Titles/Abstracts

We show that the (infinite-dimensional) space of conformally compact hyperbolic metrics on surfaces with boundary, up to diffeomorphisms fixing the boundary, has a natural symplectic structure. The action of boundary diffeomorphisms is Hamiltonian, making this space into an example of a Hamiltonian Virasoro space. (Joint work with Anton Alekseev.)

The approach to Poisson cohomology using spectral sequences is not new, but it has not been sufficiently explored. In this talk, we will present new Poisson cohomology results for a wide class of Poisson manifolds using spectral sequences. Our technique works well for a broad range of Poisson manifolds, including transversely linearizable Poisson structures, $latex b^m$-Poisson structures, and Poisson structures with transverse simple singularities of corank 2. Our construction elucidates former computations for b-Poisson manifolds (by Guillemin, Miranda, Pires, and Marcut-Osorno) using the classical technique of unfolding applied to $latex b^m$-Poisson structures. This talk is joint work with Eva Miranda.

For any complex reductive algebraic group G, one can associate a completely integrable system J_G over the coadjoint quotient of the dual of the Lie algebra, which generalizes the cotangent bundle of a torus over its Lie algebra dual. The variety J_G plays an important role in geometric representation theory and mathematical physics. In particular, it is a Coulomb branch mathematically defined by Braverman-Finkelberg-Nakajima. I will present recent results on the (homological) mirror symmetry of J_G. I will also present a loop group version of the results, which is joint work with Zhiwei Yun. The latter mirror symmetry can be viewed as a Betti Geometric Langlands correspondence in the wild setting.

Braid varieties are smooth affine varieties associated to any positive braid. Their cohomology contains information about the Khovanov-Rozansky homology of a related link. Special cases of braid varieties include Richardson varieties, double Bruhat cells, and double Bott-Samelson cells. Cluster algebras, introduced by Fomin and Zelevinsky, are a class of commutative rings which are completely determined by some combinatorial input called a seed. I'll discuss joint work with P. Galashin, T. Lam and D. Speyer in which we show the coordinate rings of braid varieties are cluster algebras; that is, braid varieties admit a cluster A-variety structure. Independent work of Casals-Gorsky-Gorsky-Le-Shen-Simental shows that braid varieties also admit a cluster Poisson structure.

Local deformations of log symplectic complex manifolds can be studied using a combinatorial gadget, called smoothing diagrams. I will discuss the rules such diagrams satisfy, mention a few classification results, and conjecture how the combinatorics of these diagrams predict the geometry of the deformed log symplectic manifolds. Part of the presented results are joint with Pym and Schedler.

We propose a new approach to building log-canonical coordinate charts for any simply-connected simple Lie group G and arbitrary Poisson-homogeneous bracket on G associated with Belavin-Drinfeld data. Given a pair of representatives r,r′ from two arbitrary Belavin–Drinfeld classes, we build a rational map from G with the Poisson structure defined by two appropriately selected representatives from the standard class to G equipped with the Poisson structure defined by the pair r,r′. In the A_n case, we prove that this map is invertible whenever the pair r,r′ is drawn from aperiodic Belavin-Drinfeld data and apply this construction to recover the existence of a regular complete cluster structure compatible with the Poisson structure associated with the pair r,r′. This is joint work with M. Shapiro and A. Vainshtein.

In this talk I will explain how to apply the techniques of partially wrapped Fukaya categories and stop removal (due to Ganatra-Pardon-Shende) to log symplectic surfaces. Every closed log symplectic surface can be decomposed into open symplectic surfaces and tubular neighbourhoods of the log degeneracy loci. Having already directly constructed the Fukaya category of any whole closed log symplectic surface, we can use this decomposition as a tool for explicit computation: While computing the whole category explicitly quickly becomes complicated for surfaces of higher genus and with more complicated degeneracy loci, the wrapped Fukaya categories of the smaller components are simpler to compute and can then be combined to obtain the Fukaya category of the whole.

Joint-work with J. Jiang (in progress) based on arXiv:2301.08706. We construct a Nash-type blowup method on arbitrary almost Lie algebroids with a specific focus on applications to foliated bivector fields, (i.e., bivector field that induces an integrable distribution in the sense of Stefan-Sussman) particularly in the context of Poisson structures. The study reveals that the cotangent Lie algebroid can be transformed through the Nash-blowup into a Lie algebroid whose anchor map is injective on an open dense subset. We provide concrete examples to elucidate these theoretical constructions.

In this talk, we will give an overview of (-1)-shifted derived Poisson manifolds in the $C^infty$-context, and discuss the quantization problem. We describe the obstruction theory and prove that the linear (-1)-shifted derived Poisson manifold associated to any $L_infty$-algebroid admits a canonical $BV_infty$ quantization. This is a joint work with Kai Behrend and Matt Peddie.

The Atiyah class of a dg manifold $(mathcal{M},Q)$ is the obstruction to the existence of an affine connection on the graded manifold $mathcal{M}$ that is compatible with the homological vector field $Q$. The Todd class of dg manifolds extends both the classical Todd class of complex manifolds and the Duflo element of Lie theory. Using Kontsevich's famous formality theorem, Liao, Xu and I established a formality theorem for smooth dg manifolds: given any finite-dimensional dg manifold $(mathcal{M},Q)$, there exists an $L_infty$ quasi-isomorphism of dglas from an appropriate space of polyvector fields $mathcal{T}_{oplus,operatorname{poly}}^{bullet}(mathcal{M})$ endowed with the Schouten bracket $[-,-]$ and the differential $[Q,-]$ to an appropriate space of polydifferential operators $mathcal{D}_{oplus,operatorname{poly}}^{bullet}(mathcal{M})$ endowed with the Gerstenhaber bracket $llbracket -,- rrbracket$ and the differential $llbracket m+Q,- rrbracket$, whose first Taylor coefficient (1) is equal to the composition of the action of the square root of the Todd class of the dg manifold $(mathcal{M},Q)$ on $mathcal{T}_{oplus,operatorname{poly}}^{bullet}(mathcal{M})$ with the Hochschild–Kostant–Rosenberg map and (2) preserves the associative algebra structures on the level of cohomology. As an application, we proved the Kontsevich–Shoikhet conjecture: a Kontsevich–Duflo type theorem holds for all finite-dimensional smooth dg manifolds. This last result shows that, when understood in the unifying framework of dg manifolds, the classical Duflo theorem of Lie theory and the Kontsevich–Duflo theorem for complex manifolds are really just one and the same phenomenon.

I will discuss an approach to classification of certain irreducible Harish-Chandra modules (that should be related to unitary representations) as quantizations of singular lagrangian subvarieties in singular symplectic varieties. The approach allows to classify these Harish-Chandra modules in terms of some basic geometric data assuming, roughly speaking, that the lagrangian subvariety in question is not too singular. This is based on some of my solo work, 1605.00592, 1810.07625, the joint monograph with Mason-Brown and Matvieievskyi, 2108.03453, and my work with Shilin Yu, 2309.11191.

I will outline the results of the preprint arXiv:2312.04859 (joint work with M. Gekhtman). We constructed compatible generalized cluster structures on the Poisson varieties $(mathrm{SL}_n^{dagger},pi_{mathbf{Gamma}}^{dagger})$ parameterized by Belavin-Drinfeld triples $mathbf{Gamma}:=(Gamma_1,Gamma_2,gamma)$. These varieties are Zariski open subsets of $mathrm{SL}_n$, and they are obtained as rational images of the Poisson dual of the Poisson-Lie group $(mathrm{SL}_n,pi_{mathbf{Gamma}})$ where $pi_{mathbf{Gamma}}$ is the Poisson bivector from the Belavin-Drinfeld class. Along the way, we also constructed a family of Poisson birational quasi-isomorphisms, which are birational maps that preserve the Poisson brackets and which are equivariant with respect to mutations.

Bethe subalgebras form a family of maximal commutative subalgebras of the Yangian Y(gl(n)) indexed by points of the Deligne-Mumford compactification of M(0,n+2). Over a point in the real locus of this moduli space, the corresponding Bethe subalgebra acts with simple spectrum on a given tame representation of Y(gl(n)). I will discuss the monodromy action on the fiber of the resulting covering, by a type of cactus group, as well as the identification of the fibers with Gelfand-Tsetlin keystone patterns. This is joint work with Anfisa Gurenkova and Leonid Rybnikov.

"Perverse coherent t-structure on the equivariant coherent sheaves on the nilpotent cone was introduced by Bezrukavnikov and used in various instances in geometric representation theory. We attempt to generalize this notion to other conical symplectic singularities. Instead of stratification by group orbits, we consider the stratification by symplectic leaves, and instead of the group-equivariance condition, we consider the equivariance over (a variant of) Poisson Lie algebroid. We discuss a potential relation of this notion to stable envelopes and representations of quantizations in positive characteristic."

I will explain how to use symplectic double groupoids to solve the problem of determining the fundamental degrees of freedom of a generalized Kähler (GK) structure. The underlying holomorphic structure of a GK manifold is characterized by a square in the double category of holomorphic symplectic double groupoids, and a GK metric is determined by a Lagrangian submanifold. This is joint work with Daniel Alvarez and Marco Gualtieri.

We construct a simplified Morse-Bott-Smale chain complex for a manifold admitting a torus action and a special type of invariant Morse-Bott functions. This simplified chain complex admits a nice correspondence with the invariant Witten instanton complex. In our settings, we allow the possibility that the unstable manifolds are nonorientable. The key to overcome this difficulty is clarifying the orthogonal representation of the Euclidean space around the critical submanifold.

TBA

Differential reduction algebras are associated to oscillator representations of a Lie (super)algebra. They act on multiplicity spaces in infinite dimensional tensor products and are h-deformations of the Weyl/Clifford superalgebras. They are also connected to solutions of the dynamical Yang-Baxter equation. I will report on joint work with Dwight Williams II, in which we compute a presentation the reduction algebra in the case of the symplectic Lie algebra sp(4). The algebras turn out to be generalized Weyl algebras and thus their irreducible representations are known. The case of gl(n) has been studied previously by Herlemont, Khoroshkin, and Ogievetsky.

Let $f:M to N$ be a fibration between two oriented manifolds of dimension m and m-2, respectively. Garcia-Naranjo, Suarez-Serrato & Vera defined a Poisson structure associated to any such fibration by comparing the pullback of the volume form on N to the volume form on M. In this talk we study a Poisson structure induced by a fibration with Lefschetz singularity. In particular, we compute the formal Poisson cohomology at the singularity. Note that generally, using the construction above, we obtain a family of Poisson structures by varying the volume forms. Using the result for cohomology, we obtain a formal classification for Poisson structures arising in such a way. This is based on joint work with L. Toussaint.

The double Dyck path algebra (Bqt) and its polynomial representation arose as a central figure in the proof of the celebrated Shuffle Theorem of Carlsson and Mellit. Despite its deep connections with important objects such as the Shuffle algebra, the DAHA, Macdonald polynomials, and the Hilbert scheme of points on the plane, very little was known of its representation theory. I will discuss joint work with E. Gorsky and J. Simental where we initiate the study and classification of the calibrated representations of Bqt. We study the hom spaces between such representations, define a 'generically' monoidal structure on them, and realize tensor products of the polynomial representation geometrically.

Any cluster Poisson variety X gives rise to its cluster symplectic double D. The latter has two real forms: the positive locus, and the unitary locus. Cluster quantization of the positive locus was done in our work with V. Fock. In the talk I will describe cluster quantization of the unitary locus, given by the representation of the corresponding *-algebra. Here are two examples: (i) The Unitary group carries a (version of) unitary double structure. (ii) The double S' of a decorated surface S carries an orientation reversing involution s. The space of antiholomorphic s-equivariant G(C)-local systems on S' is essentially the unitary double of the space of G(C)-local systems on S.

Let M be a Poisson manifold with a Hamiltonian action of a semisimple complex group G. Whittaker reduction is a type of Hamiltonian reduction along the corresponding moment map which is associated to transverse slices to the adjoint action. We introduce a multiplicative analogue of this construction. In this setting, M is a quasi-Poisson manifold whose moment map takes values in the group G, and transversal slices to the conjugation action are indexed by elements of the Weyl group. We show that M can be reduced along these slices to produce manifolds which are quasi-Poisson for the action of a reductive subgroup of G of smaller rank. This is joint work with Maxence Mayrand.

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. Time permitting, we will also discuss recent work in progress, 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.

We classify twistings of Grothendieck’s differential operators on a smooth variety X in prime characteristic p. We prove isomorphism classes of twistings are in bijection with $H^2(X, mathbb Z_p (1))$, the degree 2, weight 1 syntomic cohomology of X. We also discuss the relationship between twistings of crystalline and Grothendieck differential operators. Twistings in mixed characteristic are also analyzed.

We consider the symplectic groupoid: pairs (B,A) with A unipotent upper-triangular matrices and $Bin GL_n$ being such that $tilde A=BAB^T$ are also unipotent upper-triangular matrices. We explicitly solve this groupoid condition using Fock–Goncharov–Shen cluster variables and show that for B satisfying the standard semiclassical Lie–Poisson algebra, the matrices B, A, and $tilde A$ satisfy the closed Poisson algebra relations. Identifying entries of A and $tilde A$ with geodesic functions on a closed Riemann surface of genus $g=n−1$, we construct the geodesic function $G_B$ for geodesic joining two halves of the Riemann surface. We thus obtain the complete cluster algebra description of Teichmüller space $T_{2,0}$ of closed Riemann surfaces of genus two . We discuss also the generalization of our construction for higher genera. (based on joint paper ArXiv:2304.05580 with Misha Shapiro)

Barcodes have been discovered and rediscovered many times as a useful way to organize the information of a filtration. I will discuss an ongoing project, joint with D. Alvarez-Gavela and Y. Eliashberg, to use moduli of barcodes to encode homological fillings of Legendrian submanifolds, ie objects of Fukaya categories.

I will talk about generalizing the Chevalley restriction theorem to general algebraic varieties where the classical theorem corresponds to the case of the affine line. Using this, we can demonstrate the construction of Cherednik algebras for algebraic curves using Hamiltnoian reduction.

Locally conformal cosymplectic structures on smooth odd-dimensional manifolds were introduced by Vaisman in 1980. They are closely related to almost contact metric structures, Sasakian and locally conformal Kahler structures. In this talk, our goal is to discuss locally conformal cosymplectic structures on Lie groupoids and their infinitesimal counterparts. We will start with a brief review of locally conformal cosymplectic manifolds.

Representation-theoretic constructs such as the induction functor Ind_H^G and the space of intertwiners Hom_G have well-known geometric analogues defined using symplectic reduction. A first work (with T. Ratiu) provides similar constructions in the prequantum category, establishes “Frobenius Reciprocity” as a bijection t: Hom_G(X, Ind_H^G Y) —> Hom_H(Res^G_H X, Y), and conjectures that 1º) t is a diffeomorphism when these spaces are endowed with their natural subquotient diffeologies, 2º) t respects the reduced diffeological forms they may carry. A second work (with G. Barbieri and J. Watts) proves the conjecture, and gives new sufficient conditions for the reduced forms to exist. For example, when G is compact they always exist, and restrict to the Sjamaar-Lerman-Bates forms on each stratum of the reduced space.

The convolution algebra of a Lie groupoid can be thought of as a model for the non-commutative "functions" on the orbit space. In this way, convolution algebras provide us with a bridge between the topics of non-commutative geometry and Lie groupoid theory. The most famous example of this correspondence (the non-commutative torus) actually arises from a strict 2-group which is a special case of a double Lie groupoid. Motivated by the goal of understanding the relationship between the convolution algebras of spaces equipped with multiple products, I will discuss the analogue of Haar systems and convolution algebras for double groupoids.

Let $G$ be a reductive group such that the identity component $G^circ$ is a torus and let $V$ be a $G$-module. Let $mucolon Voplus V^asttomathfrak{g}^ast$ denote the homogeneous quadratic moment map, $N = mu^{-1}(0)$, and $N/!!/G$ the complex symplectic quotient. We will present recent results regarding the question of when two such symplectic quotients are isomorphic as Poisson varieties. Call $V$ emph{minimal} if the singular set in $N$ has codimension at least $4$. We show that if $V$ is faithful, $1$-modular, and minimal, then $N/!!/G$ determines the representation $Voplus V^ast$. If $V$ is not minimal, then $V$ and $G$ can be replaced with a minimal representation with isomorphic symplectic quotient. Similar results hold for the case of real symplectic quotients by compact Lie groups with identity component a torus with the additional hypothesis that $V$ is stable as a $G$-module. (Joint work with Hans-Christian Herbig and Gerald Schwarz)

Let $A$ be a filtered Poisson algebra with Poisson bracket $lbrace{,rbrace}$ of degree $-2$. A {it star product} on $A$ is an associative product $*: Aotimes Ato A$ given by $$a*b=ab+sum_{ige 1}C_i(a,b),$$ where $C_i$ has degree $-2i$ and $C_1(a,b)-C_1(b,a)=lbrace{a,brbrace}$. We call the product * {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 star-products for homogeneous symplectic singularities (more precisely, hyperK"ahler 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 twisted traces and 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 Daniel Klyuev, Eric Rains and Douglas Stryker.

It is well-known that the Drinfeld–Jimbo quantum groups can be viewed as quantized coordinate algebras of the dual Poisson-Lie groups. A quantum symmetric pair consists of a quantum group and a coideal subalgebra, called an i-quantum group. In this talk, I will explain that i-quantum groups can be viewed as quantized coordinate algebras of Poisson homogeneous spaces of dual Poisson groups. The consruction fits into the general picture, called the quantum duality principal. The relation with cluster algebras will be briefly discussed.