• Speaker: Theodore Voronov, University of Manchester
• Title:  From homotopy Lie brackets to thick morphisms of supermanifolds and non-linear functional-algebraic duality (NOTE UNUSUAL DAY)
• Date: 01/31/2023
• Time: 3:00 PM - 4:00 PM
• Place: C204A Wells Hall
• Contact: Michael Shapiro (mshapiro@msu.edu)

I will give a motivation for homotopy Lie brackets and the corresponding morphisms preserving brackets "up to homotopy" (more precisely, for L-infinity morphisms and L-infinity algebras), and show how to describe them using supergeometry. So, instead of a single Poisson or Lie bracket, there is a whole sequence of operations with n arguments, n=1,2,3,..., satisfying a linked infinite sequence of identities replacing the familiar Jacobi identity for a Lie bracket; and, instead of a morphism as a linear map mapping a bracket to a bracket, there is a sequence of multi-linear mappings mixing brackets with different numbers of arguments, and, in particular, the binary bracket is preserved only up to an (algebraic) homotopy. Geometrically, such a sequence of multi-linear mappings assembles into one non-linear map of supermanifolds. For the case of homotopy brackets of functions ("higher Poisson" or "homotopy Poisson" structure), this leads us to the question about a natural construction of non-linear mappings between algebras of smooth functions generalizing the usual pull-backs. I discovered such a construction some years ago. These are "thick morphisms" of (super)manifolds generalizing ordinary smooth maps. From a more general perspective, we arrive in this way at a non-linear analog of the classical functional-algebraic duality between spaces and algebras.

• Speaker: Peikai Qi, MSU
• Title: Iwasawa lambda-invariants and Massey products
• Date: 02/20/2023
• Time: 3:00 PM - 4:00 PM
• Place: C304 Wells Hall
• Contact: Preston Wake (wakepres@msu.edu)

No abstract available.

• Speaker: Ivan Loseu, Yale University
• Title: Unipotent representations and quantization
• Date: 03/13/2023
• Time: 3:00 PM - 4:00 PM
• Place: C304 Wells Hall
• Contact: Igor Rapinchuk (rapinchu@msu.edu)

This talk is aimed more at the general audience. A fundamental question in the representation theory of semisimple Lie groups is to classify their irreducible unitary representations. A guiding principle here is the Orbit method, first discovered by Kirillov in the 60's for nilpotent Lie groups. It states that the irreducible unitary representations should be related to coadjoint orbits, i.e., the orbits of the Lie group action in the dual of its Lie algebra. Passing from orbits to representations could be thought of as a quantization problem and it is known that in this setting this is very difficult. For semisimple Lie groups it makes sense to speak about nilpotent orbits, and one could try to study representations that should correspond to these orbits via the yet undefined Orbit method. These representations are called unipotent: they are expected to be nicer than general ones, while one hopes to reduce the study of general representations to that of unipotent ones. I will concentrate on the case of complex Lie groups. I will explain how recent advances in the study of deformation quantizations of singular symplectic varieties allow to define unipotent representations and obtain some results about them. The talk is based on the joint work with Lucas Mason-Brown and Dmytro Matvieievskyi.

• Speaker: Jaclyn Lang, Temple
• Title: TBA (note unusual day)
• Date: 04/21/2023
• Time: 3:00 PM - 4:00 PM
• Place: C304 Wells Hall
• Contact: Preston Wake (wakepres@msu.edu)

No abstract available.