Schedule for: 25w5442 - Around Singularities in Poisson Geometry

A set dinner is served daily between 5:30pm and 7:30pm in the Xianghu Lake National Tourist Resort.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

This talk aims to give an introduction to deformation problems with an emphasis on Poisson and related geometries. It will begin with a general overview of deformation theory, introducing differential graded Lie algebras (dgLas) and \(L_\infty-\)algebras as the central tools. We'll discuss how these algebraic structures "control" deformation problems, exploring concepts like Maurer-Cartan elements, their gauge equivalence, and the nature of infinitesimal deformations and their obstructions. Building on this foundation, we will then focus on the deformation problem of a Poisson structure. It is controlled by the associated Koszul dgLa and, in general, is obstructed. The Poisson moduli space on a compact manifold has been explicitly described in only a few instances: in the neighborhood of a symplectic structure, around a log Poisson structure (after Marcut and Osorno Torres), and around the Lie-Poisson sphere of a compact semisimple Lie algebra (after Marcut). If time permits, we will also briefly extend to the deformation theory of foliations and symplectic foliations. This talk aims to provide a solid conceptual understanding for those new to the subject and motivate further exploration of this active research area.

Building on the ideas and tools introduced in Alfonso's lecture, I will discuss the key modifications and new phenomena that arise when passing from the real to the complex setting. After defining a suitable version of Poisson cohomology, I will present the Bogomolov-Tian-Todorov theorem in the context of holomorphic symplectic manifolds. This celebrated result asserts that, under certain conditions, a first-order solution to the Maurer-Cartan equation is unobstructed -- that is, it extends to an exact solution. I will then proceed to discussing broader situations where one still encounters unobstructedness: Hitchin's theorem on deformations of holomorphic Poisson structures, a complex analogue of Marcut-Osorno Torres result from Alfonso's lecture, and finally the case of complex log symplectic manifolds with normal crossings polar divisors. Parts of this lecture are based on joint projects with Brent Pym and Travis Schedler, as well as with Jiang-Hua Lu.

Lunch is served daily between 11:30am and 1:30pm in the Xianghu Lake National Tourist Resort

The use of differential graded Lie algebras (dgLas) and L-infinity-algebras in deformation theory goes back to Kodaira and Spencer, and a general principle was postulated by Deligne, among others: Given a deformation problem, there is an L-infinity-algebra such that solutions are Maurer-Cartan elements, and equivalence of solutions is given by gauge equivalence in the L-infinity-algebra. Using the L-infinity algebra g, one can address the question of local rigidity: Given a geometric structure \(Q_0\), when are small deformations of \(Q_0\) equivalent to \(Q_0\)? More generally, we can study the question of local stability of a property: Given a geometric structure \(Q_0\), satisfying a property P, when do small deformations of \(Q_0\) also possess property P up to equivalence? We show that in examples, the property P can be encoded into the choice of a L-infinity-subalgebra h. Using this description, the stability question translates to the following: Given a Maurer-Cartan element \(Q_0\) in h, when are all Maurer-Cartan elements near \(Q_0\) in g equivalent to an element of h? Under some analytical conditions, we give a sufficient criterion for a positive answer to the stability question, which we then use to obtain a stability criterion for stability of compact leaves of Dirac structures.

Nanyan Xu: Leibniz 2-algebras, linear 2-racks and the Zamolodchikov Tetrahedron equation Xuedan Luo: Polyubles of pre-Lie bialgebras Qiao Li: Poisson relations between some singularity data of meromorphic systems Shining Wang: Relative Rota-Baxter Operators on Lie 2-groups Zihang Liu: Poisson deformation and cluster degeneration of Bott-Samelson cells

Proper maps in various categories studied in singularity theory (for example, the real analytic category) are known to be constructible, in the sense that the image of the map can be stratified in such a way that the map is a topological fiber bundle over each stratum. In this talk we will present an equivariant analogue of this fact for momentum maps of Hamiltonian actions by compact Lie groups. This provides insight into when the symplectic reduced spaces change as we vary the value at which we perform reduction. In particular, they will not change as topological spaces when the reduction value stays within a stratum. After discussing this, we will explain a stratum-wise version of the Duistermaat-Heckman linear variation theorem and, time permitting, we will touch upon an instance of a local invariant cycle type theorem for momentum maps.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

A normal form for a geometric structure is, roughly speaking, a particularly simple expression for it in a well chosen coordinate chart. Normal forms play an important role in Poisson geometry, as a way to classify the local singularities of Poisson brackets, and control the impact of these singularities on the global geometry. I will give an introduction to the subject, covering various known results and techniques for proving them, with a special emphasis on the role of cohomological techniques.

In this talk, largely based on previous work with Lima and Meinrenken, I will discuss how Euler-like vector fields can be used as a tool for obtaining normal form results in Poisson geometry (and beyond). As an application, I will illustrate the method with a normal form result for Poisson structures around cosymplectic submanifolds, originally due to Frejlich and Marcut.

We will discuss how to classify singular foliations in a formal setting, given a fixed leaf with a given transverse structure at a point. One way to classify those is via multiplicative Yang-Mills connections, a connection appearing in curved gauge theory; that is, with tools from a generalised version of gauge theory we will argue that there are not as many foliations as one might think. This is a joint work with Camille Laurent-Gengoux from Université de Lorraine.

Anastasios Fotiadis: Universal central extensions of the Lie algebra of contact vector fields. Sambit Senapati: Generalized Complex structures around symplectic leaves Lennart Obster: Linearising Lie algebroids Boris Zupancic: Weighted Blowups for Poisson Resolution. Nikolay Grantcharov: q-Differential operators on the base affine space G/U

Drinfeld classified Poisson homogeneous spaces of a Poisson Lie group in terms of Dirac structures of the Lie bialgebra. In this talk, we study homogeneous spaces of a 2-group and develop Drinfeld theorem in the Poisson 2-group context.

Given a Poisson manifold that admits a proper symplectic integration, we introduce a canonical resolution -- a certain regular Dirac manifold -- which we call the Weyl resolution. We describe this construction and use it to define the Weyl group associated with such a Poisson manifold. We will explain the geometry underlying this group, which turns out to be a Coxeter group acting by reflections on an integral affine manifold: the universal orbifold covering space of the Poisson manifold leaf space. We will illustrate these ideas with several examples. This talk is based on joint work with Marius Crainic (Utrecht) and David Martínez-Torres (Madrid).

This is an introductory lecture. Symplectic singularities play an important role in Poisson geometry, algebraic geometry, geometric representation theory and other related fields. In this lecture we describe the relationship between birational geometry and deformation theory in terms of a symplectic resolution (or more generally a {\bf Q}-factorial terminalization) of a symplectic singularity. As an application we also mention my recent work with Y. Odaka on Kaledin's conjecture on a torus action of a symplectic singularity.

This is a joint work with C. Laurent-Gengoux. We construct a finite-dimensional higher Lie groupoid integrating a singular foliation \(F\), \[ \cdots K_3 \stackrel{\scriptsize\rightrightarrows}{\scriptsize\rightrightarrows} K_2 \stackrel{\scriptsize\longrightarrow}{\rightrightarrows} K_1 \rightrightarrows M \] under the mild assumption that the latter admits a geometric resolution. More precisely, a recursive use of bi-submersions, a tool coming from non-commutative geometry and invented by Androulidakis and Skandalis, allows us to integrate any universal Lie infinity-algebroid of a singular foliation to a Kan simplicial manifold, where all components are made of non-connected manifolds which are all the same finite dimension that can be chosen to be equal to the ranks of a given geometric resolution. Its 1-truncation is the Androulidakis-Skandalis holonomy groupoid.

Many varieties, such as Poisson singularities with finitely many symplectic leaves (eg symplectic singularities) and canonical threefold singularities, have a finite stratification along which the variety is analytically locally a product. We study how to classify their crepant resolutions in terms of local resolutions of their singularities. To this end we define a constructible sheaf whose sections are locally projective crepant resolutions. We classify obstructions to extending local resolutions to global ones, as well as to extending locally projective resolutions to globally projective ones. As an application we can give general constructions of proper, nonprojective crepant resolutions of cones (particularly, quiver varieties). This is joint work with Dan Kaplan (Boston/London/Long Beach).

In this talk, we first review basic properties of the minimal nilpotent adjoint orbit in a complex simple Lie algebra, and then we state B. Kostant’s theorem on its coordinate ring. The goal of this talk is to explain how the coordinate ring of the minimal nilpotent adjoint orbit in sl_{n+1}(C) is related to the T-equivariant cohomology of the toric variety associated to the cluster fan of type C_n. This result is analogous and complementary to P. Shlykov’s theorem on the Hikita conjecture for the minimal nilpotent orbit. This talk is based on a recent joint work with Li, Yu.

We introduce two operations that can be applied to a compatible cluster structure $\Phi$ on a Poisson variety $(Y, \pi)$. (1) If there exists a point $y \in Y$ where the Poisson bivector $\pi$ vanishes, then taking the lowest degree terms of $\Phi$ gives rise to a set $\Phi^{\rm low}$ of pairwise Poisson commutative functions on the tangent space $T_yY$ equipped with the Poisson bivector $\pi_0$ which is the linearization of $\pi$. We present a sufficient condition, in terms of the degree of the log-volume form of $\Phi$, under which $\Phi^{\rm low}$ is an integrable system on $(T_yY, \pi_0)$. When $(Y, \pi)$ is the Poisson dual group of the standard Poisson Lie group ${\rm GL}_n$ and $\Phi$ is the generalized cluster structure of Gekhtman-Shapiro-Vainshtein, our operation produces an integrable system on $\mathfrak{gl}_n^*$ which is different than the celebrated Gelfand-Zeitlin integrable system. (2) If the Poisson variety $(Y, \pi)$ integrates into a symplectic groupoid $s,t: (\mathcal G, \Omega) \implies (Y, \pi)$, then $s^* \Phi \cup t^* \Phi$ is a set of log-canonical functions on $(\mathcal G, \Omega)$, which, in many examples of representation theoretical interest, can be completed to a compatible cluster structure $\Psi$. We explain how the mutable variables, frozen variables and cluster mutations of $\Psi$ are related to those of $\Phi$. When $(Y, \pi)$ is the standard Poisson Lie group ${\rm GL}_n$, our operation produces a compatible cluster structure on the Heisenberg double of ${\rm GL}_n$. Time permitting, we explain how to relate these two operations by taking the lowest degree terms of $\Psi$ along the identity section of $\mathcal G$. This is joint work with Yanpeng Li and Jiang-Hua Lu.

Given a fibration between two oriented manifolds of dimension m and m-2, respectively, one can associate a natural Poisson structure to the total space. In this talk, I will focus on the Poisson structures arising from Lefschetz fibrations. In particular, I will outline a method for computing their Poisson cohomology by combining topological data from the regular part of the fibration with the explicit local model of the Poisson structure near the singular fibers. I will also highlight some consequences of this approach and discuss open questions that emerge. This is based on work in progress with L. Toussaint.