Day 1
Francis Bischoff
Title: The derived moduli stack of logarithmic flat connections
Abstract: I will present an explicit finite-dimensional model for the derived moduli stack of flat connections on Ck with logarithmic singularities along a weighted homogeneous Saito free divisor. I will focus in particular on the example of plane curve singularities of the form xp = yq. These moduli spaces are conjectured to admit shifted Poisson structures. I will discuss this conjectural picture and present some partial results.
Song Gao
Title: Coisotropicity of Fixed points under torus action on the variety of Lagrangian subalgebras
Abstract: I will talk about my recent study of coisotropic subalgebras of Lie bialgebras. Given a complex semisim- ple Lie algebra g with adjoint group G, the set of coisotropic subalgebras of g form an algebraic variety, which is called the variety of coisotropic subalgebras. Let H be a fixed maximal torus of G. I will introduce my results on fixed points of H-action on the variety of coisotropic subalgebras. Approaches of toric varieties and algebraic groups will be used.
Yu Li
Title: Integrable systems on the dual of nilpotent Lie subalgebras and T -Poisson cluster structures
Abstract: Let g be a semisimple Lie algebra and g = n ⊕ h ⊕ n− a triangular decomposition. Motivated by a construction of Kostant-Lipsman-Wolf, we construct an integrable system on the dual space of n− equipped with the Kirillov-Kostant Poisson structure. The Bott-Samelson coordinates on the open Bruhat cell (equipped with the standard Poisson structure) makes it into a symmetric Poisson CGL extension, hence giving rise to a T-Poisson seed on it. We explain a relation between our integrable system and this T -Poisson seed. This is joint work in progress with Yanpeng Li and Jiang-Hua Lu.
Joshua Mundinger
Title: Quantization of restricted Lagrangian subvarieties in positive characteristic
Abstract: Bezrukavnikov and Kaledin introduced quantizations of symplectic varieties X in positive characteristic which endow the Poisson bracket on X with the structure of a restricted Lie algebra. We consider deformation quantization of line bundles on Lagrangian subvarieties Y of X to modules over such quantizations. If the ideal sheaf of Y is a restricted Lie subalgebra of the structure sheaf of X, we show that there is a certain cohomology class which vanishes if and only if a line bundle on Y admits a quantization.
Pavel Safronov
Title: Local structure of holomorphic Lagrangians and shifted Poisson geometry
Abstract: A Lagrangian submanifold of a real symplectic manifold has a neighborhood symplectomorphic to a neighborhood of the zero section of the cotangent bundle of the Lagrangian. Such a result is no longer true when working with complex manifolds. I will describe the obstruction in terms of a shifted Poisson structure. I will also present several related settings (Poisson-Lie groups, critical loci of Morse–Bott functions), where similar obstructions appear.
Theodore Voronov
Title: L-infinity comorphisms and duality for L-infinity algebroids and P-infinity manifolds
Abstract: For Lie algebroids, there are two notions of a morphism: "`morphisms" and "comorphisms" (due to Higgins--Mackenzie). Comorphisms are dual to Poisson maps under functorial constructions E ↦ E* and M ↦ T*M. We introduce a new notion of L-infinity comorphisms of L-infinity algebroids (also its thick version) and show how the functorial relations between Lie algebroids and Poisson manifolds extend to the categories (P-infinity manifolds, P-infinity morphisms) and (L-infinity algebroids, L-infinity comorphisms).
(Based on joint work in progress with Samuel Brady.)