I'll review the algebraic quasi-Hamiltonian construction of the Poisson structures in global Lie theory, i.e. on the moduli spaces of Stokes representations of the wild surface groupoids, on the multiplicative/Betti side of the Riemann-Hilbert-Birkhoff correspondence, complementary to the earlier holomorphic and hyperkahler approaches. They generalise the Poisson structures underlying the Drinfeld-Jimbo quantum groups. This TQFT approach "fusion of fission spaces" leads to a theory of Dynkin diagrams for the symplectic wild character varieties, viewed as global analogues of Lie groups. If time permits I'll also discuss how to characterise polystable Stokes representations as those with linearly reductive differential Galois groups (B.-Yamakawa 2023), generalising a result of Richardson.

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All sorts of algebro-geometric moduli spaces (of stable curves, stable sheaves on a CY 3-folds, flat bundles, Higgs bundles...) are best understood as objects in derived geometry. Derived enhancements of classical moduli spaces give transparent and intrinsic meaning to previously ad-hoc structures pertaining to, for instance, enumerative geometry and are indispensable for more for more advanced constructions, such as categorification of enumerative invariants and (algebraic) deformation quantization of derived symplectic structures. I will outline how to construct such enhancements for moduli spaces in global analysis and mathematical physics -that is, solution spaces of nonlinear PDEs- in the framework of derived differential geometry and discuss the elliptic representability theorem, which guarantees that, for elliptic equations, these derived moduli stacks are bona fide geometric objects (Artin stacks at worst). If time permits I'll discuss applications to enumerative geometry (symplectic Gromov-Witten and Floer theory) and derived symplectic geometry (the global BV formalism). 

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Character varieties and more generally representation varieties give many interesting Poisson structures. We describe a general procedure of obtaining (almost) Poisson structures on such varieties, as well as their deformation quantizations, by applying factorization homology to deformations of symmetric monoidal categories. In the case of the category of representations of a group, we recover the Goldman bracket and its various Poisson and quasi-Poisson generalizations. Based on a joint work with Eilind Karlsson, Corina Keller and Lukas Müller.

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I will explain a construction of a combinatorial 2d TCFT, assigning partition functions to triangulated cobordisms (as chain maps between spaces of states), in such a way that a Pachner flip induces a Q-exact change. More generally, the partition function becomes a nonhomogeneous closed cochain on the “flip complex.” One has a combinatorial counterpart of the BV operator G_{0,-} arising from evaluating the theory on a special 1-cycle on the flip complex of the cylinder. The local input for the model is a cyclic A-infinity algebra, with the operation m_3 playing the role of the BRST-primitive G of the stress-energy tensor T=Q(G).

I will also describe a way to incorporate invariance-up-to-homotopy with respect to the second 2d Pachner move (stellar subdivision/aggregation). This version of the model is based on secondary polytopes of Gelfand-Kapranov-Zelevinsky and uses a certain enhancement (by extra homotopies) version of a cyclic A-infinity algebra as input.

The talk is based on a joint work with Andrey Losev and Justin Beck, https://arxiv.org/pdf/2402.04468.pdf.

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Chaidez and Edtmair have recently found the first examples of dynamically convex domains in R4 that are not symplectomorphic to convex domains (called symplectically convex domains), answering a long-standing open question. In this talk we shall present new examples of such domains without referring to Chaidez-Edtmair’s criterion. We shall show that these domains are arbitrarily far from the set of symplectically convex domains in R4 with respect to the coarse symplectic Banach-Mazur distance by using an explicit numerical criterion for symplectic non-convexity. This is joint work with J.Dardennes, V.Ramos and J.Zhang.

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Wigner initiated the study of random matrices by studying hermitian N x N matrices equipped with a natural Gaussian measure, and he discovered remarkable features in the large N limit. In joint work with Ginot, Hamilton, and Zeinalian, we revisited this system within the framework of shifted Poisson geometry and Batalin-Vilkovisky quantization, and we found that cyclic cohomology and the LQT theorem illuminate the emergence of a meaningful large N limit. Our work can be seen as a kind of noncommutative BV formalism, and it suggests fruitful directions for future exploration. This talk will explain the relevant background from random matrices and cyclic cohomology.

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I explain several old and relatively new constructions of W - algebras , explain the ideology of screenings and how to glue vertex algebras in order to create the new ones

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Moduli spaces of logarithmic connections on Riemann surfaces have a rich geometric structure, and in particular can be made into symplectic spaces isomorphic to (complex) character varieties. The general wisdom is that the moduli spaces are obtained by gluing local pieces at each simple pole, involving choices of coadjoint orbits for a (dual) complex reductive Lie algebra: in particular quantising such orbits is a preliminary step towards quantising the moduli spaces themselves.

In this talk we will aim at a review of this story, and then describe a recent extension for irregular singular (= wild) meromorphic connections. The quantisation of the corresponding orbits is based upon a result of Alekseev--Lachowska, and is joint work with D. Calaque, G. Felder, and R. Wentworth. (The main ingredient are the Shapovalov form for certain representations of truncated current Lie algebras, generalising the (generalised) Verma modules.)

If time allows, we will also recall how the moduli spaces can be deformed, and describe the universal space of local deformations: this is joint work with P. Boalch, J. Douçot, and M. Tamiozzo.

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Mutually completely orthogonal actions on a symplectic manifold yield dual pairs of momentum maps (in the sense of Weinstein). We review several such dual pairs for diffeomorphism group actions: the Marsden-Weinstein ideal fluid dual pair, the Holm-Marsden EPDiff dual pair, and a recent dual pair involving the group of volume preserving diffeomorphisms. Coadjoint orbits of diffeomorphism groups can be obtained via symplectic reduction in these dual pairs.

This is joint work with Stefan Haller (University of Vienna) and Francois Gay-Balmaz (NTU Singapore).

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6pm Timisoara

The Kontsevich–Rosenberg principle of noncommutative algebraic geometry says that a structure on an associative algebra A has a (noncommutative) geometric meaning whenever it induces a genuine corresponding geometric structure on representation spaces. This principle led to the discovery of bisymplectic structures such that the associated representation spaces are respectively hamiltonian GLn-varieties. On the other hand, in higher algebra, one can consider Calabi–Yau structures on differential graded categories which induce in the world of derived algebraic geometry shifted symplectic structure on the respective derived moduli stacks. We show that relative Calabi–Yau structures on noncommutative moment maps give rise to (quasi-)bisymplectic structure. This is based on joint work with Bozec–Calaque.

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5pm Luxembourg

We will consider Geometric Quantization on general Kähler manifold and propose a program for compatible constructions of the quantization of functions, the Hilbert space structure, and the dependence on the choice of the Kähler structure, fixing only the underlying symplectic manifold and a prequantum line bundle.

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5pm Odense

A complexity-one space is a symplectic 2n-manifold equipped with Hamiltonian action of a torus of dimension n-1. The momentum map for such an action can be identified with n-1 real valued functions. On the other hand, an integrable system on such a manifold is the data of n functions. This motivates several natural questions: given a complexity-one space, when can an additional function be found to produce an integrable system? When can the resulting system be chosen to be toric? When can it be chosen to have no degenerate singularities?

In this talk, I will discuss answers to various versions of these questions, both in dimension four and higher. In particular, I will mention previous results of Karshon and Hohloch-Sabatini-Sepe-Symington in dimension four, and describe new results in higher dimensions. Parts of the new work that I will present are joint with Sonja Hohloch, Susan Tolman, and Jason Liu.

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10am Urbana

After a short history of Statistical Mechanics, I will recall the definition of a Gibbs state built with the Hamiltonian of a vector field defined on a symplectic manifold. First introduced by the famous American mathematician and physicist Josiah Willard Gibbs (1839--1902), Gibbs states of an isolated mechanical system are statistical states stationary in time, considered by physicists as states of thermodynamic equilibrium.

The French mathematician and physicist Jean-Marie Souriau (1922--2012) proposed a generalization of Gibbs states in which the Hamiltonian, which is a moment map of the one-dimensional group of translations in time, is replaced by a moment map of the symplectic action of a general connected Lie group on a symplectic manifold. This moment map (whose existence is assumed)  takes its values in the dual of the Lie algebra of the considered Lie group and is equivarian with respect to its action on the symplectic manifold, and an affine action on the dual of its Lie algebra whose linear part is the coadjoint action.

In fact, this generalization was already proposed by Gibbs, who shortly considered in his book the product of the one-dimensional group of translations in time and the three-dimensional group of rotations in space.

I will present the properties of these generalized Gibbs states, describe the associated generalized thermodynamic functions, in relation with the important notion of entropy. I will discuss several examples, including some cases in which generalized Gibbs states do exist on orbits of some affine actions of the considered Lie group on the dual of its Lie algebra, although no generalized Gibbs state can exist on the coadjoint orbits themselves.

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Short abstract:

Quantum Toda lattice may solved by means of the Representation Theory of semisimple Lie groups, or alternatively by using the technique of the Quantum Inverse Scattering Method. A comparison of the two approaches sheds a new light on Representation Theory and leads to   a number of challenging questions.

Extended abstract:

The fundamental works of Harish-Chandra completed by the famous Gindikin-Karpelevich formula created a sort of paradigm in Harmonic analysis on semisimple Lie groups. It became a common wisdom that the asymptotical behavior of matrix coefficients of irreducible representations, such as spherical or Whittaker functions, is described  by simple factorized formulae. Similar results hold also for Lie groups over local fields; their extension to Lie groups over the rings of adèles leads to a highly romantic Langlands program (or at least to some part of it, as exposed in the famous book of Gelfand and Pyatetsky-Shapiro). On the other hand, the eigenfunctions of the Laplace operators themselves are given by much less manageable integral expressions or infinite sums involving complicated functions of several variables. The rather radical change of this paradigm brought about by the Quantum Inverse Scattering Method consists in the following main points:

(i) There exists a representation (in the sense of Dirac) in which the algebras of quantum integrals of motion for the Toda lattice, or of the Sutherland system are freely generated by first order difference operators.

(ii) In this representation the common eigenvectors of these algebras (alias, Whittaker or spherical functions) are  decomposable, i.e., they are products of functions of one variable.

(iii) The transformation operators which relate this representation to the standard coordinate representation may be constructed recursively and amount to a sequence of ordinary Fourier transforms.

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5pm Bourgogne

Deformation quantization is a general framework for quantizing classical mechanical systems by deforming the classical observable algebra into a non-commutative algebra, with the deformation parameter playing the role of the reduced Planck constant hbar. In a formal setting, Kontsevich's Formality Theorem provides a formula to quantize any Poisson structure on R^n. However, the weights appearing in this formula make it hard to understand its convergence properties, and the existence of "strict deformation quantizations", where the deformation parameter can be evaluated to the actual physical value of hbar, remains a widely open problem. 

In this talk, I will present a combinatorial approach to the formal deformation quantization of polynomial Poisson structures on R^n, due to Barmeier--Wang, which does not involve any weights. This makes the convergence properties more accessible, and yields strict quantizations for several polynomial Poisson structures on R^n. A particularly striking example is the quantization of the constant Poisson structure on R^2 perturbed by a quadratic term, which results in a quantum Weyl algebra that is, in some sense, much "bigger" than the usual Weyl algebra. 

This talk is based on joint work with S. Barmeier.

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5pm Hannover

We will start explaining that (quasi-)hamiltonian reduction can be understood as a particular instance of lagrangian intersections, within the framework of shifted symlectic geometry. This will naturally lead to consider shifted versions of reduction. We will finally state our main result, saying that the derived critical locus of a function defined on a quotient space X/G can be obtained as a shifted reduction of the derived critical locus of the corresponding function on X. This somehow follows from a more general fact, stating that shifted reduction commutes with lagrangian intersection. 

The talk is based on a joint paper with Mathieu Anel.

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5pm Montpellier

The Kontsevich-Tamarkin approach gives deformation quantisations for all Poisson structures on manifolds and smooth varieties (algebroids in the latter case),  but the argument does not extend to singular schemes and spaces. I will explain how a modified argument exploiting an involution of the Hochschild complex gives A-infinity deformation quantisations of (0-shifted) Poisson structures whenever the cotangent complex is perfect (in particular, for LCI singularities and for dg manifolds).

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In this talk, based on joint work with Stephane Geudens and Marco Zambon, we develop the deformation theory of symplectic foliations, i.e. regular foliations equipped with a leafwise symplectic form. The main result is that each symplectic foliation is attached with a cubic L∞ algebra controlling its deformation problem. Indeed, we establish a one-to-one correspondence between the small deformations of a given symplectic foliation and the Maurer–Cartan elements of the associated L∞ algebra. Further, we prove that, under this one-to-one correspondence, the equivalence by isotopies of symplectic foliations agrees with the gauge equivalence of Maurer–Cartan elements. Finally, we show that the infinitesimal deformations of symplectic foliations can be obstructed.

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5pm Salerno

In this talk, we will explore the relationship between the geometry and topology of a complexity-one four-manifold and the combinatorial data that encode it. We will use a generators-and-relations description for the even part of the equivariant cohomology of the manifold to see what geometric aspects the equivariant cohomology determines. Namely, it allows us to reconstruct the diffeotype but not the complex structure. The talk will be driven by specific examples. It is based on joint work with Liat Kessler; and with Liat Kessler and Susan Tolman.

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11am Ithaca

This talk gives an introduction and overview of recent developments around so-called hypersemitoric systems. These are two degree of freedom integrable Hamiltonian systems on 4-dimensional symplectic manifolds with possibly mild degeneracies where one of the integrals gives rise to an effective Hamiltonian S^1-action, i.e., these systems have a global S^1-symmetry.

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5pm Antwerp

I will start with explaining how to view naturally Contact Geometry as a chapter of Symplectic Geometry. This understanding of contact structures serves for general contact structures and is very effective in applications. Instead of ad hoc definitions in the contact case, we have therefore obvious concepts coming from the standard symplectic picture. In particular, the contact Hamiltonian Mechanics, extensively studied nowadays, can be fully expressed in terms of the traditional symplectic one. Also a natural understanding of contact reductions with respect to group actions comes easily from the Marsden-Weinstein-Meyer symplectic reduction.

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5pm Warsaw

The theory of graded manifolds is a very important tool in Poisson geometry. In our talk we will compare the concepts of a supermanifold and a graded manifold. Further we will discuss the following question.

In geometry there is a well-known notion of a covering space. A classical example is the following universal covering: $p:\mathbb R \to S^1$, $t \mapsto \exp (it)$. Another example of this notion is a flat covering or torsion-free covering in the theory of modules over a ring. All these coverings satisfy some common universal properties. In the paper ``Super Atiyah classes and obstructions to splitting of supermoduli space'', Donagi and Witten suggested a construction of a first obstruction class for splitting a  supermanifold. It appeared that an infinite prolongation of the Donagi-Witten construction satisfies universal properties of a covering. In other words this is a covering of a supermanifold in the category of graded manifolds.

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1pm Belo Horizonte

We begin with a cluster algebra solution of the problem of symplectic groupoid: what are log-canonical coordinates on the manifolds of pairs (B,A) where B is an SL_N matrix, A is unipotent upper-triangular matrix, and à = BAB^T is again a unipotent upper-triangular matrix. Solutions obtained possess a natural Poisson algebra and, upon identification of entries of A and à with geodesic functions on a Riemann surface, we give a cluster algebra description of Teichmuller spaces of closed Riemann surfaces of genus  g ⩾ 2. We have a complete cluster algera description for the case g = 2 and germs of such description for higher genera. Time permitting, I'll discuss quantization and a relation to Cherednick's DAHA.

Based on the forthcoming joint paper with Misha Shapiro.

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7pm Moscow

The classical construction of characteristic classes via the Chern-Weil morphism can be seen as a special instance of more abstract construction due to Lecomte, which assigns certain cohomology classes to every extension of Lie algebras together with a representation. In this talk we explain a result which upgrades Lecomte's construction to the setting of L-infinity algebras and representations up to homotopy. As an application, we introduce a Chern-Weil map for principal 2-bundles with connection. This is a joint work with Sebastian Herrera (Sao Paulo).

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