# GLOBAL POISSONWEBSPACE

### Talks in Poisson Geometry for a Global Audience

#### GLOBAL POISSON WEBINAR SCHEDULE

Thursdays
Western Hemisphere Edition: 5pm (Geneva time)
Eastern Hemisphere Edition: 2pm (Geneva time)
All talks are 1 hour long.
Virtually hosted by the University of Geneva

#### AIM

The Global Poisson Webinar aims at presenting talks in the field of Poisson geometry, understood in a very broad sense. We welcome suggestions of talks for this seminar which should be sent to one of the seminar organisers.

Zoom access data is distributed via emails to the Global Poisson Mailing List.

# GLOBAL POISSON CALENDAR

The best way to stay up to date with Global Poisson events is to subscribe to our Global Poisson Calendar using your preferred calendar app:

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## On Cosymplectic Groupoids

Symplectic groupoids are known to be global objects associated with Poisson manifolds while contact groupoids are global counterparts of Jacobi manifolds. Since both contact and cosymplectic structures (in the sense of P. Libermann) can be considered as odd-dimensional analogues of symplectic structures, it is natural to consider cosymplectic structures on Lie groupoids. Recall that a cosymplectic structure on a smooth $2n+1$-dimensional manifold $M$ is given by a closed 1-form $\eta$ and a closed 2-form $\omega$ such that $\omega^n \wedge \eta$ is a volume form. The notion of a cosymplectic structure on a groupoid requires some compatibility with the groupoid structure, namely, the multiplicativity of the cosymplectic tensors. Consequently, cosymplectic groupoid structures are related to Spencer operators. In this talk, we will first review cosymplectic manifolds. Then we will discuss cosymplectic groupoids and their infinitesimal counterparts.

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## Poisson geometry of big quantum supergroups at roots of unity

In a celebrated sequence of works from the early 1990s, De Concini, Kac and Procesi constructed a Poisson geometric framework for the study of the irreducible representations of big quantum groups at roots of unity, based on the notion of Poisson orders. We will describe an extension of this framework to big quantum supergroups. The latter have 18 different kinds of Serre relations on up to 4 generators and the original approach of De Concini, Kac and Procesi using reductions to low rank situations does not apply. Instead, we use general arguments with Nichols algebras, Yetter-Drinfeld modules and perfect pairings between restricted and non-restricted integral forms. These are shown to control the underling Poisson-Lie groups without the need of any concrete computations of Poisson brackets. We will provide an intuitive introduction to all of the above notions. This is a joint work with Nicolas Andruskiewitsch and Ivan Angiono (University of Cordoba).

## Wonderful compactification of a Cartan subalgebra of a semisimple Lie algebra

Let $H$ be a Cartan subgroup of a semisimple algebraic group $G$ over the complex numbers. The wonderful compactification $\bar H$ of $H$ was introduced and studied by De Concini and Procesi. For the Lie algebra $\mathfrak h$ of $H$, we define an analogous compactification $\bar {\mathfrak h}$ of $\mathfrak h$, to be referred to as the wonderful compactification of $\mathfrak h$, as a subvariety of a variety of Lagrangian subalgebras. We will describe various properties of the cohomology of $\bar {\mathfrak h}$. In particular, we will connect the Betti numbers of $\bar {\mathfrak h}$ with some classical combinatorial sequences (Stirling numbers, Whitney numbers of the Dowling lattice, etc.). The ring structure of the cohomology of $\bar {\mathfrak h}$ will be explained in terms of the intersection lattice of the Coxeter hyperplane arrangement. This is joint work with Sam Evens.

## Geometry of the individual terms in the Duistermaat-Heckman formula

Viktor Ginzburg-Guillemin-Karshon used symplectic cobordism to interpret each term in the alternating-sum-of-cones version of the Duistermaat-Heckman formula. I'll recall the algebraic origins of this formula, and give a different geometric interpretation where each term is itself a Duistermaat-Heckman measure, but of a "characteristic cycle" in the cotangent bundle. The main ingredient is the additivity of Chern-Schwartz-MacPherson classes (and I'll recall a construction of these, due to Victor "Ginsburg").

## Almost complex structures, transverse complex structures and $(p,0)$ Dolbeault cohomology

An almost complex structure $j$ on a manifold $M$ is integrable if and only if its Nijenhuis tensor $N^j$ vanishes, and this is true iﬀ the distribution $T_j^{1,0} \subset TM^\mathbb{C}$ is involutive. When it is not integrable, we relate properties of $M$ to properties of distributions associated to $j$. In particular, we study conditions for $j$ to deﬁne a complex tranverse structure and we relate the transverse $(p, 0)$-Dolbeault cohomology to the $(p, 0)$ generalized Dolbeault cohomology of $(M, j)$ as introduced recently by Cirici and Wilson.

## Chiral symplectic leaves and arc spaces of Slodowy slices

In this talk, I will present various applications of the notion of chiral symplectic leaves: to quasi-lisse vertex algebras, to the arc spaces of Slodowy slices, to the affine W-algebra at the critical level and the Feigin-Frenkel center, etc. This is based on several joint works with Tomoyuki Arakawa.

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## Homotopy fiber product of manifolds

A main motivation of developing derived differential geometry is to deal with singularities arising from zero loci or intersections of submanifolds. Both zero loci and intersections can be considered as fiber products of manifolds which may not be manifolds. Thus, we extend the category of differentiable manifolds to a larger category in which one has "homotopy fiber products". In this talk, I would like to show a construction, using vector bundles, sections and connections, of homotopy fiber products of manifolds and explain structures behind the construction. The talk is mainly based on a joint work with Kai Behrend and Ping Xu.

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## On the variety of coisotropic subalgebras

I will discuss some results due mostly to my students Nicole Kroeger and Doan Le on classifying coisotropic subalgebras in a complex semisimple Lie algebra with standard Lie bialgebra structure. This work builds on previous results of Zambon, and uses my previous work with Jiang-Hua Lu on the variety of Lagrangian subalgebras, along with additional results of Lu on spherical conjugacy classes.

## Some remarks on Lagrangian intersections in the algebraic case

Some years ago, in joint work with B. Fantechi, we constructed brackets on the higher structure sheaves of Lagrangian intersections, and compatible Batalin-Vilkovisky operators, when certain orientations are chosen (see our contribution to Manin’s 70th birthday festschrift). This lead to a de-Rham type cohomology theory for Lagrangian intersections. In the interim, much progress has been made on a better understanding of the origin of these structures, and some related conjectures have been proved. We will explain some of these results.

## Two field-theoretic viewpoints on the Fukaya-Morse $A_\infty$-category

We study an enhanced version of the Morse degeneration of the Fukaya $A_\infty$-category with higher compositions given by counts of gradient flow trees. The enhancement consists in allowing morphisms from an object to itself to be chains on the manifold. Higher compositions correspond to counting Morse trees passing through a given set of chains. We provide two viewpoints on the construction and on the proof of the $A_\infty$-relations for the composition maps. One viewpoint is via an effective action for the BF theory computed in a special gauge. The other is via higher topological quantum mechanics. This is a report on a joint work with O. Chekeres, A. Losev and D. Youmans, preprint available at arXiv:2112.12756.

## Vector bundles over Lie groupoids and related structures

The differentiation of a Lie groupoid yields a Lie algebroid and the transverse geometry of a Lie groupoid is encoded in a differentiable stack. These two constructions admit partial inverses, thus setting a bridge between the theories of algebroids and stacks, which has shown to be useful when dealing for instance with representations and cohomology. In this talk, I will overview vector bundles over Lie groupoids, Lie algebroids, and differentiable stacks, explain their key role in Poisson and Dirac geometries, discuss their behavior when crossing through that bridge, and mention some of my contributions to the subject.

## Nijenhuis Geometry and its applications

The talk is an introduction to Nijenhuis Geometry, a new challenging area in Differential Geometry that studies local and global properties of geometric structures given by a field of endomorphisms with vanishing Nijenhuis torsion. This topic is located on the crossroad of Geometry, Mathematical Physics and Algebra as Nijenhuis structures naturally appear in many seemingly unrelated areas such as bi-Hamiltonian integrable systems and compatible Poisson structures (both finite and infinite-dimensional), projective geometry, theory of left-symmetric and Frobenius algebras and others. The talk is based on a recent series of papers by V. Matveev, A. Konyaev and the speaker.

## Higher holonomy and representations up to homotopy

Given a connection for a smooth vector bundle $p:E\to M$, parallel transport with respect to smooth paths in the base space $M$ provides a correspondence between smooth vector bundles with flat connection on $M$ and representations of $\pi_1(M)$ . Based in part on earlier groundbreaking work of K.T. Chen, recently this correspondence has been enhanced to the level of smooth paths (not homotopy classes) in the base space $M$ and differential graded vector bundles with generalized flat connections. Classical parallel transport with respect to smooth paths in the base space $M$ and the correspondence with representations of $\pi_1(M)$ will be recalled briefly, but no familiarity with differential graded vector bundles with generalized flat connections will be assumed.

## Moment maps for non-reductive group actions in Kähler geometry

When a complex reductive group $G$ acts linearly on a projective variety $X$, the GIT quotient $X//G$ can be identified with a symplectic quotient of $X$ by a Hamiltonian action of a maximal compact subgroup $K$ of $G$. Here the moment map takes values in the (real) dual of the Lie algebra of $K$, which embeds naturally in the complex dual of the Lie algebra of $G$ (as those complex linear maps taking real values on $\mathfrak{k}$). The aim of this talk is to discuss an analogue of this description for GIT quotients by suitable non-reductive actions, where the analogue of the moment map takes values in the complex dual of the Lie algebra of the non-reductive group. This is joint work with Gergely Berczi.

## Symplectic reduction along a submanifold

Noether's perspective on conserved quantities gives rise to quotient constructions in symplectic geometry. The most classical such construction is Marsden-Weinstein-Meyer reduction, while more modern variants include Ginzburg-Kazhdan reduction, Kostant-Whittaker reduction, Mikami-Weinstein reduction, symplectic cutting, and symplectic implosion.

I will outline a generalization of the quotient constructions mentioned above. This generalization will be shown to have versions in the smooth, holomorphic, complex algebraic, and derived symplectic contexts. As a corollary, I will derive a concrete and Lie-theoretic construction of "universal" symplectic quotients.

This represents joint work with Maxence Mayrand.

## On tropical Poisson-Lie theory

For a compact Lie group $K$ with the standard Poisson structure, we first construct a tropical version for the dual Poisson-Lie group $K^\ast$. This construction will then help us 1) to establish a relation between $K^\ast$ and the Langlands dual group $G^\vee$ of the complexification $G:=K^\mathbb{C}$; 2) to construct an exhaustion by symplectic embeddings of toric domains for each regular coadjoint orbit of $K$. We combine ideas from Poisson-Lie groups, cluster algebras and the geometric crystals of Berenstein-Kazhdan.

The talk is based on joint works with A. Alekseev, A. Berenstein, B. Hoffman, and J. Lane.

## Lie groupoids in fluid dynamics

In 1966, V. Arnold showed that the Euler equation describing the motion of an ideal fluid on a Riemannian manifold can be regarded as the geodesic flow of a right-invariant metric on the Lie group of volume-preserving diffeomorphisms. This insight turned out to be indispensable for the study of Hamiltonian properties and conservation laws in hydrodynamics, fluid instabilities, topological properties of flows, as well as a powerful tool for obtaining sharper existence and uniqueness results for Euler-type equations. However, the scope of application of Arnold’s approach is limited to problems whose symmetries form a group. At the same time, there are many problems in fluid dynamics, such as free boundary problems, fluid-structure interactions, as well as discontinuous fluid flows, whose symmetries should instead be regarded as a groupoid. In the talk, I will discuss an extension of Arnold's theory from Lie groups to Lie groupoids. The example of vortex sheet motion (i.e. fluids with discontinuities) will be addressed in detail. The talk is based on ongoing work with B. Khesin.

## Complexity one Hamiltonian torus actions

I will report on my classification, joint with Sue Tolman, of Hamiltonian torus actions with two dimensional quotients.

## Poisson brackets on projective bundles

To a holomorphic Poisson bracket on a complex projective bundle, one can associate a co-Higgs field on the corresponding holomorphic vector bundle. I will discuss how one can recover the Poisson bracket from its co-Higgs field. Using the BNR spectral correspondence for co-Higgs fields, I will explain how to classify Poisson brackets on CP2-bundles over CP1. The talk is based on my PhD thesis.

## Elliptic chiral homology and quantum master equation

We present an effective BV quantization theory for chiral deformation of two dimensional conformal field theories. We explain a connection between the quantum master equation and the chiral homology for vertex operator algebras. As an application, we construct correlation functions of the curved beta-gamma/b-c system and establish a coupled equation relating to chiral homology groups of chiral differential operators. This can be viewed as the vertex algebra analogue of the trace map in algebraic index theory.

## On almost toric degenerations of projective varieties and applications to Hamiltonian torus actions

Roughly speaking, a toric degeneration of a variety $X$ is a (flat) one-parameter family of irreducible varieties $X_t$ such that for nonzero $t$, $X_t$ is isomorphic to $X$ and $X_0$ is a (not necessarily normal) toric variety. I will present the recent result that any projective variety has an "almost" toric degeneration and will discuss applications in constructing Hamiltonian torus actions as well as estimating Gromov widths. I will try to cover needed definitions, motivations and background in the talk. This is a joint work with Chris Manon and Takuya Murata.

## Embedding ellipsoids into Hirzebruch surfaces

This talk will report on joint work with Magill and Weiler concerning the question of when an ellipsoid symplectically embeds into the one-point blowup of $\mathbb{CP}^1$. The precise size of the blowup has a great effect on the corresponding embedding capacity function. Indeed, as discovered in earlier work with collaborators Bertozzi, Holm, Maw, Mwakyoma, Pires, and Weiler, for certain blowup parameters there are infinitely many significant obstructive classes, which implies that the capacity function has a staircase. We have now found that the set of these parameters, though still not fully understood, displays some very interesting symmetries and recursive patterns.

## Stable Weinstein geometry through localizations

Much of computational math is formula-driven, while much of categorical math is formalism-driven. Mirror symmetry is rich in part because many of its results are driven by both. With the advent of stable-homotopy-theoretic invariants in symplectic geometry--such as Nadler-Shende's microlocal categories and (on the horizon) spectrally enriched wrapped Fukaya categories--there has been a real need for better-behaved formalisms in symplectic geometry. (This is because, now-a-days, much of stable homotopy theory is possible only thanks to extremely well-constructed formalisms.) In this talk, we will talk about recent success in constructing the formalism, especially in the setting of certain non-compact symplectic manifolds called Weinstein sectors. The results have concrete geometric consequences, like showing that spaces of embeddings of these manifolds map continuously to spaces of maps between certain invariants. (And in particular, leads to higher-homotopy-group generalizations, in the Weinstein setting, of the Seidel homomorphism, similar to works of Savelyev and Oh-Tanaka.) The main result we'll discuss is that the infinity-category of stabilized sectors can be constructed using the categorically formal process of localization. Most of what we discuss is joint with Oleg Lazarev and Zachary Sylvan.

## Non-archimedian deformation of Landau-Ginzburg potentials and Gelfand-Cetlin systems

Using the bulk-deformation of Floer cohomology by Schubert classes and non-Archimedean analysis of Fukaya--Oh--Ohta--Ono's bulk-deformed potential function, we prove that every complete flag manifold with a monotone Kirillov--Kostant--Souriau symplectic form carries a continuum of non-displaceable Lagrangian tori which degenerates to a non-torus fiber in the Hausdorff limit. This talk is based on a joint work with Yunhyung Cho and Yoosik Kim.

## Thick morphisms of supermanifolds and bracket structures

A “thick morphism” of supermanifolds is a generalization of a smooth map that I introduced in 2014. It is NOT a map, but it induces a pull-back of smooth functions. A peculiar feature of such pull-back is that it is NONLINEAR --- actually, it is a formal mapping of the algebras of smooth functions regarded as infinite-dimensional (super)manifolds. Compare with ordinary pull-backs, which are algebra homomorphisms, in particular linear. In the talk, I will give the definition of thick morphisms and explain the construction of nonlinear pull-backs. Actually, because of the non-linearity, there are two parallel versions of thick morphisms and the corresponding pull-backs: “bosonic” (acting on even functions) and “fermionic” (acting on odd functions). Each of them gives rise to a formal category containing the category of ordinary maps. My original motivation was constructing L-infinity morphisms for homotopy Poisson or homotopy Schouten brackets. Thick morphisms also make it possible to give adjoints for nonlinear vector bundle maps (useful for L-infinity algebroids). There is a nonlinear analog of “functional-algebraic duality” with certain “nonlinear algebra homomorphisms” taking place of ordinary homomorphisms. In the bosonic case, thick morphisms also have a quantum version given by particular Fourier integral operators, which provide L-infinity morphisms for “quantum brackets” generated by BV- type operators.

## KMS Functionals in Poisson Geometry

In this talk I will report on some old results about KMS states in symplectic geometry and present new results in the general Poisson case. The classical KMS condition captures thermodynamical states in classical mechanical systems as a semi-classical limit of the (original) quantum KMS condition used in algebraic quantum field theory. In the symplectic case the classification of KMS functionals is rather simple. In the general Poisson case, the investigation of the KMS condition for volume forms can be seen as one of the main motivations for the definition of the modular class by Alan Weinstein. Considering more general functionals gives new and interesting structures where in some simple cases a full classification is available. While the classical situation is already very rich, the quantization of classical KMS states is yet to be explored. The results are a joint work with Nicolò Drago.

## PAST WEBINARS

#### SPRING2021

Richard Melrose
Sergei Tabachnikov
Xiaomeng Xu
Edward Witten
Chenchang Zhu
David Martínez Torres
Francis Bischoff
Zheng Hua
Tobias Diez
Claudia Scheimbauer
Vladimir Fock
Yunhe Sheng
Marco Zambon
Vladimir Rubtsov
Yakov Eliashberg

#### SUMMER2021

Nigel Higson
Linhui Shen
Adriano Tomassini
Francesco Bonechi
Ana Balibanu
Xiao-Jun Chen
Charlotte Kirchhoff-Lukat
Pedro Frejlich

#### SPRING 2020

Rui Loja Fernandes
Eckhard Meinrenken
Pavel Etingof
Alan Weinstein

#### SUMMER 2020

Brent Pym
Eva Miranda
Nicolai Reshetikhin
Tudor Ratiu
Maxim Kontsevich
Boris Khesin
Xiang Tang
Michael Gekhtman
Camille Laurent-Gengoux
Susan Tolman
Marta Mazzocco
Reyer Sjamaar
Marco Gualtieri
Henrique Bursztyn
Pavel Safronov

#### AUTUMN2020

Silvia Sabatini
Jiang-Hua Lu
Nigel Hitchin
Alejandro Cabrera
Pavol Ševera
Alberto Cattaneo
Leonid Polterovich
Luca Vitagliano
Ezra Getzler
Megumi Harada
Ioan Marcut
Lisa Jeffrey
Marius Crainic

24 June 2021

## The bundle picture of Poisson transversals

In this talk, we describe the nonlinear Grassmannian $PT(M,\pi)$ of all closed Poisson transversals of a given Poisson manifold $(M,\pi)$, and show that the tautological bundle over it carries a canonical coupling Dirac structure. Our main result is that a choice of invariant volume form on the ambient manifold induces a weak symplectic structure on the nonlinear Grassmannian, which is a coadjoint orbit for the (infinitesimal) action of a certain central extension of the Hamiltonian group -- generalizing the result of Haller-Vizman in the symplectic case. This is joint work with I. Marcut.

17 June 2021

## Exploring the modular class of Dirac structures

The concept of modular class is best known for Poisson structures, but is naturally defined for any Lie algebroid: It is a class in the first Lie algebroid cohomology. Poisson structures as Lie algebroids have the special feature that their dual is isomorphic to the tangent bundle and thus representatives are vector fields, which allows for the definition of the so-called modular foliation, locally spanned by Hamiltonian vector fields and the modular vector field. This modular foliation can in turn be viewed as the foliation of a Poisson structure on the total space of the real line bundle $\det (T^\ast M)$ (Gualtieri-Pym). In this talk, I will show how to extend these concepts to general real or complex Dirac structures in exact Courant algebroids and discuss the information contained in the modular class of a Dirac structure in some non-Poisson examples. (This is joint work in progress with Ralph Klaasse.)

10 June 2021

## Batalin-Vilkovisky and gravity algebras on Poisson manifolds with semisimple modular symmetry

In this talk, we study the "twisted" Poincare duality of smooth Poisson manifolds, and show that, if the modular symmetry is semisimple, that is, the modular vector is diagonalizable, there is a mixed complex associated to the Poisson complex which, combining with the twisted Poincare duality, gives a Batalin-Vilkovisky algebra structure on the Poisson cohomology, and a gravity algebra structure on the negative cyclic Poisson homology. This generalizes the previous results obtained by Xu et al for unimodular Poisson algebras. We also show that these two algebraic structures are preserved under Kontsevich's deformation quantization, and in the case of polynomial algebras they are also preserved by Koszul duality. This talk is based on a joint work with Liu, Yu and Zeng.

3 June 2021

## Steinberg slices and group-valued moment maps

We define a class of transversal slices in spaces which are quasi-Poisson for the action of a complex semisimple group $G$. This is a multiplicative analogue of Whittaker reduction. One example is the multiplicative universal centralizer of $G$, which is equipped with the usual symplectic structure in this way. We construct a smooth partial compactification of $Z$ by taking the closure of each centralizer fiber in the wonderful compactification of $G$. By realizing this partial compactification as a transversal in a larger quasi-Poisson variety, we show that it is smooth and log-symplectic.

27 May 2021

27 May 2021

## Bihamiltonian systems and invariant polynomials

Motivated by the problem of quantization of the symplectic groupoid we study a class of bihamiltonian systems defined on compact hermitian symmetric spaces. Indeed, a Poisson Nijenhuis (PN) structure defines a (singular) real polarization of the symplectic groupoid integrating any of the Poisson structures appearing in the bihamiltonian hierarchy. Despite its singularity, this polarization leads to the quantization of complex projective spaces. We will discuss in some detail a way to discuss this polarization in terms of invariant polynomials of a certain Thimm chain of subalgebras. This approach works for the classical cases; time permitting, I will discuss some partial results about the exceptional cases.

20 May 2021

## $\overline{\partial}$-Harmonic forms on compact almost Hermitian manifolds

Let $M$ be a smooth manifold of dimension $2n$ and let $J$ be an almost-complex structure on $M$. Then, $J$ induces on the space of forms $A^\bullet(M)$ a natural bigrading, namely $A^\bullet(M)=\bigoplus_{p+q=\bullet}A^{p,q}(M).$ Accordingly, the exterior derivative $d$ splits into four operators $d:A^{p,q}(M)\to A^{p+2,q-1}(M)\oplus A^{p+1,q}(M)\oplus A^{p,q+1}(X)\oplus A^{p-1,q+2}(M)$ $d=\mu+\partial+\overline{\partial}+\bar\mu,$ where $\mu$ and $\bar\mu$ are differential operators that are linear over functions. Let $g$ be a Hermitian metric on $(M,J)$. Denote by $\Delta_{\overline{\partial}}:=\overline{\partial}\,\overline{\partial}^*+\overline{\partial}^*\overline{\partial}$ the $\overline{\partial}$-Laplacian. Then $\Delta_{\overline{\partial}}$ is an elliptic differential operator. We study the space of $\overline{\partial}$-harmonic forms on $(M,J,g)$. Some explicit examples will be discussed. Special results are obtained for $\dim_\mathbb{R} M=4$. This a joint work with Nicoletta Tardini.

13 May 2021

## Moduli spaces of $G$-local systems and Poisson geometry

Let $G$ be a split semi-simple algebraic group over $\mathbb{Q}$. We introduce a natural cluster structure on moduli spaces of framed $G$-local systems over surfaces with marked points. As a consequence, the moduli spaces of $G$-local systems admit natural Poisson structures, and can be further quantized. We will study the principal series representations of such quantum spaces. If time permits, I will discuss its applications in the study of quantum groups. This talk will mainly be based on joint work with A.B. Goncharov (arXiv:1904.10491).

6 May 2021

## An introduction to the hypoelliptic Laplacian

Jean-Michel Bismut's hypoelliptic Laplacian is a one-parameter family of linear differential operators that interpolates between the Laplacian and the geodesic flow.  It may be constructed in a variety of contexts, but in this lecture I shall concentrate on symmetric spaces. Here a special mechanism comes into play, as a result of which the heat traces associated to all the operators in the family remain constant throughout the interpolation.  By studying the limits at both ends of the family, remarkable formulas are obtained, including for example the Selberg trace formula.  All this requires a heavy dose of analysis in the spirit of, but more complicated than, the local index theory of Dirac operators. But in this talk I shall mostly ignore the analysis and concentrate on a few basic ideas, in the hope that they may eventually lead to a more geometric understanding of the hypoelliptic Laplacian.

29 April 2021

## Topology of the space of tight contact structures on $\mathbb{R}^3$.

30 years ago I proved that any tight contact structure on $\mathbb{R}^3$ is equivalent to the standard one. In the same paper I suggested that one can establish along the same lines the contractibility of the space of  fixed at infinity tight contact structure on $\mathbb{R}^3$. Recently we proved this claim in our joint work with N. Mishachev. The proof is based on the study of topology of 1-dimensional foliations  and functions on the 2-sphere.

22 April 2021

## Associative Yang-Baxter equation: from double Poisson structures to modular forms

I shall give a survey of various avatars of Associative Yang-Baxter Equations from (double) Poisson structure existence conditions to a form of the trisecant Fay identity and as some equations on generating functions for period polynomials of (quasi-)modular forms.

15 April 2021

## Deformations of Lagrangian submanifolds in log-symplectic geometry

Log-symplectic manifolds constitute a class of Poisson manifolds that in many respects behave like symplectic ones. We address the question of whether Lagrangian submanifolds and their deformations are as well-behaved as in symplectic geometry. Since the case of Lagrangians transversal to the singular locus is well understood, we focus on Lagrangian submanifolds contained in the singular locus. We establish a normal form theorem around such submanifolds, and show that their deformations are governed by a DGLA. The latter allows to draw geometric consequences: we discuss when a Lagrangian admits deformations not contained in the singular locus, and we give precise criteria for unobstructedness of first order deformations.

This talk is based on joint work with Stephane Geudens.

8 April 2021

## Deformations, cohomology and homotopy of relative Rota-Baxter Lie algebras

Rota-Baxter operators were originally defined on a commutative associative algebra by Rota. Then it was defined on Lie algebras as the operator form of the classical Yang-Baxter equation. Kupershmidt introduced a more general notion called O-operator (later called relative Rota-Baxter operator) for arbitrary representation. Rota-Baxter operators have fruitful applications in mathematical physics. We determine the  L-infty-algebra that characterizes relative Rota-Baxter Lie algebras as Maurer-Cartan elements. As applications, first we determine the L-infty-algebra that controls deformations of a relative Rota-Baxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying Lie algebra and representation by the dg Lie algebra controlling deformations of the relative Rota-Baxter operator. Then we define the  cohomology  of relative Rota-Baxter Lie algebras and relate it to their infinitesimal deformations.  In particular the cohomolgoy of Rota-Baxter Lie algebras and triangular Lie bialgebras are given. Finally we introduce the notion of homotopy relative Rota-Baxter operators and show that the underlying structure is pre-Lie-infinity algebras. This talk is based on joint works with Chenming Bai, Li Guo, Andrey Lazarev and Rong Tang.

1 April 2021

## Momentum map of general relativity

We study an approach to general relativity using vielbein with values in a Clifford algebra. This approach allows to simplify computations and in particular define a hidden $\mathfrak{sl}(2) \times \mathfrak{sl}(2)$ symmetry (and even affine $\mathfrak{sl}(4)$ one in the Kaehler case).  This formalism allows to compute in simple terms the phase space of the theory and the action of the diffeomorphisms on it. The main feature of this situation is that diffeomorphisms do not form a group, but a groupoid. We will discuss the reason for this situation and suggest an analogue of the momentum map. Joint work with P. Goussard.

25 March 2021

## Derived symplectic geometry and AKSZ topological field theories

Derived algebraic geometry and derived symplectic geometry in the sense of Pantev-Toen-Vaquié-Vezzosi allows for a reinterpretation/analog of the classical AKSZ construction for certain $\sigma$-models. After recalling this procedure I will explain how it can be extended to give a fully extended oriented TFT in the sense of Lurie with values in a higher category whose objects are $n$-shifted symplectic derived stacks and (higher) morphisms are (higher) Lagrangian correspondences. It is given by taking mapping stacks with a fixed target building and describes "semi-classical TFTs". This is joint work in progress with Damien Calaque and Rune Haugseng.

18 March 2021

## Group-valued momentum maps for diffeomorphism groups

In mathematical physics, some conserved quantities have a discrete nature, for example because they have a topological origin. These conservation laws cannot be captured by the usual momentum map. I will present a generalized notion of a momentum map taking values in a Lie group, which is able to include discrete conversed quantities. It is inspired by the Lu-Weinstein momentum map for Poisson Lie group actions, but the groups involved do not necessarily have to be Poisson Lie groups. The most interesting applications include momentum maps for diffeomorphism groups which take values in groups of Cheeger-Simons differential characters. As an important example, I will show that the Teichmüller space with the Weil-Petersson symplectic form can be realized as symplectic orbit reduced space.

11 March 2021

## Semiclassical limits of Feigin-Odesskii elliptic algebras via derived geometry

In 1980s, Feigin and Odesskii constructed the elliptic algebras $Q_{n,k}(C,\eta)$ generalizing the construction of Sklyanin and Cherednik. Here n,k are coprime positive integers, $C$ is a complex elliptic curve and $\eta$ is a point on $C$. Elliptic algebras are quantization of polynomial algebras. They are conjectured to be regular in the sense of Artin and Schelter for all parameters.  Homological and representation theoretical properties of elliptic algebras are studied via Poisson geometry of their semiclassical limits. We will discuss various results about these Poisson structures, e.g. classification of symplectic leaves, bihamiltonian structures and so on. The main technical tool is derived geometry, in particular the work of Calaque-Pantev-Toen-Vaquie-Vezzosi. This is based on the joint work with Alexander Polishchuk.

4 March 2021

## Lie Groupoids and differential equations

This talk will discuss applications of Lie groupoids to the study of differential equations with singularities. Several classes of singular differential equations, or flat connections, can be recast as representations of Lie algebroids, and by integration, correspond to Lie groupoid representations. This perspective allows us to introduce new tools to the study of these equations. In this talk, I will give an overview of this approach, with a focus on the case of differential equations with logarithmic singularities along certain (possibly singular) submanifolds that are associated to reductive groups. Whereas the traditional approach to classification relies heavily on the use of power series, I will explain how the use of Lie groupoids gives rise to a more geometric approach.

25 February 2021

## Coregular submanifolds and Poisson submersions

This talk discusses aspects of the theory of submanifolds and submersions in Poisson geometry. In the first part we present the general picture concerning manifolds which inherit a Poisson structure from an ambient Poisson manifold, and among those, we select a class (coregular submanifolds) which have particularly nice functorial properties. The second part is devoted to Poisson submersions with coregular fibers. Coregular submersions restrict nicely over symplectic leaves in the base (coupling property), and we determine when they split into commuting vertical and horizontal Poisson structures. In the last part we present instances in which such coregular Poisson submersions appear. Our illustrations all revolve around Poisson actions of Poisson-Lie groups. This is joint work with L. Brambila and P. Frejlich.

18 February 2021

## Classifying space $BG$ as a symplectic stack

It is probably well known to people who know it well that $BG$ carries a sort of symplectic structure, if the Lie algebra of $G$ is quadratic Lie algebra.  In this talk, we explore various differential-geometric (1-group, 2-group, double-group) models to realise this (2-shift) symplectic structure in concrete formulas and show the equivalences between them.

In the infinite dimensional models (2-group, double-group), Segal's symplectic form on based loop groups turns out to be additionally multiplicative or almost so. These models are equivalent to a finite dimensional model with Cartan 3-form and Karshon-Weinstein 2-form via Morita Equivalence. All these forms give rise to the first Pontryagin class on $BG$. Moreover, they are related to the original invariant pairing on the Lie algebra through an explicit integration and Van Est procedure. Finally, as you might have guessed, the associated String group $BString(G)$ may be seen as a prequantization of this symplectic structure. From the math-physics point of view, what is behind is the Chern-Simons sigma model.

11 February 2021

## Quantization by Branes and Geometric Langlands

In this talk, which is based on work with D. Gaiotto, I will explain a quantum field theory perspective on recent developments in the geometric Langlands program by P. Etinghof, E. Frenkel, and D. Kazhdan (see their paper https://arxiv.org/abs/1908.09677).

28 January 2021

## Stokes phenomenon and quantum Ginzburg-Weinstein isomorphisms

This talk first gives an introduction to the Stokes matrices of meromorphic linear systems of ordinary differential equations. It then uses the quantum Stokes matrices to construct the quantization of a family of Ginzburg-Weinstein isomorphisms from ${\frak g \frak l}_n^*$ to the dual Poisson Lie group ${\rm GL}_n^*$ found by Boalch. In the end, it gives explicit formula for the quantization, as special Drinfeld isomorphisms from the quantum group $U_\hbar({\frak g \frak l}_n)$ to the classical $U({\frak g \frak l}_n)$, and briefly discusses the relation with representation theory of quantum groups.

21 January 2021

## Cross-ratio dynamics on ideal polygons

Define a relation between labeled ideal polygons in the hyperbolic space by requiring that the complex distances (a combination of the distance and the angle) between their respective sides equal c; the complex number c is a parameter of the relation. This defines a 1-parameter family of maps on the moduli space of ideal polygons in the hyperbolic space (or, in its real version, in the hyperbolic plane). I shall discuss complete integrability of this family of maps and related topics, including its connection with the Korteweg-de Vries equation.

14 January 2021

## Resolution of Lie algebroids and quantization

I will give an overview of what is known about the resolution of Lie algebroids -- limited for the most part to the geometric case' of a subalgebra of the Lie algebra of vector fields on a manifold. This gives a direct quantization with corresponding algebras (and modules) of pseudodifferential operators. In particular I will make the case that the notion of a groupoid is inadequate here even though there is as yet no precise replacement for it.

17 December 2020

## From Poisson Geometry to (almost) geometric structures

I will report on an approach to general geometric structures (with an eye on integrability) based on groupoids endowed with multiplicative structures; Poisson geometry (with its symplectic groupoids, Hamiltonian theories and Morita equivalences) will provide us with some guiding principles. This allows one to discuss general "almost structures" and an integrability theorem based on Nash-Moser techniques (and this also opens up the way for a general "smooth Cartan-Kahler theorem"). This report is based on collaborations/discussions with Francesco Cataffi (almost structures), Ioan Marcut (Nash-Moser techniques), Maria Amelia Salzar (Pfaffian groupoids).

10 December 2020

## Flat connections and the $SU(2)$ commutator map

This talk is joint work with Nan-Kuo Ho, Paul Selick and Eugene Xia. We describe the space of conjugacy classes of representations of the fundamental group of a genus 2 oriented 2-manifold into $G:=SU(2)$.

1. We identify the cohomology ring and a cell decomposition of a space homotopy equivalent to the space of commuting pairs in $SU(2)$.
2. We compute the cohomology of the space $M:=\mu^{-1}(-I)$ where $\mu: G^4 \to G$ is the product of commutators.
3. We give a new proof of the cohomology of $A:=M/G$, both as a group and as a ring. The group structure is due to Atiyah and Bott in their landmark 1983 paper. The ring structure is due to Michael Thaddeus 1992.
4. We compute the cohomology of the total space of the prequantum line bundle over $A$.
5. We identify the transition functions of the induced SO(3) bundle $M\to A$.

To appear in QJM (Atiyah memorial special issue). arXiv:2005.07390

3 December 2020

## Poisson non-degeneracy of the Lie algebra $\mathfrak{sl}(2,\mathbb{C})=\mathfrak{so}(3,1)$

In this talk, I will revisit the classical problem of linearizing Poisson structures around fixed points, introduced by Alan Weinstein. If the isotropy Lie algebra at the fixed point is semi-simple, the problem has been settled in most cases, through the works of Conn, Weinstein, Monnier and Zung. The lowest dimensional semi-simple Lie algebra for which the problem was still open is $\mathfrak{sl}(2,\mathbb{C})=\mathfrak{so}(3,1)$. Together with my PhD student Florian Zeiser we have shown that $\mathfrak{sl}(2,\mathbb{C})$ is the first non-compact semi-simple Lie algebra that is "Poisson non-degenerate", in the sense that a version of Conn's theorem holds for this Lie algebra. I will explain the main ingredients of the proof.

26 November 2020

## Newton-Okounkov bodies, integrable systems, and convergence of polarizations

Let $X$ be a smooth irreducible complex algebraic variety of dimension $n$ and $L$ a very ample Hermitian line bundle. In this talk I will recount, in very broad strokes, two interconnected stories related to the symplectic geometry of $X$. The first story is that the theory of Newton-Okounkov bodies, and the toric degenerations to which they give rise, can provide -- in rather general situations -- constructions of integrable systems on $X$. The main tool in the first story is the gradient-Hamiltonian vector field. The second story concerns the independence of polarization'' issue which arises in the theory of geometric quantization. Specifically, given a toric degeneration of $(X,L)$ satisfying some technical hypotheses, we construct a deformation $\{J_s\}$ of the complex structure on $X$ and bases $B_s$ of $H^0(X, L, J_s)$ so that $J_0$ is the standard complex structure and, in the limit as $s \to \infty$, the basis elements approach dirac-delta distributions centered at Bohr-Sommerfeld fibers of the moment map associated to the integrable system on $X$ (constructed using the first story). This significantly generalizes previous results in geometric quantization proving independence of polarization between Kahler quantizations and real polarizations.

19 November 2020

## Classical field theory, variational calculus, and the Batalin-Vilkovisky formalism

The Batalin-Vilkovisky formalism extends Noether's approach to classical field theories, which is restricted to the Euler-Lagrange locus (or "on-shell", as physicists say), off-shell. This is of course important in the study of quantization of field theories, since the quantized theory is not restricted to the Euler-Lagrange locus.

In the variational calculus, the action functional is the integral of a local expression in the fields and their derivatives. The symmetries of the action may be expressed by the classical Batalin-Vilkovisky master equation, which is a Maurer-Cartan equation for functionals of the classical fields, ghost fields expressing the symmetries of the theory, and certain auxilliary fields known as antifields.

The Batalin-Vilkovisky formalism has a natural extension in functionals are lifted to densities. In the first part of today's talk, I explain this extension. which relies on the Soloviev bracket in the variational calculus, originally introduced in the study of general relativity.

Symmetries of a field theory involving diffeomorphisms of the world sheet do not really fit into the formalism of the variational calculus. In my article "Covariance in the Batalin-Vilkovisky formalism",  I explain how to take into account such symmetries of the world sheet by incorporating a curvature term into the Batalin-Vilkovisky master equation, associated to a differential graded Lie algebra with curvature. This construction is the subject of the second part of the talk.

I will finish with a few words on the work of Bonechi, Cattaneo, Qiu and Zabzine, who studied the extension of our formalism to quantum field theory.

12 November 2020

## Calculus up to Homotopy on the Space of Solutions of a PDE

Every partial differential equation (PDE) can be encoded in a geometric object, what is sometimes called a diffiety, which is a submanifold of an appropriate type in an infinite jet space. There is a Lie algebroid naturally attached to a diffiety, and the associated Lie algebroid cohomology contains important coordinate independent information on the PDE: variational principles, symmetries, conservation laws, recursion operators, etc. To some extent these cohomologies can also be interpreted as vector fields, differential forms, tensors, etc. on the space of solutions. This interpretation is supported by the fact that we find the appropriate algebraic structures in cohomology. I will review this theory and show that those algebraic structures do actually come from homotopy algebras at the level of cochains, confirming an old conjecture of A. M. Vinogradov that “the calculus on the space of solutions of a PDE is a calculus up to homotopy”.

5 November 2020

## Approximate representations and quantization

We discuss some links between Ulam-type stability for  algebras and groups ("approximate representations are close to genuine representations")  and quantization, with applications to classification of quantizations and Hamiltonian actions of finitely presented groups. (with L.Charles, L.Ioos, D.Kazhdan).

29 October 2020

22 October 2020

## Quantization of Poisson Hopf algebras and moduli of flat connections

I will describe a universal quantization of Poisson Hopf algebras using simplicial methods, i.e. nerves of Hopf algebras (a joint work with Jan Pulmann). The motivation for this method comes from moduli spaces of flat connections on surfaces with decorated boundaries (an older joint work with David Li-Bland).

15 October 2020

## Semiclassical aspects of quantization: Local symplectic groupoids, generating functions and the Poisson sigma model

The aim of this talk is to present three results related to local symplectic groupoids in connection to quantization of the underlying Poisson manifold. We first review the notion of a generating function S for such local symplectic groupoids and outline the first result stating that such S always exist and how to construct them. When the Poisson manifold is a coordinate space, we provide an explicit (integral) formula for S. The second result makes reference to quantization: we show that the formal Taylor expansion S_K of the coordinate S yields the tree-level part of Kontsevich's quantization formula, as first studied by Cattaneo-Dherin-Felder. We also sketch how the (non-formal) analytic formula for S actually "explains" the graph structure of S_K, using Butcher series techniques. Finally, the third result relates S to the functional perspective underlying the Poisson Sigma Model: we can recover S by evaluating a functional on a set of solutions ("semiclassical fields") for a system of PDEs on a disk, which we also show how to solve (non-perturbatively). We comment on conclusions and further directions at the end.

8 October 2020

## Teichmueller spaces and the geometry of geodesics

The talk concerns a moduli space of representations of the fundamental group of a compact surface into the group of Hamiltonian diffeomorphisms of $S^1\times \mathbb{R}$. The motivation comes from applying the ideas of  Higgs bundles for $\mathrm{SL}(N,\mathbb{R})$ with $N$ equal to infinity.

24 September 2020

## Some examples of algebraic symplectic groupoids

We construct Poisson and symplectic groupoids over a class of polynomial Poisson structures on C^n whose total spaces are certain configuration spaces of flags. This is joint work with Victor Mouquin and ShiZhuo Yu.

17 September 2020

## Some topological properties of monotone complexity one spaces

In symplectic geometry it is often the case that compact symplectic manifolds with large group symmetries admit indeed a Kähler structure. For instance, if the manifold is of dimension $2n$ and it is acted on effectively by a compact torus of dimension $n$ in a Hamiltonian way (namely, there exists a moment map which describes the action), then it is well-known that there exists an invariant Kähler structure. These spaces are also called complexity-zero spaces, where the complexity is given by $n$ minus the dimension of the torus. In this talk I will explain how there is some evidence that a similar statement holds true when the complexity is one and the manifold is monotone (the latter being the symplectic analog of the Fano condition in algebraic geometry), namely, that every monotone complexity-one space is simply connected and has Todd genus one, properties which are also enjoyed by Fano varieties.

These results were largely inspired by a conjecture posed by Fine and Panov in 2010, by work of Lindsay and Panov of 2019, and it is joint with Daniele Sepe.

13 August 2020

## Complexified Floer homology and skein modules

A few years ago Abouzaid and Manolescu have constructed an $SL(2, \mathbb{C})$ Floer homology of 3-manifolds using techniques from derived geometry. Just like the usual instanton Floer homology, it can be viewed as a mathematical formalization of the vector space of states in a topologically twisted super Yang—Mills theory. In this talk I will explain a connection of this invariant to another purely topological invariant known as the skein module, defined in terms of embedded links. The relationship goes through the theory of deformation quantization modules and along the way we establish a GAGA theorem for DQ modules on symplectic manifolds with a log symplectic compactification. This is a report on work in progress joint with Sam Gunningham.

6 August 2020

## Morita equivalence of formal Poisson structures and links with deformation quantization

The classical notion of Morita equivalence of algebras has a geometric version for Poisson manifolds (due to Xu), defined in terms of Weinstein's dual pairs. A natural question is whether these two parallel Morita theories could be related by quantization. Motivated by this question, this talk will discuss an extension of Morita equivalence of Poisson manifolds to the setting of {\em formal} Poisson structures, and present a result characterizing Morita equivalent formal Poisson structures vanishing in zeroth order in terms of `B-field transformations'' (joint work with I. Ortiz and S. Waldmann). Using the correspondence between formal Poisson structures and star products (due to Kontsevich), this result leads to a concrete link between Morita equivalence in Poisson geometry and noncommutative algebra via deformation quantization.

30 July 2020

## Branes in symplectic groupoids

After an introduction to coisotropic A-branes in symplectic manifolds and their role in mirror symmetry, I will explain how the problem of holomorphic quantization of Poisson brackets may be recast, and in some cases solved, as a problem of computing morphisms between coisotropic branes in symplectic groupoids.   This is joint work with Francis Bischoff and Joshua Lackman

23 July 2020

## Reduction and quantization for log symplectic manifolds

Kirillov's orbit method suggests that the classical analogue of a representation of a Lie group $G$ is a Hamiltonian $G$-action on a symplectic manifold $M$. The classical analogue of isotypical subspaces should then be the symplectic quotients of $M$. This "quantization commutes with reduction" problem was articulated in the 80's by Guillemin and Sternberg and solved by them in the context of Kaehler quantization and homogeneous quantization. In the 90's Meinrenken solved an index-theoretic version of the problem. Yi Lin, Yiannis Loizides, Yanli Song, and I have been trying to extend Meinrenken's theorem to log symplectic manifolds, and this talk will be a progress report on our work.

16 July 2020

## Quantum uniformisation and CY algebras

In this talk, I will discuss a special class of quantum del Pezzo surfaces. In particular I will introduce the generalised Sklyanin-Painlevé algebra and characterise its PBW/PHS/Koszul properties. This algebra contains as limiting cases the generalised Sklyanin algebra, Etingof-Ginzburg and Etingof-Oblomkov-Rains quantum del Pezzo and the quantum monodromy manifolds of the Painlevé equations.

9 July 2020

## Beyond semitoric

A compact four dimensional completely integrable system $f \colon M \to \mathbb R^2$ is {\bf semitoric} if it has only non-degenerate singularities, without hyperbolic blocks, and one of the components of $f$ generates a circle action.  Semitoric systems have been extensively studied and have many nice properties: for example, the preimages $f^{-1}(x)$  are all  connected.  Unfortunately, although there are many interesting examples of semitoric systems, the class has some limitation.  For example, there are blowups of $S^2 \times S^2$ with Hamiltonian circle actions which cannot be extended to semitoric systems.  We expand the class of semitoric systems by allowing certain degenerate singularities, which we call {\bf ephemeral} singularities.  We prove that the preimage $f^{-1}(x)$ is still connected for this larger class.  We hope that this class will be large enough to include not only all compact four manifolds with Hamiltonian circle actions, but more generally all complexity one spaces. Based on joint work with D. Sepe.

2 July 2020

## About singular leaves of singular foliations

Joint work with Leonid Ryvkin. For singular foliations, e.g. symplectic leaves of a Poisson structure or Lie group orbits, the dimension of the leaves may vary: When it does, the leaf is said to be singular. We will explain why (formal) neighborhoods of simply connected leaves have surprisingly simple local models. This is in sharp contrast with Poisson structures or Lie algebroids. We will derive some consequences (sometimes conjectural) of these facts in terms of first return map, Androulidakis-Skandalis holonomy groupoid, and the universal Q-manifold that Lavau, Strobl and myself have previously associated to a singular foliation.

25 June 2020

## Generalized cluster structures related to the Drinfeld double of $\mathrm{GL}(n)$

As is well-known, cluster transformations in cluster algebras of geometric type are often modeled on determinant identities, such  short Plucker  relations, Desnanot-Jacobi identities and their generalizations. I will present a construction that plays a similar role in a description of generalized cluster transformations and discuss its applications to generalized cluster structures in $\mathrm{GL}(n)$ compatible with a certain subclass of Belavin-Drinfeld Poisson-Lie backers, in the Drinfeld double of $\mathrm{GL}(n)$ and in spaces of periodic difference operators. Based on a joint work with M. Shapiro and A. Vainshtein.

18 June 2020

## An index theorem on the tempered dual of a real reductive Lie group

Let $G$ be a (real reductive) Lie group. The tempered dual of $G$ is the space of isomorphism classes of irreducible unitary $G$-representations that are contained in the (left) regular representation of $G$ on $L^2(G)$. In this talk, we will report our study  on the geometry of the tempered dual. As an application, we will present an index theorem for proper cocompact $G$-actions. This talk is based on the joint works with Peter Hochs, Markus Pflaum, Hessel Posthuma, and Yanli Song.

11 June 2020

## Hamiltonian geometry of compressible fluids

We describe a geometric framework to study Newton's equations on infinite-dimensional configuration spaces of diffeomorphisms and smooth probability densities. It turns out that several important PDEs of hydrodynamical origin can be described in this framework in a natural way. In particular, the so-called Madelung transform between the Schrödinger-type equations on wave functions and Newton's equations on densities turns out to be a Kähler map between the corresponding phase spaces, equipped with the Fubini-Study and Fisher-Rao information metrics. This is a joint work with G.Misiolek and K.Modin.

4 June 2020

## Quantum minimal surface and noncommutative Kaehler geometry

I will talk about several interrelated topics, based on works 1903.10792 and 2003.03171. Minimal surfaces in Euclidean space can be approximated (in sense of Berezin-Toeplitz quantization) by representations of Yang-Mills algebra given by relations $\forall i\,\sum_j[X_j,[X_j,X_i]]=0$ where $X_i$ are self-adjoint operators. Similarly, complex affine curves are approximated by representations of hermitian Yang-Mills algebra $\sum_k [Z_k^\dagger,Z_k]=\hbar\cdot id$ where $Z_i$ are commuting operators (but not self-adjoint in general). I will explain how the latter equation appears in the context of a version of Kaehler geometry for noncommutative algebras.

9 April 2020

## Local models around Poisson submanifolds

Given a Poisson manifold and a Poisson submanifold we discuss the existence of a 1st order local model around the submanifold. We introduce a class of Poisson submanifolds for which such a local model exists, generalizing Vorobjev's local model for symplectic leaves. We give a general linearization result which includes as special cases the Conn's linearization theorem for fixed points and the Crainic-Marcut linearization theorem for symplectic leaves. On the global side, our results can be thought of as a coisotropic embedding theorem for over-symplectic groupoids into symplectic groupoids. This talk is based on joint on-going work with Ioan Marcut.

16 April 2020

## Van Est Differentiation and Van Est Integration

The classical Van Est theory relates the smooth cohomology of Lie groups with the cohomology of the associated Lie algebra. Some aspects of this theory generalize to Lie groupoids and their Lie algebroids. In this talk, we revisit the van Est theory using the Perturbation Lemma from homological algebra. This leads to precise descriptions of the van Est differentiation and integration at the level of cochains. The talk is based on recent work with Maria Amelia Salazar.

23 April 2020

## Short star-products for filtered quantizations

Let $A$ be a filtered Poisson algebra with Poisson bracket $\lbrace{,\rbrace}$ of degree $-2$. A star product on $A$ is an associative product $*: A\otimes A\to A$ given by $a*b=ab+\sum_{i\ge 1}C_i(a,b),$ where $C_i$ has degree $-2i$ and $C_1(a,b)-C_1(b,a)=\lbrace{a,b\rbrace}$. We call the product *  even if $C_i(a,b)=(-1)^iC_i(b,a)$ for all $i$, and call it short if $C_i(a,b)=0$ whenever $i>{\rm min}({\rm deg}(a), {\rm deg}(b))$.

Motivated by three-dimensional $N=4$ superconformal field theory, In 2016 Beem, Peelaers and Rastelli considered short even star-products for homogeneous symplectic singularities (more precisely, hyperKähler cones) and conjectured that that they exist and depend on finitely many parameters. We prove the dependence on finitely many parameters in general and existence for a large class of examples, using the connection of this problem with zeroth Hochschild homology of quantizations suggested by Kontsevich.

Beem, Peelaers and Rastelli also computed the first few terms of short quantizations for Kleinian singularities of type A, which were later computed to all orders by Dedushenko, Pufu and Yacoby. We will discuss some generalizations of these results.

This is joint work with Douglas Stryker.

30 April 2020

## Failure of twisted Poisson property for monopole plasma

Although the dynamical system for a charged particle in a continuous background distribution of magnetic monopoles is given by a twisted Poisson structure, that for a plasma of such particles is not. (Joint work with Manuel Lainz and Cristina Sardón)

7 May 2020

## Holonomic Poisson manifolds

Holonomicity is a new sort of nondegeneracy condition for holomorphic Poisson structures, closely related to the notion of a log symplectic form, and intimately connected with the geometry of Weinstein's modular vector field.  It encompasses many natural Poisson structures arising in gauge theory, representation theory, and algebraic geometry.  The motivation for the definition comes from deformation theory: a Poisson manifold is holonomic when its space of deformations is "as finite-dimensional as possible", in a sense I will make precise during the talk (via D-modules).  I will describe the basic theory and examples of holonomic Poisson manifolds, along with some concrete classification results, including the discovery of many new irreducible components of the moduli space of Poisson fourfolds.  This talk is based on joint works with Schedler, and Matviichuk--Schedler.

14 May 2020

## From b-Poisson manifolds to singular contact structures

Taking as starting point motivating examples from celestial mechanics and fluid dynamics, we introduce the odd-dimensional counterpart of b-Poisson/log-symplectic structures as Jacobi structures with transversality conditions. We discuss the basic theory and some constructions. In particular,  we prove that a connected component of a  convex hypersurface of a contact manifold can be realized as a connected component of the critical set of a $b^m$-contact structure. In dimension 3, this construction yields the existence of a generic set of surfaces $Z$ such that the pair $(M,Z)$ is a $b^{2k}$-contact manifold and $Z$ is its critical hypersurface.

We also consider classical problems in Hamiltonian/Reeb dynamics and address the Weinstein conjecture on the existence of periodic orbits of the Reeb vector field in this singular set-up. We end up this talk with some applications of this singular Weinstein conjecture to the motivating examples discussed at the beginning.

This is joint work with Cédric Oms.

21 May 2020

## Integrable systems of Calogero-Moser type and moduli spaces of flat connections

The talk will be focused on spin Calogero-Moser systems related to symmetric spaces. They have natural generalizations related to moduli spaces of flat connections.

28 May 2020

## Differential character valued momentum maps and the Teichmüller space

It is well-known that the actions of several diffeomorphism groups of geometric interest do not admit momentum maps. The definition of the Teichmüller space via Riemannian geometry strongly suggest that it is a symplectic reduced space. I will present an extension of the classical momentum map which always exists for actions of diffeomorphism groups possessing the crucial Noether property. This extended momentum map has no longer values in (pre)duals of Lie algebras; its values are in differential character groups. This extended momentum map encodes discrete topological information, something the classical momentum map cannot do. In order to focus the presentation, the Teichmüller space will serve as the example of this theory. The talk is based on joint work with Tobias Diez from TU Delft.

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