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Conference in honour of Michèle Vergne's 80th birthday
Programme
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Conference openning:
Olivier Biquard, Paul-Emile Paradan, Michèle Vergne
The Video of the Opening Ceremony: video
The File of Michèle Vergne's speech: speech
Anton Alekseev (Université de Genève)
Title: Many lives of Kashiwara-Vergne equations slides videos
Abstract: In 1978, Kashiwara and Vergne stated a problem on properties of the Baker-Campbell-Hausdorff series which consists of a pair of mysteriously looking linear equations on two elements of a free Lie algebra with two generators. A positive solution of the Kashiwara-Vergne (KV) problem implies the Duflo isomorphism between the center of the universal enveloping algebra and the ring of invariant polynomials.
In this talk, we review some of the interpretations and applications of the KV equations. In particular, the first KV equation can be restated in terms of the Goldman bracket on homotopy classes of oriented loops on a sphere with three holes. And the second equation leads to the notion of divergence in non-commutative calculus. Among other things, we will present some new uniqueness results for non-commutative divergence cocycles.
The talk is based on joint works with N. Kawazumi, Y. Kuno, E. Meinrenken, F. Naef and C. Torossian.
Alexander Barvinok (University of Michigan)
Title: Some quick formulas for the volumes of and the number of integer points in higher-dimensional polyhedra
slides videos
Abstract: Given a polyhedron P, defined as the intersection of an affine subspace and the non-negative orthant of R^n, we want to estimate the volume of $P$ and the number of integer points in P. Although the latter problem is #P-hard, I will present an approach, based on the maximum entropy principle rooted in statistical physics, which in certain situations provides a quick and reasonably accurate estimate. This is based on a joint work with J.A. Hartigan and on a joint work with M. Rudelson.
Jean-Michel Bismut (Université Paris Scalay)
Title: From localization formulas to orbital integrals slides videos
Abstract: I will explain how my work was directly influenced by Nicole Berline and Mich le Vergne's results and ideas on localization formulas in equivariant cohomology and on representation theory. In particular, the geometric evaluation of semi-simple orbital integrals associated with the Casimir operator can be viewed in part as a formal consequence of their formulas.
Benjamin Enriquez (Université de Strasbourg)
Title: On polylogarithm functions on Riemann surfaces slides videos
Abstract: The hyperlogarithm (HL) functions are a class of functions on the punctured complex plane, which were introduced in relation with monodromy computations (Poincar , Lappo-Danilevskii). They were recently used (Brown) in relation with the problem of identification of multiple zeta values with certain classes of periods (Goncharov-Manin) as well as with Feynman integral computations in QFT (Brown, Panzer). Elliptic analogues of the HL functions were also introduced and applied to QFT computations (Brown-Levin, Broedel-Duhr-Dulat-Tancredi). We introduce and study analogues of the algebras of HL functions for an arbitrary affine complex curve. Time permitting, we will explain the relation with the construction of an alternative analogue of the algebra of HL functions, recently proposed by d'Hoker-Hidding-Schlotterer (joint work w. F. Zerbini).
Giovanni Felder (ETH Zurich)
Title: Hypergeometric integrals, hook formulas, and Whittaker vectors slides videos
Abstract: The classical hook-length formula in combinatorics relates the number
of standard Young tableaux with given shape to the product of
hook-lengths of the Young diagram. A generalization to skew Young
diagrams is due to Naruse, who obtains it as a limit of a multivariate
generalization arising from equivariant Schubert calculus.
We show that these combinatorial objects appear in a relation between
multidimensional hypergeometric integrals motivated by 3D mirror
symmetry and in the diagonalization of the action of the centre of
U(gl_n) on the space of Whittaker vectors in the tensor product of
dual Verma modules with fundamental modules.
The talk is based on joint work with Andrey Smirnov, Vitaly Tarasov
and Alexander Varchenko.
Ezra Getzler (Northwestern University)
Title: Moment maps in multisymplectic geometry and differential characters slides videos
Abstract: A moment map for a Lie group G acting on a symplectic manifold (M,w) is a G-equivariant Poisson morphism from the underlying Poisson manifold of M to the Poisson manifold g*, where g is the Lie algebra of G. Equivalently, a moment map is a G-equivariant morphism from g to the Poisson Lie-algebra (O(M),{-,-}) of M. Multisymplectic geometry provides a framework for adapting symplectic geometry to classical field theories (and potentially also to quantum field theories). In multisymplectic geometry, the differential form "w" is a closed (n+1)-form, non-degenerate in the sense that it defines an injective map from the vector fields to n-forms. (Symplectic geometry is the case n=1.) Rogers introduced the analogue of the Poisson Lie-algebra for multisymplectic manifolds: it is a Lie n-algebra, or L-infinity algebra concentrated in homological degrees [0,n), and a moment map is a G-equivariant morphism of L-infinity algebra to it from g.
A better way of thinking of moment maps is as a lift of the action of G to the prequantization line bundle of (M,w), in other words, a lift of "w" to a G-equivariant differential character. In this talk, we recast Roger's multisymplectic moment maps, further developed by Callies, Fr gier, Rogers and Zambon, in terms of equivariant differential characters. As an example, we discuss the invariance of Chern-Simons theory under diffeomorphisms.
Pascale Harinck (École Polytechnique)
Title: Local zeta functions for a class of p-adic symmetric spaces slides videos
Abstract: Our goal is to generalize Tate's functional equation for local zeta functions to a class of p-adic symmetric spaces associated with commutative regular prehomogeneous spaces. First, we establish the classification of such irreducible prehomogeneous space (G,V) and give an explicit description of the open orbits of G in V. Each open orbit is a symmetric space G/Hp, p=1,...,r and there exist minimal principal series which are distinct from Hp for all p. We associate zeta functions to these data and prove an explicit functional equation satisfied by these zeta functions. These results are the analogue in the p-adic case of those of N. Bopp and H. Rubenthaler obtained in the real setting. When G has a unique open orbit G/H in V, as well as the S-parabolic subgroups (where S is the involution defining H), we prove an abstract functional equation for the zeta functions associated to irreducible $H$-distinguished representations of G. In the group case
G/H = Go x Go/diag, our zeta functions coincide with those of Godement-Jacquet. I will explain perspectives in the case of a unique open orbit. The talk is based on joint work with H. Rubenthaler.
Nigel Higson (Penn State University)
Title: Tempered representations and pseudodifferential operators on symmetric spaces slides videos
Abstract: A beautiful but difficult theorem of David Vogan asserts that if G is a real reductive group, and if K is a maximal compact subgroup, then every irreducible representation of K occurs as a minimal K-type in precisely one so-called tempiric representation of G, and that each tempiric representation of G has a unique minimal K-type. As I shall explain, this links the representation theories of G and K in a way that is very much reminiscent of the Connes-Kasparov isomorphism in C*-algebra K-theory. Vogan has asked if there is a way to reformulate the Connes-Kasparov isomorphism to bring it still closer to his theorem. I shall present an answer to Vogan's question that uses order zero pseudodifferential operators on the symmetric space for G. In fact, in real rank one, the reformulated Connes-Kasparov isomorphism essentially implies Vogan's theorem. This is joint work with Peter Debello.
Louis Ioos (Université Cergy Pointoise)
Title: A Riemann-Roch formula for singular symplectic reductions slides videos
Abstract: Given a Hamiltonian action of a Lie group G on a symplectic manifold,
the Quantization commutes with Reduction principle ([Q,R]=0) of Guillemin-Sternberg
states that the space of G-invariants of the quantization of this manifold coincides
with the quantization of its symplectic reduction by G. This principle provides in
particular a geometric approach to the study of the representation theory of G.
In this talk, I will consider the case where G is a circle and where the symplectic
reduction is a compact singular symplectic space, then present an approach to
establish this principle based on the Berline-Vergne formula and the asymptotics
of the Witten integral. This talk is based on a joint work in collaboration with Benjamin
Delarue and Pablo Ramacher.
Karola Mészáros (Cornell University)
Title: Flow polytopes in algebra and combinatorics slides videos
Abstract: The flow polytope F_G(a) consists of all nonnegative flows on the edges the graph G with netflows at each vertex prescribed by the coordinates of an integer netflow vector "a". Baldoni and Vergne (2008) established that the Ehrhart polynomial of flow polytopes, which in special cases is an evaluation of the Kostant partition function, can be deduced from their volume function, generalizing Lidskii's work. In the past decade flow polytopes have made appearances in other algebraic contexts, such as in the study of Schubert and Grothendieck polynomials. This talk will survey a selection of results about the ubiquitous flow polytopes.
Eckhard Meinrenken (University of Toronto)
Title: Symplectic geometry of the moduli space of hyperbolic 0-metrics slides videos
Abstract: A hyperbolic 0-metric on a compact surface with boundary is a
hyperbolic metric on the interior, with a boundary behaviour similar to
that of the Poincare metric on the upper half plane. The
infinite-dimensional Teichm ller space of such metrics is defined as the
quotient by the identity component of diffeomorphisms fixing the boundary.
We show that these spaces have natural symplectic structures, and are
examples of Hamiltonian Virasoro spaces. Based on joint work with Anton
Alekseev.
Eva Miranda (UPC Barcelona)
Title: Using polytopes to understand quantization slides videos
Abstract: We will present a simple quantization method for toric manifolds. Our approach connects Bohr-Sommerfeld leaves to integer points located in the image of the moment map and to formal geometric quantization as investigated by Mich le Vergne. This method is suitable for symplectic manifolds and specific classes of Poisson manifolds.
Nicolas Ressayre (Université Lyon 1)
Title: The Belkale-Kumar cohomology of complete flag manifolds slides videos
Abstract: Joint work with Luca Francone. In 2006, Belkale-Kumar defined a new product on the cohomology groups of flag manifolds G/P. This product was proved to govern the geometry of the eigencone (associated to the complex reductive group G). They conjectured that, in the case of a complete flag manifold
G/B, the structure constants are either 0 or 1. Several partial results was obtained ever since. In this talk, we will present a new and complete proof of this conjecture. Our proof uses, along the way, a purely combinatorial result on root systems: we will adress the question to find a satisfactory proof of it.
Silvia Sabatini (University of Cologne)
Title: Topological properties of (tall) monotone complexity one spaces slides videos
Abstract: In symplectic geometry it is often the case that compact symplectic manifolds with large group symmetries admit indeed a K hler 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 K hler structure. These spaces are called symplectic toric manifolds or also 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 are largely inspired by the Fine-Panov conjecture and are in collaboration with Daniele Sepe [2].
Moreover, with Isabelle Charton and Daniele Sepe [1], we completely classify monotone complexity one space that are "tall" (no reduced space is a point), and prove that the torus action extends to a full toric action, that each of these spaces admits a K hler structure and that there are finitely many such spaces, up to a notion of equivalence that will be introduced in the talk. References:
[1] I. Charton, S. Sabatini, D. Sepe, "Compact monotone tall complexity one T-spaces", arXiv:2307.04198
[2] S. Sabatini, D. Sepe, "On topological properties of positive complexity one spaces", Transformation Groups 9 (2020).
Shu Shen (Sorbonne Université)
Title: Coherent sheaves, superconnection, and the Riemann-Roch-Grothendieck formula slides videos
Abstract: In this talk, I will explain a construction of Chern character for coherent sheaves on a closed complex manifold with values in Bott-Chern cohomology. I will also show a corresponding Riemann-Roch-Grothendieck formula, which holds for general holomorphic maps between closed non-K hler manifolds. Our proof is based on two fundamental objects : the superconnection and the hypoelliptic deformations. This is a joint work with J.-M. Bismut and Z. Wei (arXiv:2102.08129).
Andras Szenes (Université de Genève)
Title: Intersection cohomologies of toric varieties and moduli spaces of vector bundles slides videos
Abstract: I will present joint work with Camilla Felisetti and Olga Trapeznikova on the intersection cohomologies of toric varieties of Lie type, and the related description of the intersection cohomologies of the moduli spaces of vector bundles on Riemann surfaces.
Eric Vasserot (Université Paris Cité)
Title: A geometric realization of the center of the small quantum group slides videos
Abstract: We propose a new geometric model for the center of the small quantum group using the
cohomology of certain affine Springer fibers. More precisely, we establish an isomorphism between the
equivariant cohomology of affine Springer fibers for a split element and the center of the deformed
graded modules for the small quantum group, and an embedding from the invariant part of
the nonequivariant cohomology under the action of the extended affine Weyl group to the G-invariant
part of the center of the small quantum group, which we conjecture to be an isomorphism.
Michael Walter (Ruhr University Bochum)
Title: (Hidden) Symmetries of Computational Problems slides videos
Abstract: Many computational problems have underlying (and often hidden) symmetries. Revealing these symmetries can be an essential tool for finding faster algorithms and obtaining new structural insights. I will give a gentle introduction to these connections, survey some applications (from algebra and combinatorics all the way to computational complexity and quantum information), and sketch how optimization in curved spaces has recently led to some significant progress.
Hang Wang (East China Normal University)
Title: Topological K-theory for discrete groups and index theory slides videos
Abstract: For a countable discrete group we construct an explicit Chern morphism from the Left-Hand side of the Baum-Connes assembly map (known as the topological K-theory for the discrete group where wrong way functoriality in equivariant K-theory was used for its construction), to the periodic cyclic homology of the group algebra. This morphism allows in particular to give a proper and explicit formulation for a Chern-Connes pairing. This theorem gives an explicit geometric index theoretical formula for the above pairing in terms of pairings of invariant forms, associated to geometric cycles and given in terms of delocalized Chern and Todd classes, and currents naturally associated to group cocycles. This is joint work with Paulo Carrillo-Rouse and Bai-Ling Wang. We shall also discuss some development for generalizing the pairing to connected Lie groups.
Weiping Zhang (Chern Institute of Math)
Title: Deformations of Dirac operators slides videos
Abstract: We will describe some applications of deformed Dirac operators in geometry and topology.
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