Mini-Workshop

Symmetry & Reduction in Poisson & Related Geometries

May 20-24, 2024

Department of Mathematics, University of Salerno

Description

This mini-workshop aims at discussing the most recent results around coisotropic reduction in Poisson and related geometries, and their algebraic counterparts, both in the classical and the quantum setting. The bulk of the event consists of three mini-courses on different aspects of the topic. The mini-courses will be complemented by invited talks. Notice that this is the only event during the intensive period that will take place at the University of Salerno, in Fisciano, rather than at the University of Napoli! Moreover, notice that on Sunday, May 19, there will be no busses to reach Fisciano. If you need any help to reach your accomodation please contact us.

Speakers

Nicola Ciccoli (University of Perugia)
Peter Crooks (Utah State University)
Sylvain Lavau (Ruđer Bošković Institute Zagreb)
Antonio Miti (Sapienza Università di Roma)
Praful Rahangdale (Paderborn University)
Leonid Ryvkin (University Claude Bernard Lyon 1)
Matthias Schötz (IMPAN Warsaw)
Stefan Waldmann (University of Würzburg)
Janina Bernardy (Max Planck Institute Bonn)

Schedule

The lectures will take place in room P18 on the top floor of the F3 building on the campus of the University of Salerno.

You can find here a PDF file with the complete program of the miniworkshop.

Mini-courses

Nicola Ciccoli (University of Perugia)

Title: Poisson structures on quotients

Abstract: We will describe the theory of Poisson homogeneous spaces of Poisson-Lie groups, together with their classification in terms of orbits on the algebraic variety of Langrangian subalgebras of the double and their symplectic integration. The role of coisotropic subgroups will be emphasized. Time permitting we will add details on generalizations to quasi-Poisson spaces.

Leonid Ryvkin (University Claude Bernard Lyon 1) & Antonio Michele Miti (Sapienza Università di Roma)

Title: Reduction of (multi)-symplectic observables

Abstract: Let M be a manifold with a geometric structure and sufficiently nice G a symmetry group, often the geometric structure can be transferred to M/G. In (multi-)symplectic geometry, reduction procedures permit to transfer the differential form to an even smaller space. However, all approaches working directly on the space have very strong regularity requirements. We present an approach to reducing the algebra of (multi-)symplectic observables for general (covariant) moment maps, without any regularity assumptions on the level sets (and the symmetries). Even in the well-studied symplectic case, this construction is distinct from pre-existing ones. Based on joint work with Casey Blacker.

Stefan Waldmann (University of Wurzburg)

Title: Reduction of Star Products

Abstract: In my minicourse I will outline several reduction procedures for star products. To this end, I will give a short introduction to the motivations and the main results in (formal) deformation quantization including a discussion of symmetries. In geometric mechanics one knows several reduction schemes based either on symmetries like the Marsden-Weinstein reduction or on the usage of coisotropic submanifolds to name a few. For all these reduction schemes a quantum analog within deformation quantization is highly desirable. This leads to questions about invariance properties of star products, quantization of submanifolds etc. Recently, in joint work with Chiara Esposito and Marvin Dippell, we generalized the geometric situation to an algebraic framework of constraint algebras for which a systematic reduction is inherent. I will outline the deformation theory of such constraint algebras and discuss the relevant cohomologies.

Talks

Janina Bernardy (Max Planck Institute Bonn)

Title: Homotopy reduction of multisymplectic structures in Lagrangian field theory

Abstract: While symplectic geometry is the geometric framework of classical mechanics, the geometry of classical field theories is governed by multisymplectic structures. In multisymplectic geometry, the Poisson algebra of Hamiltonian functions is replaced by the L∞-algebra of Hamiltonian forms introduced by Rogers in 2012. The corresponding notion of homotopy momentum maps as morphisms of L∞-algebras is due to Callies, Frégier, Rogers, and Zambon in 2016. I will report on the development of the notion of homotopy reduction using these higher structures, in particular on the definition of the homotopy zero locus of a local homotopy momentum map in Lagrangian field theory. This is joint work with Christian Blohmann.

Peter Crooks (Utah State University)

Title: Generalized Hamiltonian reduction and Moore-Tachikawa varieties

Abstract: Marsden and Weinstein's approach to Hamiltonian reduction is a mainstay of modern research at the interface of Poisson geometry, representation theory, and theoretical physics. Generalizations of this approach include Mikami-Weinstein reduction, symplectic cutting, symplectic implosion, and Ginzburg-Kazhdan reduction by abelian group schemes. The last of these was motivated by conjectural solutions to the decade-old Moore-Tachikawa conjecture on topological quantum field theories (TQFTs).

Sylvain Lavau (Ruđer Bošković Institute Zagreb)

Title: The Bott connection generates gauge transformations

Abstract: This talk is about connecting the Bott connection associated to regular foliations and the gauge generators of coisotropic (first-class) submanifolds. The annihilator bundle of a smooth involutive regular distribution is a coisotropic submanifold (constraint surface) of the cotangent bundle. One observes that the generators of this involutive distribution (vector fields tangent to the foliation) can be lifted to first-class constraints defining the constraint surface. Then, we show that the gauge orbits generated by these constraints correspond to the sections of the annihilator bundle which are horizontal with respect to the Bott connection. This allows to characterize the zero-th group of the foliated cohomology as the gauge invariant linear functions on the annihilator bundle.

Praful Rahangdale (Paderborn University)

Title: Poisson Lie groups, Lie bialgebras, the Yang-Baxter equation, and Manin triples

Abstract: In this talk, I will present the work by V. G. Drinfeld (1983) which gives the one-to-one correspondence between finite dimensional connected simply connected Poisson Lie groups and finite dimensional Lie bialgebras. Then, I will show the correspondence between Lie bialgebras, the classical Yang-Baxter equation, and Manin triples. In the end, I will discuss attempts made and difficulties arisen in the context of infinite dimensional Poisson Lie groups (modeled on locally convex topological vector spaces).

Matthias Schötz (IMPAN Warsaw)

Title: Symmetry Reduction of States

Abstract: We develop a general theory of symmetry reduction of states on (possibly non-commutative) ∗-algebras that are equipped with a Poisson bracket and a Hamiltonian action of a commutative Lie algebra 𝔤. The key idea here is that the correct notion of positivity on a ∗-algebra A is not necessarily the algebraic one whose positive elements are the sums of Hermitian squares, but can be a more general one that depends on the example at hand, like pointwise positivity on ∗-algebras of functions or positivity in a representation as operators. The notion of states (normalized positive Hermitian linear functionals) on A thus depends on this choice of positivity on A, and the notion of positivity on the reduced algebra should be such that its states are obtained as reductions of certain states on A. After laying out the general setup I will then discuss several examples, like smooth functions on a Poisson manifold M, the reduction of the Weyl algebra, and an example from deformation quantization.

Registration

There is no registration fee but registration is required. To register send an e-mail to mdippell@unisa.it (Marvin Dippell).

Other Participants

Smail Chemikh
Hai Chau Nguyen
Thomas Weber

More Info

Here is a map with some relevant places in and outside the Campus of the University of Salerno, where the mini-workshop will take place. For more information contact us at mdippell@unisa.it (Marvin Dippell).

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