well behaved, countable-dimensional Lie algebras.  In this talk, we shall discuss the definition of these Lie algebras as well as their important subalgebras---Borel subalgebras, Cartan subalgebras, and splitting maximal toral subalgebras.  We shall also dwell on some representations of these Lie algebras if time permits. 

I will discuss some ideas from a recent breakthrough in the field of Lie superalgebras:

https://arxiv.org/abs/1610.08470

No prior knowledge on Lie superalgebras will be assumed. 

The classical Wedderburn-Artin theorem describes associative rings A for which the category of A-modules is semisimple. This turns out to be a left-right symmetric property: the category of left A-modules is semisimple if and only if the category of right A-modules is. In this talk, I will present a generalization of the Wedderburn-Artin theory to topololgical associative rings R in which open right ideals form a base of neighborhoods of zero. The talk will start with a discussion of split and semisimple abelian categories and end with a description of topological rings R for which the category of left R-contramodules is split (or equivalently, semisimple) or, equivalently, the category of discrete right R-modules is split (or equivalently, semisimple).

In my previous talk, I described the partial L-functions of cuspidal automorphic representations and said that their conjectured meromorphic continuation is important for the theory of automorphic forms. I then talked about integral representations as a method for proving this meromorphic continuation.

In this talk (which will be mostly independent of the previous one), I will describe a softened version of functoriality (which is the main theme in the Langlands program) and how L-functions help to detect and construct instances of functoriality.

In this talk, we will present an overview of the doubling construction for integral representations of L-functions. An explicit construction will be presented for quasi-split special orthogonal groups.