# Programme

 Monday 15/7 Tuesday 16/7 Wednesday 17/7 Thursday 18/7 Friday 19/7 8:00-9:00 Registration 9:00-9:50 Vergne Nonnenmacher Meinrenken Zelditch Mathai 10:00-10:50 Liu Shen Finski Klevtsov Song Coffee Coffee Coffee Coffee Coffee 11:30-12:20 Sabatini Trelat Jeffrey Schubert Miranda Lunch Lunch Lunch Lunch Lunch 14:00-14:50 Karshon Charles Free afternoon Hsiao Departure 15:00-15:50 Le Floch Ioos Herrmann Coffee Coffee Coffee 16:30-17:20 Paradan Kordyukov Uribe 19:30 Dinner at Paffgen

Laurent Charles (IMJ, Paris 6)
Title:   From Weyl law to area law for entanglement entropy.
Abstract:   The Weyl law gives the mean distribution of eigenvalues of an operator in terms of its symbol. I will discuss the case of discontinuous symbols, for truncated Toeplitz matrices and for Berezin-Toeplitz operators. I will then present some applications to complex random matrices: typically, in the Ginibre ensemble, we estimate the variance for the number of eigenvalues in a smooth domain. Another application is a mathematical proof of the area law for the entanglement entropy for fermions in quantum Hall effect.

Siarhei Finski (IMJ, Paris 7)
Title:  Riemann-Roch-Grothendieck theorem for families of curves with hyperbolic cusps and its applications to the moduli space of curves    slides
Abstract:   We'll present a refinement of Riemann-Roch-Grothendieck theorem on the level of differential forms for families of curves with hyperbolic cusps. The study of spectral properties of the Kodaira Laplacian on a Riemann surface, and more precisely of its determinant, lies in the heart of our approach. When our result is applied directly to the moduli space of punctured stable curves, it expresses the extension of the Weil-Petersson form (as a current) to the boundary of the moduli space in terms of the first Chern form of a Hermitian line bundle, which provides a generalisation of a result of Takhtajan-Zograf. Our result also implies some bounds on the growth of the Weil-Petersson form near the compactifying divisor of the moduli space of punctured stable curves, which permit us to give a new approach to some well-known results of Wolpert on the Weil-Petersson geometry of the moduli space of curves.

Hendrik Herrmann (Wuppertal)
Title:  Szego kernel asymptotics on Sasakian manifolds
Abstract:   We consider weighted sums of Szego kernels which are equivariant under the flow of the Reeb vector field and study their asymptotic behaviour when the range of the sums become large. As an application we obtain information about the properties of the related weighted Kodaira map. This is a joint work with Chin-Yu Hsiao and Xiaoshan Li.

Title:  Geometric quantization on CR manifolds    slides
Abstract:   Let $X$ be a compact connected orientable CR manifold of dimension greater than five with the action of a connected compact Lie group $G$. Assuming that the Levi form of $X$ is positive definite near the inverse image $Y$ of $0$ by the momentum map and that the tangential Cauchy-Riemann operator has closed range on the reduction $Y/G$, we prove that there is a canonical Fredholm operator between the space of global $G$-invariant $L^2$ CR functions on $X$ and the space of global $L^2$ CR functions on the reduction $Y/G$. This is a joint work with Xiaonan Ma and George Marinescu.

Louis Ioos (Tel Aviv)
Title:  An operational point of view on Berezin-Toeplitz quantization
Abstract:   In this talk, I will discuss Berezin-Toeplitz quantization of compact symplectic manifolds from the point of view of quantum measurement theory. I will give a semi-classical estimate for the associated quantum noise, then present applications to Donaldson's program in Kahler geometry. This talk is based on a joint work with Victoria Kaminker, Leonid Polterovich and Dor Shmoish.

Lisa Jeffrey (University of Toronto)
Title:  Geometric quantization of reduced space of product of orbits    slides
Abstract:  We describe the geometric quantization of a symplectic quotient of a product of coadjoint orbits of $SU(n)$.

Yael Karshon (University of Toronto)
Title:  Geometric quantization with metaplectic-c structures
Abstract:   In the classical geometric quantization procedure with the half-form correction, one cannot quantize a complex projective space of even complex dimension (there is no "half form bundle"), and one cannot equivariantly quantize any symplectic toric manifold (there is no "equivariant half form bundle"). This can be remedied by using metaplectic-c structures to incorporate the "half form correction" into the prequantization stage. This procedure goes back to work of Harald Hess from the late 1970s, and it has the potential to generalize and improve upon recent results in geometric quantization.

Semyon Klevtsov (Cologne)
Title:  Laughlin states on Riemann surfaces    slides
Abstract:   Laughlin state is an $N$-particle wave function, describing the fractional quantum Hall effect (FQHE). We define and construct Laughlin states on genus-$g$ Riemann surface, prove topological degeneracy and discuss adiabatic transport on the corresponding moduli spaces. Mathematically, the problems around Laughlin states involve subjects as asymptotics of Bergman kernels for higher powers of line bundle on a surface, large-$N$ asymptotics of Coulomb gas-type integrals, vector bundles on moduli spaces.

Yuri A. Kordyukov (Ufa, Russia)
Title:  Trace formula for the magnetic Laplacian    slides
Abstract:   We will discuss the Guillemin-Uribe trace formula, which relates some asymptotic spectral invariants of the magnetic Laplacian with geometric invariants of the associated magnetic geodesic flow. First, we will explain this formula. Then we will describe concrete examples of its computation for constant curvature surfaces with constant magnetic fields and for the Katok example.

Yohann Le Floch (Universite de Strasbourg)
Title:  Classical and quantum semitoric systems    slides
Abstract:   Semitoric systems form a particular class of integrable Hamiltonian systems with $S^1$-action on four-manifolds. I will review these systems, present some results and conjectures regarding their inverse spectral theory, and, if time permits, discuss some recent progress towards some fully explicit construction of such systems starting from part of their symplectic invariants. This is based on joint work with J. Palmer (Rutgers University), A. Pelayo (UC San Diego) and S. Vu Ngoc (Universite Rennes 1).

Bingxiao Liu (MPIM, Bonn)
Title:  Asymptotics of equivariant analytic torsion on compact locally symmetric spaces
Abstract:   In many cases, using the Cheeger-Mueller/Bismut-Zhang theorem, the size of torsion subgroups in the cohomology of a closed manifold can be studied by computing the Ray-Singer real analytic torsion. Mueller initiated the study of asymptotic analytic torsions associated with a family of flat homogeneous vector bundles on compact locally symmetric spaces. Bismut, Ma and Zhang considered the analytic torsion forms in the more general context, and they introduced the W-invariants as the leading terms of the asymptotic analytic torsion forms. Here, we consider the leading term in the asymptotic expansion of the equivariant analytic torsions on a compact locally symmetric space, which suggests an extension of Bismut-Ma-Zhang's results to the case of equivariant analytic torsion. For that purpose, we use an explicit geometric formula for the twisted orbital integrals and the twisted Selberg's trace formula.

Title:  Quantization of certain q-Hamiltonian G-spaces and twisted G-equivariant KK-theory for noncompact Lie groups.
Abstract:   I will talk about ongoing joint work with Alex Fok, partly based on arxiv:1903.05298.

Eckhard Meinrenken (Toronto)
Title:  Verlinde formulas for nonsimply connected groups    slides
Abstract:   In 1999, Fuchs and Schweigert proposed formulas of Verlinde type for moduli spaces of surface group representations in compact nonsimply connected Lie groups. I'll explain a proof of a symplectic version of their conjecture for surfaces with at most one boundary component.

Eva Miranda (UPC Barcelona)
Title:  Quantization, singularities and symmetries in interaction    slides
Abstract:   I shall describe several approaches to quantization in the presence of singularities where the torus actions play a central role. The singularities are present either on the polarization or the symplectic structure that blows up in a controlled way along a critical set. This talk is based on several joint works with Guillemin, Hamilton, Presas, Reshetikhin (in progress), Solha and Weitsman.

Stephane Nonnenmacher (Paris Sud)
Title:  Delocalization of Anosov Eigenmodes
Abstract:   The eigenmodes of the Laplacian on a smooth compact Riemannian manifold $(M,g)$ can exhibit various localization properties in the high frequency regime, which strongly depend on the properties of the geodesic flow on $(M,g)$. I will focus on "quantum chaotic" situations, namely assume that the geodesic flow is strongly chaotic (Anosov); this is the case if the sectional curvature of $(M,g)$ is everywhere negative. The Quantum Ergodicity theorem shows that almost all eigenmodes become equidistributed on $M$ in the the high-frequency limit. This theorem leaves the possibility for sequences of exceptional eigenmodes, with different localization behaviours. The Quantum Unique Ergodicity conjecture states that such exceptional eigenstates do not exist. A less ambitious purpose is to constrain the possible localization behaviours, using the chaotic properties of the classical flow. I will report on recent progresses in the two-dimensional case. Generalizing a previous result by Dyatlov-Jin in the constant negative curvature case, we show that the eigenmodes cannot concentrate on a proper subset of $M$; more precisely, any semiclassical measure (a probability measure on $S^*M$, which encodes the asymptotic microlocal properties of a subsequence of eigenmodes) must have full support on $S^*M$. The proof uses methods of semiclassical analysis, and a Fractal Uncertainty Principle due to Bourgain-Dyatlov. The result should be generalizable to quantized Anosov symplectomorphisms defined on a compact 2-dimensional phase space, typically the quantized hyperbolic symplectomorphisms of the 2-torus ("quantum Arnold's cat map") and their nonlinear perturbations. Joint work with Semyon Dyatlov and Long Jin.

Title:  Horn's problem for pseudo-hermitian matrices    slides
Abstract:   Let $G/K$ be an irreducible Hermitian symmetric space of non-compact type. We consider the "causal cone" $C$, which is a $G$-invariant closed convex cone in the dual of $Lie(G)$. Let $T$ be a maximal torus of $K$ and let $C_0\subset Lie(T)^*$ be the intersection of $C$ with a Weyl chamber. The Horn set is then defined as the following subset of $C_0^3$: $$Horn(G/K): = \{(a, b, c) \in C_0^3, Ga \subset Gb + Gc \}.$$ In this talk, we will explain why $Horn(G/K)$ is a closed convex cone and how we can calculate the equations of its faces. In the particular case where $G =U(p,q)$, we will show that the equations of the faces can be obtained recursively, as in the classical Horn's problem.

Silvia Sabatini (Cologne)
Title:  Generalizing the Mukai conjecture
Abstract:   The Mukai conjecture is an inequality involving the second Betti number $b_2$ and the index $k$ of a Fano variety $M$, the index being the largest integer dividing the first Chern class of the tangent bundle, which is not zero in the Fano case. More precisely it asserts that $$b_2(k-1)< n+1 \quad (*),$$ where $n$ denotes the complex dimension of $M$. It gives, in particular, an upper bound on the second Betti number, whenever the index is greater than one. The goal of this ongoing project, joint with Alexander Caviedes Castro and Milena Pabiniak, is to see to which extent this inequality holds in the category of compact monotone symplectic manifolds endowed with a Hamiltonian $S^1$-action. These spaces naturally carry almost complex structures which are compatible with the symplectic structure. We first provide a new criterion, in terms of the indices of certain line bundles, that implies (*) requiring only a choice of an almost complex structure (i.e., not necessarily integrable). In particular, this new approach generalizes the methods used so far to prove the Mukai conjecture in certain cases. We apply this to give an alternative proof of the Mukai conjecture for toric Fano varieties and for flag varieties. We also provide conditions in terms of equivariant cohomology that ensure inequality (*) to hold for compact symplectic manifolds supporting a Hamiltonian $S^1$-action with isolated fixed points.

Roman Schubert (University of Bristol)
Title:  Open Quantum Systems and Symplectic Geometry.    slides
Abstract:   I will give an overview of two areas in the theory of open quantum systems where interesting geometric structures appear in the semiclassical limit. The first area is the evolution of wave packets when we allow to Hamiltonian to be non-Hermitian, in this case the evolution of the centre of the wave packet is governed by a combination of a Hamiltonian vectorfield and a gradient vectorfield which are coupled by a complex structure on phase space. The second area is the Lindblad equation which describes the time evolution of the density operator in a quantum system coupled to an environment (in particular in situations where the influence of the environment can be treated as noise). In the semiclassical limit the leading order terms in the Lindblad equation can be written in the Hoermander "sum of squares" of vectorfields form known from second order hypoelliptic equations, and the geometry of the vectorfields and their commutators is related to decoherence. It turns out that these two areas are related, as we can formally interpret the Lindblad equation as a Schroedinger equation with non-hermitian Hamiltonian on the space of Hilbert-Schmidt operators.

Shu Shen (IMJ, Paris 6)
Title:  Analytic torsion, Cheeger-Mueller theorem, and Fried conjecture.
Abstract:   The relation between the spectrum of the Laplacian and the dynamical flow on a closed Riemannian manifold is one of the central themes in differential geometry. Fried conjectured a relation between the analytic torsion, which is an alternating product of regularized determinants of the Hodge Laplacians, and the Ruelle dynamical zeta function. We will formulate and show this conjecture for Morse-Smale flow. Our proof relies on Cheeger-Mueller/Bismut-Zhang theorem. This is a joint work with Jianqing Yu.

Yanli Song (Washington University)
Title:  Cyclic Cocycles, Higher Index and Higher Orbtial Integral
Abstract:   Connes-Moscovici's $L^2$-index theorem on the homogeneous space $G/K$ says that the $L^2$- index of an equivariant Dirac operator is equal to the Plancherel measure (or formal degree) of the corresponding discrete series representation of $G$. However, this index always vanishes when the group $G$ has no discrete series, or equivalently is of non-equal rank. To generalize Connes-Moscovici's result to the non-equal rank case, one needs a generalization of the $L^2$ index. In this talk, I will discuss how to construct higher cyclic cocycles using parabolic subgroups, and compute the higher index pairing using Harish-Chandra's Plancherel theorem. This is joint work with Xiang Tang.

Emmanuel Trelat (LJLL, Paris 6)
Title:  Spectral analysis of sub-Riemannian Laplacians and Weyl measure    slides
Abstract:   In a series of works on sub-Riemannian geometry with Yves Colin de Verdiere and Luc Hillairet, we study spectral properties of sub-Riemannian Laplacians, which are hypoelliptic operators. The main objective is to obtain quantum ergodicity results, what we have achieved in the 3D contact case. In the general case we study the small-time asymptotics of sub-Riemannian heat kernels. We prove that they are given by the nilpotentized heat kernel. In the equiregular case, we infer the local and microlocal Weyl law, putting in light the Weyl measure in sR geometry. This measure coincides with the Popp measure in low dimension but differs from it in general. We prove that spectral concentration occurs on the shief generated by Lie brackets of length $r-1$, where $r$ is the degree of nonholonomy. In the singular case, like Martinet or Grushin, the situation is more involved but we obtain small-time asymptotic expansions of the heat kernel and the Weyl law in some cases. Finally, we give the Weyl law in the general singular case, under the assumption that the singular set is stratifiable.

Alejandro Uribe (Michigan)
Title:  The Hermite calculus of isotropic states and applications.    slides
Abstract:   I will first describe a general construction of quantum states associated to isotropic submanifolds, both in cotangent and symplectic contexts. Examples include a new class of squeezed spin coherent states. Such states have symbols that are symplectic spinors, and I will sketch their symbol calculus. The main applications I will discuss are to the so-called Hermite operators, whose kernels are isotropic distributions associated to submanifolds of the diagonal. Examples include certain mixed states, for which we obtain a Szego limit theorem.

Michele Vergne (Paris 7)
Title:  Asymptotic distributions associated to transversally elliptic operators
Abstract:   I will report on a common work with Yiannis Loizides, and Paul-Emile-Paradan. Let $G$ be a compact Lie group, and $M$ be a (possibly non compact) spin manifold with line bundle $L$, and vector bundle $E$. We study the behavior when $k\to \infty$ of the multiplicity function $m_G(\lambda,k,E)$ of the equivariant Dirac operator twisted by $L^k\otimes E$. More generally, for $m(k,\lambda)$ a function of $(k,\lambda)$, we study the asymptotic expansion of the distribution ${\Theta(m;k)(\varphi)}=\sum_{\lambda }m(k,\lambda)\varphi \big(\tfrac{\lambda}{k}\big)$ We prove that, under some properties of piecewise quasi polynomial behavior, $m$ is uniquely determined by its asymptotic expansion. Applications are the functoriality of the restriction to a subgroup $H$ of the index of particular $G$-transversally elliptic operators.

Steve Zelditch (Northwestern University)
Title:  Probabilistic analogies in Kahler spectral asymptotics    slides
Abstract:   Spectral asymptotics of a Toeplitz Hamiltonian concerns the semi-classical asymptotics of the density of states $\rho_k(z)$ corresponding to eigenvalues in a window of energies $[E_1, E_2]$. If the point $z$ lies on the energy surface at the edge, $H(z) = E_2$, then the density of states abruptly changes from almost 1 to almost 0 near $z$. The transition has a universal shape reminiscent of the CLT (central limit theorem) if the level set is regular. If it is critical, the shape is different and depends on the type of critical point. In the case of toric Kahler manifolds, the analogy to the CLT is even stronger, and actually is the CLT for the Fubini-Study metric on the hyperplane bundle over $\mathbb{CP}^m$. My talk will survey a variety of results in Kahler asymptotic analysis which are analogues of theorems in probability.