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Programme
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Wolfram Bauer (Leibniz Universität, Hannover)
Title: Uniform continuity and Toeplitz quantization on $\mathbb{C}^n$ or bounded symmetric domains slides
Abstract: I will report some progress on Toeplitz quantization for the Bergman space over bounded symmetric domains or
the Segal-Bargmann space of Gaussian square integrable entire functions. Asymptotic relations in Rieffel's definition of deformation quantization are extended to classes of i.g. non-continuous and possibly unbounded operator symbols. In case of $\Omega= \mathbb{C}^n$ and for the Segal-Bargmann space these results can be formulated within the Fock quantization algebra via an oscillation condition.
The resulting estimates are useful in the analysis of Toeplitz $C^*$-algebras and their irreducible representations after a reduction of dimension through a "quantization effect" for Toeplitz operators with i.g. non-differentiable symbols. If time permits I will mention related criteria on boundedness and compactness of Toeplitz operators and results on the essential spectrum of elements
in Toeplitz $C^*$-algebras.
This talk is based on joint work with L.A. Coburn (SUNY at Buffalo), R. Hagger (U. Kiel) and N. Vasilevski (CINVESTAV, Mexico).
Jean-Michel Bismut (Paris-Saclay)
Title: Loop groups, coadjoint orbits, and localization formulas slides
Abstract: Witten and Atiyah have explained the formal role of the equivariant
cohomology of the loop space of a manifold to give an
interpretation of the index of a Dirac operator in connections with
localization formulas in equivariant cohomology.
If $G$ is a compact Lie group, the finite dimensional representations
of $G$ can be obtained by quantization of the coadjoint orbits. The
corresponding character formula have two forms: the Lefschetz
formula, that express the character as a finite sum indexed by the
Weyl group, the Kirillov formula, which is an integral of a differential
form on the coadjoint orbit. These two kinds of formulas can be
directly related by the Duistermaat-Heckman, Berline-Vergne
localization formulas
in equivariant cohomology.
By following arguments of I. Frenkel, Atiyah, and ourselves, we will
explain how the heat kernel of the group $G$ can be interpreted as a
formula of Duistermaat-Heckman for the loop group $LG$. We will
show how the formula that expresses the heat kernel in terms of the
coroot lattice can be viewed as a consequence of an infinite
dimensional DV, BV formula, via a construction involving the
hypoelliptic Laplacian.
Sheng-Fu Chiu (NCTS)
Title: Quantum Speed Limit and Relative Categorical Energy slides
Abstract: Heisenberg's Uncertainty Principle is one of the most celebrated features of quantum mechanics, which states that one cannot simultaneously obtain the precise measurements of two conjugated physical quantities such as the pair of position and momentum or the pair of electric potential and charge density. Among the different formulations of this fundamental quantum property, the uncertainty between energy and time has a special place. This is because the time is rather a variable parametrizing the system evolution than a physical quantity waiting for determination. Physicists working in quantum information theory have understood this energy-time relation by a universal bound of how fast any quantum system with given energy can evolve from one state to another in a distinguishable (orthogonal) way. Recently, there have been many arguing that this bound is not a pure quantum phenomenon but a general dynamical property of Hilbert space. In this talk, in contrast to the usual Hilbert space formalism, we will provide a dual viewpoint of this evolutional speed limit based on a persistence-like distance of the derived category of sheaves : during a fixed time period what is the minimal energy needed for a system to evolve from one sheaf to a status that is distinguishable from a given subcategory? As an application, we will show that such categorical energy with respect to open subsets gives rise to a nontrivial lower bounded of Hofer displacement energy.
Siarhei Finski (École Polytechnique)
Title: Submultiplicative norms on section rings slides
Abstract: A graded norm on a section ring of a polarised projective manifold is called submultiplicative if the norm of products of holomorphic sections is no bigger than their products of norms. Such norms arise naturally in complex geometry and functional analysis. In the former context, they appear in the study of holomorphic extension problems, submultiplicative filtrations (related to K-stability and non-Archimedean pluripotential theory) and Narasimhan-Simha pseudonorms. In the latter context, they appear in the study of projective tensor norms on polynomial rings.
We show that submultiplicative norms on section rings of polarised projective manifolds are asymptotically equivalent to sup-norms associated with metrics on the polarising line bundle. We then derive several applications of this result to the aforementioned problems.
Véronique Fischer (University of Bath)
Title: Quantizations on Lie groups slides
Abstract: A first simple definition of quantization is an operation which associates an operator to a symbol.
In this talk, we will discuss two symbolic quantizations in this sense, that arise naturally on Lie groups. As applications, I will present an overview of recent developments in pseudo-differential theory on compact and nilpotent Lie groups, together with ensuing results in the global analysis of these groups.
Sergei Gukov (California Institute of Technology)
Title: Quantum algebra and quantum invariants from
quantization of Coulomb branches slides
Abstract: After a brief review of the "brane quantization" approach,
introduced in a joint work with Witten over ten years ago, I will discuss
some of the more recent applications to quantum algebra and quantum
topology. In particular, we will see how this approach allows to understand
representation theory of interesting algebras and produce new invariants of
3-manifolds via geometry. The talk is based on the following recent work:
Branes and DAHA Representations
https://arxiv.org/abs/2206.03565
Rozansky-Witten geometry of Coulomb branches and logarithmic
knot invariants https://arxiv.org/abs/2005.05347.
Peter Hochs (Radboud University)
Title: Equivariant analytic torsion and an equivariant Ruelle dynamical zeta function slides
Abstract: Analytic torsion was introduced by Ray and Singer as a way to realise Reidemeister-Franz torsion analytically. (The equality was independently proved by Cheeger and Müller.) The Ruelle dynamical zeta function is a topological way to count closed curves of flows on compact manifolds. The Fried conjecture states that, for a suitable class of flows, the Ruelle dynamical zeta function has a well-defined value at zero, and that the absolute value of this value equals analytic torsion. With Hemanth Saratchandran, we define equivariant versions of analytic torsion and of the Ruelle dynamical zeta function, for proper actions by locally compact groups, with compact quotients. These have some natural fundamental properties, generalising properties of their non-equivariant counterparts. The resulting equivariant version of Fried's conjecture does not hold in general, but it does hold in some classes of examples. This motivates the search for general conditions under which the equivariant Fried conjecture is true.
Chin-Yu Hsiao (Institute of Mathematics, Academia Sinica)
Title: Semi-classical Toeplitz operators and geometric quantization on CR manifolds and complex manifolds with boundary slides
Abstract: I will first introduce my results about geometric quantization on CR manifolds and complex manifolds with boundary (joint with Ma, Marinescu, Huang, Li, Shao). Then, I will mention $G$-invariant semi-classical Toepltiz operators on CR manifolds and complex manifolds with boundary and giving another new viewpoint of geometric quantization on CR manifolds and complex manifolds with boundary.
Louis Ioos (Université de Cergy-Pontoise)
Title: Quantization commutes with Reduction for singular circle actions
Abstract: Given the Hamiltonian action of a Lie group G on a symplectic manifold M,
the principle of Quantization commutes with Reduction, due to Guillemin and
Sternberg, states that the space of G-invariants of the quantization of M coincides
with the quantization of its symplectic reduction by 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 localization formula and the
asymptotics of the Witten integral. This is based on a joint work in collaboration
with Benjamin Delarue and Pablo Ramacher.
Yael Karshon (University of Toronto/Tel-Aviv University)
Title: Geometric quantization of Lagrangian torus fibrations
slides
Abstract: By the Arnold-Liouville theorem, a regular completely integrable system
on a compact symplectic manifold M is Lagrangian torus fibration over
an integral affine manifold B.
When M is pre-quantized, we give a simple proof that its Riemann-Roch
number coincides with its number of Bohr-Sommerfeld leaves. This can be
viewed as an instance of the mysterious "independence of polarization"
phenomenon of geometric quantization.
The proof uses the following fact, whose proof is trickier than one might expect:
An integral affine structure on a manifold B determines a smooth
measure on B; a so-called integral-integral affine structure determines
a notion of integer points in B.
We show that, for a compact integral-integral affine manifold,
the total volume is equal to the number of integer points.
This is joint work with Mark Hamilton and Takahiko Yoshida.
Laurent La Fuente (Université libre de Bruxelles)
Title: On formal moment maps
Abstract: I will present a construction, using deformation quantization, that recovers and deforms classical moment maps on infinite dimensional spaces. The main tool is the notion of formal connection, introduced by Andersen-Masulli-Schätz, applied to bundles of Fedosov star product algebras. The examples I will discuss are the Cahen-Gutt moment map on the space of symplectic connections, the Donaldson moment map on diffeomorphisms group and the Donaldson-Fujiki moment map on the space of compatible almost complex structures on a symplectic manifold.
Yohann Le Floch (IRMA, Université de Strasbourg)
Title: Random holomorphic sections and Berezin-Toeplitz operators slides
Abstract: I will discuss a joint work with Michele Ancona (Université Côte d'Azur), in which we study the zero loci of certain sections of a large power of some complex line bundle over a compact Kähler manifold. These sections are obtained by applying a fixed Berezin-Toeplitz operator to random holomorphic sections, and their expected zero loci reflect some properties of the principal symbol of the operator.
Bingxiao Liu (Cologne)
Title: Random holomorphic sections on noncompact complex manifolds slides
Abstract: For a compact Kähler manifold, by considering the high tensor powers of a prequantum line bundle, Shiffman and Zelditch (1999) proved the equidistribution of the zeros of random holomorphic sections in the semiclassical limit. Since then, several generalizations and extensions of this result have been made in different geometric or probabilistic settings. Notably, the large deviation estimate and hole probability associated with the random zeros were obtained for compact Hermitian manifolds. In this talk, I will present a generalization of these results to the case of noncompact complex manifolds. Specifically, we will discuss the general construction of the Gaussian random holomorphic sections for Hermitian holomorphic line bundles on a noncompact Hermitian manifold. Our primary focus is on the scenarios where the space of square integrable holomorphic sections is infinite-dimensional. Then we investigate the behaviours of their zeros in the semiclassical limit, covering topics such as equidistribution, large deviation estimate and hole probability. This talk is based on the joint work with Alexander Drewitz and George Marinescu.
Ursula Ludwig (Münster)
Title: Analytic and topological torsion on singular spaces
Abstract: The famous theorem of Cheeger and Müller states the equality between the analytic (or Ray-Singer) torsion and the topological torsion of a smooth compact manifold equipped with a unitary flat vector bundle. Using local index techniques and the Witten deformation Bismut and Zhang gave the most general comparison theorem of torsions for a smooth compact manifold.
The aim of this talk is the generalisation of the Cheeger-Müller theorem to the context of isolated conical singularities.
Qiaochu Ma (Penn State University)
Title: Semiclassical analysis, geometric representation and quantum ergodicity. slides
Abstract: Quantum Ergodicity (QE) is a classical topic in spectral geometry, which states that on a compact Riemannian manifold whose geodesic flow is ergodic with respect to the Liouville measure, the Laplacian has a density one subsequence of eigenfunctions that tends to be equidistributed. In this talk, we present the QE for a series of unitary flat bundles using a mixture of semiclassical and geometric quantizations. We shall see that even if analytically unitary flat bundles are almost the same as the trivial bundle, the nontrivial holonomy geometrically provides extra interesting phenomena.
Eva Miranda (Universitat Politècnica de Catalunya)
Title: Quantizing singular symplectic manifolds slides
Abstract: I will explore multiple strategies for quantizing singular symplectic
manifolds, with a particular focus on b-symplectic and folded symplectic
manifolds, and examine their interconnections.
This is based on joint works with Victor Guillemin and Jonathan Weitsman
(some of them in progress).
Stéphane Nonnenmacher (Paris-Saclay)
Title: Random eigenstates of the Quantum Cat Map slides
Abstract: (joint with Nir Schwartz) A quantized hyperbolic automorphisms of the 2-torus, also known as
a "Quantum Cat Map", is a popular toy model of quantized chaotic
system. It can be defined through various quantization procedures,
including a Toeplitz quantization on spaces of holomorphic sections; one ends up with a ladder of unitary operators ("propagators") of dimension $N$, the semiclassical limit corresponding to large values of $N$.
The representation theoretic properties of these operators allow to diagonalize them explicitly, producing explicit families of quantum chaotic eigenstates. One can test various conjectures of Quantum Chaos on these eigenstates, like the Quantum Unique Ergodicity conjecture, or the asymptotic Gaussianity of the vector components.
A peculiarity of this system is the presence, in the semiclassical limit, of very large spectral multiplicities. It is known that these large multiplicities allow
to construct eigenstates which are not equidistributed in the semiclassical limit, thus contradicting a general version of Quantum Unique Ergodicity (QUE).
In the present work we use these spectral multiplicities in a different manner: namely, we consider random eigenbases of the unitary propagators, in this context of large
multiplicities. We prove that, almost surely, a sequence of random eigenbases satisfies QUE, including at some microscopic scales. We also show that the local statistics of
these random eigenstates converges to that of standard Gaussian random
states (in particular, the distribution of their components is asymptotically Gaussian).
Paul-Emile Paradan (Université de Montpellier)
Title: Moment polytopes in real symplectic geometry slides
Abstract: The aim of this talk is to explain how to parameterize the equations of the moment polytope associated with the action of a compact Lie group on a Kähler manifold: we adapt the techniques developed by Nicolas Ressayre for polarized algebraic varieties. We will show how to apply this tool to the case where the group and the manifold are provided with an involution. This "involution" framework will be illustrated using the convex cones naturally associated with isotropic representations of symmetric spaces.
Silvia Sabatini (Cologne)
Title: Topological properties of (tall) monotone complexity one spaces slides
Abstract: In symplectic geometry it is often the case that compact symplectic
manifolds with large group symmetries admit indeed a Kähler 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 Kähler 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.
Moreover, with Isabelle Charton and Daniele Sepe, 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 Kähler structure and that there
are finitely many such spaces, up to a notion of equivalence that will be
introduced in the talk.
Nikhil Savale (Cologne)
Title: Bochner Laplacians and Bergman kernels for families slides
Abstract: We generalize earlier joint results with Marinescu to families of Bochner
Laplacians. This particularly leads to the fiberwise expansion for families
Bergman kernels of horizontally semi-positive index bundles. The proof uses
Ma-Zhang's description for the curvature of the index bundle as a fiberwise
Toeplitz operator. Based on joint work with X. Ma and G. Marinescu.
Shu Shen (IMJ-PRG)
Title: Coherent sheaves, superconnection, and the Riemann-Roch-Grothendieck formula. slides
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-Kahler 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
Alejandro Uribe (University of Michigan)
Title: Propagation of wave packets in complex polarizations, and open
questions.
Abstract: I will define classes of (semi-classical) wave packets in
complex polarizations with a symbol calculus, and state results on their
Hermitian and non-Hermitian propagation (in the latter case, construction
of quasi-modes). This is joint work with R. Bhattacharyya and D. Burns. I
will then present similar results in other contexts (quantum Zeno dynamics,
and, time permitting, symplectic cobordisms), where there remain intriguing
open questions.
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