Schedule for: 22w5035 - Poisson Geometry, Lie Groupoids and Differentiable Stacks

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

In both the differential and the algebraic context there are various desingularization procedures for geometric structures. Some resolve singular foliations to regular ones, and others resolve singular varieties to smooth ones; in a few instances, both phenomena happen simultaneously. We will survey some examples of desingularization with a view towards Poisson geometry, and underline some of the connections between them. They include blow-up constructions for proper Lie groupoids, alterations of semisimple Lie algebras in real and complex representation theory, Dirac desingularizations of Poisson manifolds of compact type, and resolutions of algebraic varieties with symplectic singularities.

In both the differential and the algebraic context there are various desingularization procedures for geometric structures. Some resolve singular foliations to regular ones, and others resolve singular varieties to smooth ones; in a few instances, both phenomena happen simultaneously. We will survey some examples of desingularization with a view towards Poisson geometry, and underline some of the connections between them. They include blow-up constructions for proper Lie groupoids, alterations of semisimple Lie algebras in real and complex representation theory, Dirac desingularizations of Poisson manifolds of compact type, and resolutions of algebraic varieties with symplectic singularities.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

I will present the notion of cohomology of a Lie algebroid relative to a Lie subgroupoid and explain some of the basic properties of this notion. As an application I will show an intrinsic and simple way to define and compute characteristic classes of representations of Lie algebroids.

Poisson cohomology is a natural invariant of every Poisson manifold, obtained from the complex of multivector fields and the differential obtained from the Schouten-Nijenhuis bracket and the Poisson bivector. The cohomology groups play an important role in questions such as linearization and deformations, but are quite difficult to compute in general. For the linear Poisson structure on the dual of Lie algebra, Poisson cohomology can be understood as Lie algebra cohomology with coefficients in the smooth functions on the dual. In this talk we present a description of the Poisson cohomology groups associated to all 3-dimensional Lie algebras and outline a prove of the statements. This is based on joint work with D. Hoekstra and I. Marcut.

Let G be a complex simple group of type ADE. Let $\beta$ be a positive braid whose Demazure product is the longest Weyl group element. The braid variety $M(\beta)$ generalizes many well known varieties, including positroid cells, open Richardson varieties, and double Bott-Samelson cells. We provide a concrete construction of the cluster structures on $M(\beta)$, using the weaves of Casals and Zaslow. We show that the coordinate ring of $M(\beta)$ is a cluster algebra, which confirms a conjecture of Leclerc as special cases. As an application, we show that $M(\beta)$ admits a natural Poisson structure and can be further quantized. This talk is based on joint work with Roger Casals, Eugene Gorsky, Mikhail Gorsky, Ian Le, and Jose Simental.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A log symplectic pair is a pair $(X,Y)$ consisting of a smooth projective variety $X$ and a divisor $Y$ in $X$ so that there is a non-degenerate log 2-form on $X$ with poles along $Y$. I will discuss the relationship between log symplectic pairs and so-called ``good degenerations'' of hyperkaehler varieties, and how this naturally leads to a class of log symplectic pairs called (simple normal crossings) log symplectic pairs of pure weight. I will give examples of families of log symplectic pairs of pure weight; one coming from elliptic curves, and one coming from a hybrid toric/cluster construction. Finally, I will explain that if $Y$ is a snc divisor, the cohomology of a log symplectic pair $(X,Y)$ is incredibly restricted. In particular, if there are $\dim(X)$ components of Y meeting in a point, the cohomology ring of $(X,Y)$ has the curious hard Lefschetz property of Hausel and Rodriguez--Villegas.

In this talk, I will review some recent results on local normal forms for complex log-symplectic manifolds, and their applications to Poisson deformations and integrations. In this context, I will motivate and state a conjectural geometric criterion for finite dimensionality of the Poisson cohomology. Emphasized examples will include log-symplectic manifolds with normal crossings polar divisors, and Hilbert schemes of Poisson surfaces.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Throughout the past years there has been much interest in studying Poisson structures which can be described as singular symplectic structures using Lie algebroids. In this talk we will study (self-crossing) elliptic symplectic structures, which are symplectic structures which degenerate in a quadratic fashion along a codimension-two submanifold which self-transverse intersections. These can arise as the real part of a holomorphic log symplectic manifold, but can also exist on non-complex manifolds. Some elliptic symplectic structures correspond to generalized complex structures. Using a connected sum procedure we will provide many examples of elliptic symplectic structures on manifolds which neither admit complex nor elliptic symplectic structures. Finally, we will mention how this theory can be used to extend T-duality (a version of mirror symmetry) to non-principal torus bundles. Joint work with Gil Cavalcanti and Ralph Klaasse.

A generalized Kähler structure gives rise to a pair of holomorphic Manin triples. In previous work, which focused on the case where one of the generalized complex structures is symplectic (the symplectic type case), it was shown that these Manin triples are related by a holomorphic symplectic Morita equivalence equipped with a smooth Lagrangian bisection (Bischoff-Gualtieri-Zabzine). We review this result and then proceed to extend it to the case of arbitrary generalized Kähler structures. This will involve a novel ``Bi-Poisson'' approach to generalized Kähler geometry, as well as the deformation theory of holomorphic Poisson groupoids.

A generalized Kähler structure gives rise to a pair of holomorphic Manin triples. In previous work, which focused on the case where one of the generalized complex structures is symplectic (the symplectic type case), it was shown that these Manin triples are related by a holomorphic symplectic Morita equivalence equipped with a smooth Lagrangian bisection (Bischoff-Gualtieri-Zabzine). We review this result and then proceed to extend it to the case of arbitrary generalized Kähler structures. This will involve a novel ``Bi-Poisson'' approach to generalized Kähler geometry, as well as the deformation theory of holomorphic Poisson groupoids.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, I plan to motivate the problem of constructing a category of generalized complex branes by relating it to the deformation quantization of holomorphic Poisson varieties. I will then outline a conjectural construction for the category of branes that makes use of the A-model of an associated symplectic groupoid. Finally, I will explain how this can be implemented in the case of quantizing toric Poisson varieties. This is joint work with Marco Gualtieri (arXiv:2108.01658).

In this talk, I will define a notion of Lagrangian intersection Floer cohomology for orientable log symplectic surfaces using the notion of ``crossing lunes'' between different symplectic components, and show that for a single Lagrangian, this reduces to log de Rham cohomology. I will discuss what this low-dimensional toy model can hopefully teach us about stable generalized complex structures.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will give an overview of different approaches and motivations for the study of higher cotangent bundles.

I propose a homotopy-theoretic toolkit for constructing explicit integrations and differentiations with nice geometric properties in higher Lie theory. The ``proof of concept'' is joint work in progress with Jesse Wolfson on Lie's Third Theorem for Lie $n$-algebras. Building on E. Getzler's and A. Henriques' work, we prove that every finite-type Lie $n$-algebra integrates to a finite dimensional Lie $n$-group. The proof exploits techniques from manifold topology, as well as abstract homotopy theory. Our proposed inverse to this construction is a differentiation functor that borrows heavily from the work of A. Beilinson on Chern-Weil theory, and the work of J. Pridham on the cosimplicial Dold-Kan correspondence.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We shall use the language of shifted symplectic structures to explain (1) the recent integration of Poisson homogeneous spaces by Bursztyn-Iglesias-Lu and also (2) the nature of quasi-Poisson manifolds and their integrations. This is based on a joint work with H. Bursztyn and M. Cueca.

The Hochschild cohomology of a differential graded algebra or more generally of a differential graded category admits a natural map to the graded center of its derived category: the characteristic homomorphism. We interpret it as an edge homomorphism in a spectral sequence, which allows to calculate the characteristic homomorphism in many interesting settings in geometry, physics and topology. To illustrate this, we will discuss several concrete examples like coherent sheaves over algebraic curves, as well as examples related to cochains of classifying spaces of Lie groups, free loop spaces and string topology. This is joint work with M. Szymik (Sheffield and NTNU Trondheim).

In this talk we introduce Morse Lie groupoid morphisms and we study their main properties, e.g. Morita invariance, establishing the foundations of Morse theory on Lie groupoids/stacks. We will show that on any Riemannian groupoid in the sense of del Hoyo and Fernandes, one can use gradient flows to understand how the topology of a Lie groupoid changes after crossing a critical sublevel. For a given Riemannian groupoid together with a Morse Lie groupoid morphism, we introduce a Morse-like double complex whose cohomology is isomorphic to the Bott-Shulman cohomology of the Lie groupoid. Examples will be discussed with focus on symplectic geometry. The talk is based on joint work with Fabricio Valencia (Sao Paulo).

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

We define Courant-Nijenhuis (CN) algebroids using the formalism of generalized derivations on vector bundles. From this approach, we introduce a definition of Dirac-Nijenhuis (DN) structures which recovers the Poisson-Nijenhuis structures when the Courant algebroid is the standard bundle $TM\oplus T^*M$. We explore the example of holomorphic Courant algebroids and apply this notion of CN algebroids to Kahler geometry. Moreover, we revisit the theory of Manin triples for Lie algebroids, and establish a relation between Lie-Nijenhuis bialgebroids, CN algebroids and DN structures.

In this talk I will discuss a class of diffeological groupoids for which there is a differentiation procedure. We will see that this category provides us with a “coarse” model for studying integration of Lie algebroids. In particular, we will see how one can relate the classical Weinstein groupoid of a Lie algebroid to such structures. Time permitting, we may also discuss further generalizations which seem to lead into the direction of a nice Lie theory for certain diffeological groupoids and sheaves of Lie-Rinehart algebras.

This is joint work in progress with Yi Lin, Yiannis Loizides, and Yanli Song. I will state a ``quantization commutes with reduction'' theorem for manifolds equipped with Riemannian foliations and compatible transverse symplectic structures. In this context, ``quantization'' means the index of a transverse spin-c Dirac operator, and ``reduction'' means symplectic reduction with respect to a transverse Lie algebra action. The theorem applies for instance to toric quasifolds, which are a stacky version of toric symplectic manifolds.