Workshop on Geometric Quantization (18w5182)
The Banff International Research Station will host the "Workshop on Geometric Quantization" workshop from April 15th to April 20th, 2018. Following the principle of quantum mechanics, the aim of geometric quantization is to associate to a classical phase space, described by a symplectic manifold M , a quantized version Q(M) given by a Hilbert space. In this procedure, the Poisson bracket of functions on M, regarded as classical observables, should correspond to the commutator of self-adjoint operators, regarded as quantum observables. Furthermore, an action of a group G by symmetries of M should be implemented as a unitary representation on the quantum Hilbert space Q(M). The philosophy of geometric quantization has been used in a variety of contexts, with remarkable and often surprising consequences. This workshop at BIRS will bring together mathematicians working on these topics and with different techniques such as topological K-theory, analytic estimates, C∗ -algebras, representation theory. The philosophy of geometric quantization will act as a focal point for the interaction between all of these areas. The workshop will provide an excellent opportunity for experts working on different aspects of the theory to exchange ideas, leading to fresh insights and new developments. The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).
22
2018
24
19 hours 43 minutes
22 results
49:49
Varghese, Mathai2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
1:08:26
Schick, Thomas2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
50:29
2Marinescu, GeorgeWe study the Berezin-Toeplitz quantization using as quantum space the space of eigenstates of the renormalized Bochner Laplacian on a symplectic manifold, corresponding to eigenvalues localized near the origin. We show that this quantization has the correct semiclassical behavior and construct the corresponding star-product. This is joint work with L. Ioos, W. Lu and X. Ma.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
56:07
1Rodsphon, RudyIn the early eighties, Connes developed his Noncommutative Geometry program, mostly to extend index theory to situations where usual tools of differential topology are not available. A typical situation is foliations whose holonomy does not necessarily preserve any transverse measure, or equivalently the orbit space of the action of the full group of diffeomorphisms of a manifold. In the end of the nineties, Connes and Moscovici worked out an equivariant index problem in these contexts, and left a conjecture about the calculation of this index in terms of characteristic classes. The aim of this talk will be to survey the history of this problem, and explain partly our recent solution to Connes-Moscovici's conjecture, focusing on the part concerning `quantization'. No prior knowledge of Noncommutative Geometry will be assumed, and part of this is joint work with Denis Perrot.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
50:20
Rochon, FrédéricGiven a finite dimensional irreducible complex representation of G=SOo(d,1), one can associate a canonical flat vector bundle E together with a canonical bundle metric h to any finite volume hyperbolic manifold X of dimension d. For d odd and provided X satisfies some mild hypotheses, we will explain how, by looking at a family of compact manifolds degenerating to X in a suitable sense, one can obtain a formula relating the analytic torsion of (X,E,h) with the Reidemeister torsion of an associated manifold with boundary. As an application, we will indicate how, in the arithmetic setting, this formula can be used to derive exponential growth of torsion in cohomology for various sequences of congruence subgroups. This is a joint work with Werner Mueller.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
1:08:38
Melrose, Richard2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
50:01
Bismut, Jean-MichelThe hypoelliptic Laplacian gives a natural interpolation between the Laplacian and the geodesic flow. This interpolation preserves important spectral quantities. I will explain its construction in the context of compact Lie groups: in this case, the hypoelliptic Laplacian is the analytic counterpart to localization in equivariant cohomology on the coadjoint orbits of loop groups. The construction for noncompact reductive groups ultimately produces a geometric formula for the semisimple orbital integrals, which are the key ingredient in Selberg trace formula. In both cases, the construction of the hypoelliptic Laplacian involves the Dirac operator of Kostant.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
45:17
1Tang, XiangFor a compact Lie group action on a smooth manifold, we will introduce a complex of basic relative forms on the inertia space, which was originally constructed by Brylinski. We will explain how basic relative forms can be used to study the Hochschild homology of the convolution algebra. This is work in progress with Markus Pflaum and Hessel Posthuma.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
48:03
Savale, NikhilFor manifolds including metric-contact manifolds with non-resonant Reeb flow, we prove a Gutzwiller type trace formula for the associated magnetic Dirac operator involving contributions from Reeb orbits on the base. As an application, we prove a semiclassical limit formula for the eta invariant.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
43:16
3Hochs, PeterLet G be a real semisimple Lie group, and K
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
50:25
2Stolz, Stephan2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
51:59
Puchol, MartinConsider an action of a connected compact Lie group on a compact complex manifold M, and two equivariant vector bundles L and E on M, with L of rank 1. The purpose of this talk is to establish holomorphic Morse inequalities, analogous to Demailly's one, for the invariant part of the Dolbeault cohomology of tensor powers of L, twisted by E. To do so, we define a moment map μ by the Kostant formula and then the reduction of M under a natural hypothesis on μ−1(0). Our inequalities are given in term of the curvature of the bundle induced by L on this reduction, in the spirit of "quantization commutes with reduction".
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
36:58
4Wang, HangK-theory of reduced group C∗-algebras and their trace maps can be used to study tempered representations of a semisimple Lie group from the point of view of index theory. For a semisimple Lie group, every K-theory generator can be viewed as the equivariant index of some Dirac operator, but also interpreted as a (family of) representation(s) parametrised by A in the Levi component of a cuspidal parabolic subgroup. In particular, if the group has discrete series representations, the corresponding K-theory classes can be realised as equivariant geometric quantisations of the associated coadjoint orbits. Applying orbital traces to the K-theory group, we obtain a fixed point formula which, when applied to this realisation of discrete series, recovers Harish-Chandra's character formula for the discrete series on the representation theory side. This is a noncompact analogue of Atiyah-Segal-Singer fixed point theorem in relation to the Weyl character formula. This is joint work with Peter Hochs.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
45:09
Song, YanliIn 1980s, Connes and Moscovici studied index theory of G-invariant elliptic pseudo-differential operators acting on non-compact homogeneous spaces. They proved a L2 -index formula using the heat kernel method, which is related to the discrete series representation of Lie groups. In this talk, I will discuss the orbital integral of heat kernel and its relation with Plancherel formula. This is a generalization of the analytic index studied by Connes-Moscovici to the limit of discrete series case. In a recent work by Hochs-Wang, they obained a fixed point theorem for the topogical side of the index.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
1:07:38
1Higson, NigelThis is an expository talk about C*-algebra K-theory for reductive groups. I’ll try to explain what it is, what it actually says about representation theory, and what else it suggests about representation theory, at least to a willing mind. The story begins with Harish-Chandra’s parametrization of the discrete series representations, and the realization of discrete series representations using the Dirac operator. I’ll discuss these things, and then touch on other parts of Harish-Chandra’s theory of tempered representations that are prominent from the K-theoretic point of view.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
54:27
3Karshon, YaelIn 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"). I will describe a geometric quantization procedure that uses metaplectic-c structures to incorporate the "half form correction" into the prequantization stage and that does apply to these examples. This follows work of Harald Hess from the late 1970s, with recent contributions of Jennifer Vaughan.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
44:10
Kottke, ChrisComplex line bundles are classified naturally up to isomorphism by degree two integer cohomology H2, and it is of interest to find geometric objects which are similarly associated to higher degree cohomology. Gerbes (of which there are various versions, due respectively to Giraud, Brylinski, Hitchin and Chattergee, and Murray) provide a such theory associated to H3. Various notions of"higher gerbes" have also been defined, though these tend to run into technicalities and complicted bookkeeping associated with higher categories. We propose a new geometric version of higher gerbes in the form of "multi simplicial line bundles", a pleasantly concrete theory which avoids many of the higher categorical difficulties, yet still captures key examples including the string (aka loop spin) obstruction associated to 12 p1 in H4. In fact, every integral cohomology class is represented by one of these objects in the guise of a line bundle on the iterated free loop space equipped with a "fusion product" (as defined by Stolz and Teichner and further developed by Waldorf) for each loop factor.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
55:20
2Waldorf, Konrad2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
42:41
Loizides, YiannisI will describe a map from `D-cycles' for the twisted K-homology of a compact, connected, simply connected Lie group to the Verlinde ring. The induced map on K-homology is inverse to the Freed-Hopkins-Teleman isomorphism. An application is to show that two options for `quantizing' a Hamiltonian loop group space are compatible with each other. This talk is partly based on joint work with Eckhard Meinrenken and Yanli Song.
2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
1:04:49
1Vergne, Michèle2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
1:11:18
1Gukov, Sergei2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery
1:07:48
3Freed, Daniel2018Banff International Research Station (BIRS) for Mathematical Innovation and Discovery