Jordan Watts' Homepage

The Session on Poisson Geometry, Diffeology, and Singular Spaces Session was originally planned as a special session at the 2024 Fall Southeastern AMS Sectional Meeting on October 5th and 6th at Georgia Southern University in Savannah, Georgia, USA. Unfortunately, this meeting was cancelled due to the aftermath of Hurricane Helene. The organisers decided to move the talks online. The speakers, titles, abstracts, and more appear below.

The Talks

Thomas Baird (Memorial University of Newfoundland)

Title: Anti-symplectic involutions of the Hilbert scheme of points on a symplectic surface

Abstract: Let be a smooth quasi-projective complex surface. The Hilbert scheme of points in , denoted $S^{[n]}$ , is a smooth -dimensional variety which contains the variety of distinct unordered points as a dense open subvariety. If is a symplectic, then $S^{[n]}$ is naturally symplectic.

Given an anti-symplectic involution of , there is an induced involution on $S^{[n]}$ whose fixed point locus is a smooth Lagrangian submanifold. In this talk I explain how to calculate its cohomology and mixed Hodge structure. For the special case $S=\mathbb{C}^2$ , this is done using a Morse theory argument borrowed from Ellingsrud-Stromme. For the general case, we adapt an approach due to Gottsche-Soergel involving perverse sheaves.

Video

Gabriele Barbieri (Università di Pavia & Università di Milano-Bicocca)

Title: Frobenius Reciprocity and Diffeological Reduction

Abstract: Frobenius Reciprocity was established in a recent paper by T. Ratiu and F. Ziegler as a bijection between certain symplectically reduced spaces, which are not necessarily manifolds. They also conjectured that is a diffeological diffeomorphism when these spaces are endowed with their natural subquotient diffeology and that it respects the reduced diffeological 2-forms they may carry. Our joint work with J. Watts and F. Ziegler proves this conjecture and provides new sufficient conditions for the existence of reduced forms: it is enough for the group action to be strict, or locally free, or proper. In the latter case, the global diffeological form restricts to the Sjamaar-Lerman-Bates 2-form on each stratum of the stratified reduced space.

Slides | Video

Ivan Contreras (Amherst College)

Title: Poisson Sigma Models, the Symplectic Category and 2-Segal Sets

Abstract: 2-dimensional Topological Quantum Field Theories (TQFT) can be classified via (commutative) Frobenius algebras. In this talk, we will describe Frobenius objects in a category where the objects are sets and the morphisms are isomorphism classes of spans, which is a suitable model for the symplectic category. our key result is that it is possible to construct a simplicial set (with some additional conditions) that encodes the data of the Frobenius structure. The simplicial sets that arise in this way can be characterized by a few relatively nice conditions. As an application (in progress), we recover a Frobenius structure from the reduced, and possibly singular, reduced phase space of the Poisson sigma model. This is based on work with Rajan Mehta and Molly Keller (arXiv:2106.14743), and Mehta and Walker Stern (arXiv:2311.15342).

Slides | Video

Tom Gannon (University of California Los Angeles)

Title: Proof of the Ginzburg-Kazhdan Conjecture

Abstract: The main theorem of this talk will be that the affine closure of the cotangent bundle of the basic affine space (also known as the universal hyperkahler implosion) has symplectic singularities for any reductive group, where essentially all of these terms will be defined in the course of the talk. After discussing some motivation for the theory of symplectic singularities, we will survey some of the basic facts that are known about the universal hyperkahler implosion and discuss how they are used to prove the main theorem. We will also discuss a recent result, joint with Harold Williams, which identifies the universal hyperkahler implosion in type A with a Coulomb branch in the sense of Braverman, Finkelberg, and Nakajima, confirming a conjectural description of Dancer, Hanany, and Kirwan.

Slides | Video

Derek Krespki (University of Manitoba)

Title: Exact Courant algebroids and symmetries of bundle gerbes

Abstract: Let be a smooth manifold and let $\chi\in\Omega^3(M)$ be closed differential form with integral periods. We show the Lie 2-algebra $\mathbb{L}(C_\chi)$ of sections of the $\chi$ -twisted Courant algebroid $C_\chi$ on is quasi-isomorphic to the Lie 2-algebra of connection-preserving multiplicative vector fields on an S^1 -bundle gerbe with connection (over ) whose 3-curvature is $\chi$ .

Slides | Video

Christopher Seaton (Skidmore College)

Title: Isomorphisms of linear symplectic torus quotients

Abstract: Let be a compact Lie group such that the identity component $K^\circ$ is a torus and let be a unitary -module considered as a real symplectic vector space. Let $ho\colon V o\mathfrak{k}^*$ denote the homogeneous quadratic moment map, let $Z= ho^{-1}(0)$ , and then Z/K is the symplectic quotient corresponding to the representation . There are many examples of non-isomorphic representations and yielding symplectic quotients Z/K and Z'/K' that are symplectomorphic via a regular symplectomorphism, a symplectomorphism preserving their algebraic structures. We will present recent results addressing the question of when this can occur. Call minimal if the singular set of $ho_{\mathbb{C}}^{-1}(0)$ has codimension at least , where $ho_{\mathbb{C}}\colon V\oplus V^* o\mathfrak{k}_{\mathbb{C}}^*$ denotes the complexificiation of . We show that if is minimal and stable as a representation of the complexification $K_{\mathbb{C}}$ of , then Z/K determines the representation . If is not minimal, then and can be replaced with a minimal representation without changing the symplectic quotient.

Slides | Video

Joel Villatoro (Indiana University)

Title: An overview of Lie groupoids and differentiable stacks

Abstract: Lie groupoids are a kind of "horizontal" generalization of the notion of a Lie group. One particularly important application of Lie groupoids is that they can be useful models for many kinds of singular spaces (i.e. differentiable stacks) which appear in differential geometry. In this talk I'll give a brief overview of the theory of Lie groupoids and their relationship with differentiable stacks.

Video

Joel Villatoro (Indiana University)

Title: Integration of Lie algebroids via diffeological spaces

Abstract: In this talk I will give a brief overview of the integration problem for Lie algebroids. In contrast to the theory of Lie groups and Lie algebras, it turns out that some Lie algebroids are not integrable by Lie groupoids. I will explain how, using diffeological spaces, one can work around this technical difficulty and see that every Lie algebroid can be integrated by a diffeological Lie groupoid.

Video

Jordan Watts (Central Michigan University)

Title: An overview of diffeology

Abstract: Diffeology is a language developed by Souriau and Iglesias-Zemmour (among others) in the 1980s to address differential-geometric questions involving infinite dimensions, singular quotient spaces, and irrational tori. In this talk, we will go over the basics of this theory, with examples coming from symplectic geometry, and Poisson geometry, and the study of Lie groupoids.

Slides | Video

Florian Zeiser (University of Illinois)

Title: About Poisson cohomology and linearization for $\mathfrak{sl}(2,\mathbb{R})$

Abstract: One of the most basic questions one can ask for any geometric structure is that of a normal form, i.e. are there coordinates such that the structure admits a nice form.

In Poisson geometry, due to Weinstein’s splitting theorem, the question of normal forms comes down to the question of linearization, i.e. given a Poisson structure which vanishes at a point, does there exist a diffeomorphism which identifies the Poisson structure with its linear approximation? Infinitesimally, this problem is controlled by the second Poisson cohomology group of the linear Poisson structure.

Weinstein showed that for Poisson structures whose linear part is associated to the Lie algebra $\mathfrak{sl}(2,\mathbb{R})$ such a diffeomorphism does not always exist. However, a computation of the second Poisson cohomology group suggests that all such deformations aren't unimodular.

In this talk we review the question above and show that this is indeed the case, i.e. any Poisson structure which is unimodular and whose linear part is associated to $\mathfrak{sl}(2,\mathbb{R})$ is linearizable. If time permits, we'll mention generalizations of this result and open questions arising from it.

Video

François Ziegler (Georgia Southern University)

Title: The de Rham cohomology of a Lie group modulo a dense subgroup

Abstract: Let be a dense subgroup of a Lie group with Lie algebra $\mathfrak{g}$ . We show that the (diffeological) de Rham cohomology of G/H equals the Lie algebra cohomology of $\mathfrak{g}/\mathfrak{h}$ , where $\mathfrak{h}$ is the ideal $\{Z\in\mathfrak{g} : e^{tZ}\in H ext{ for all } t\in\mathbb{R}\}$ . Based on arXiv:2407.07381.

Slides | Video

The Organisers

Yi Lin (Georgia Southern University)
Jordan Watts (Central Michigan University)
François Ziegler (Georgia Southern University)

Thanks to upmath.me for converting LaTeX into MathML!