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Programme
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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.
Chin-Yu Hsiao (Academia Sinica, Taipei)
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.
Varghese Mathai (University of Adelaide)
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.
Paul-Emile Paradan (Universite de Montpellier)
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.
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