David Li-Bland

Gone Fishing 2014 Schedule

All Talks will take place in 740 Evans Hall.

Gone Fishing 2014 Abstracts

Yanli Song: Dirac induction and quantization of conjugacy classes.

We generalize the Dirac induction to the homogeneous setting of coadjoint orbits of loop group. With this, we give an explicit construction of the quantization of conjugacy classes as elements in the equivariant twisted K-homology.

Aissa Wade: Infinitesimal automorphisms of Poisson manifolds, matched pairs of Lie algebroids, and integrability.

Given a Poisson manifold M together with an infinitesimal automorphism E, there is a procedure for building a matched pair of Lie algebroids such that the associated bicrossed product determines a Lie algebroid structure of the first jet bundle J¹M of M which, depends only on the cohomology class of E. We will discuss the integrability question for such Lie algebroid structures on the first jet bundle of M. Moreover, we will explain how to extend this procedure to Lie algebroids other than the cotangent Lie algebroid of a Poisson manifold.

Notes: http://math.berkeley.edu/~libland/gone-fishing-2014/wade.pdf

Cornelia Vizman: Coadjont orbits of diffeomorphism groups via dual pairs.

Spaces of closed submanifolds, called nonlinear Grassmannians, admit Frechet manifold structures. Connected components of the codimension two nonlinear Grassmannian can be realized as coadjoint orbits of (central extensions of) the group of (exact) volume preserving diffeomorphisms.

In this talk we present other coadjoint orbits that can be described as nonlinear Grassmannians, but this time each submanifold is endowed with an extra structure (a differential form). We obtain them by symplectic reduction related to one leg of a dual pair of momentum maps. It is known that, in a finite dimensional dual pair of momentum maps coming from two commuting Hamiltonian actions, symplectic reduction for one of the groups leads to coadjoint orbits of the other group. We give a weaker version of this theorem that works in infinite dimensional dual pairs too. First we apply it to the ideal fluid dual pair and get coadjoint orbits of (central extension of) the Hamiltonian group: nonlinear Grassmannians of weighted isotropic submanifolds of the given symplectic manifold. Applied to another dual pair, it provides coadjoint orbits of (central extensions of) the group of (exact) volume preserving diffeomorphisms: nonlinear Grassmannians of codimension one submanifolds that enclose a constant volume, each endowed with a closed 1-form.

Henry Jacobs: A “whirls within whirls” particle methods for incompressible fluids via dual pairs.

One of the maddening aspect of ideal fluids is that of “self-similarity”. One cause of this self-similarity is the nested structure of the diffeomorphism group. In this talk I will present a particle method for fluids in which the particles carry finite dimensional models of the diffeomorphism group, in an attempt to model this “whirls within whirls” picture of fluids. If we consider a kinetic energy Hamiltonian on the space dual to the Lie algebra of divergence free vector-fields, the corresponding Hamiltonian equations of motion is a evolutionary PDE which serve as a model for an incompressible ideal fluid. We can build a dual pair with finite dimensional Clebsh/Symplectic variables by considering the left action of volume preserving diffeomorphism on points in Rⁿ, and the right action of the trivial group. Using this dual pair, we can pull-back our fluid model, to obtain particle like solutions to the equations of motion. We can enrich this picture by considering the left action of volume preserving diffeomorphism on a source fiber of the jet-groupoid, and the right action of the isotropy group of the given source fiber. By pulling back the dynamics on the fluid via this dual pair we obtain particle like solutions to fluid equations wherein the particles have internal symmetries (given by jet groups), and conserved momenta. Furthermore, the conserved momenta can be related to Kelvin’s circulation theorem (i.e. the conservation of circulation).

Preprint: http://math.berkeley.edu/~libland/gone-fishing-2014/jacobs.pdf

Yi Lin: Hard Leschetz theorem for K-contact manifolds.

Recently, Cappelletti-Montano, De Nicola, and Yudin proved a Hard Lefschetz theorem for the De Rham cohomology of compact Sasakian manifolds, and proposed an associated notion of Lefschetz contact manifolds. In this talk, we discuss a new approach to the Hard Lefschetz theorem for Sasakian manifolds using the formalism of odd dimensional symplectic geometry. This leads to a more general Hard Lefschetz theorem for K-contact manifolds, and provides us a sufficient and necessary condition for a finitely presentable group to be the fundamental group of a Lefschetz contact five manifolds. As an application, we show how to use our methods to construct simply-connected K-contact manifolds which do not support any Sasakian structures. This in particular answers an open question asked by Boyer and late Galicki.

Preprint: http://arxiv.org/abs/1311.1431.

Francois Ziegler: Quantum States Localized on Lagrangian Submanifolds.

Let X be a symplectic manifold and Aut(L) the automorphism group of a Kostant-Souriau line bundle on X. *Quantum states for X*, as defined by J.-M. Souriau in the 1990s, are certain positive-definite functions on Aut(L) or, less ambitiously, on any “large enough” subgroup G of Aut(L). This definition has two major drawbacks: when G = Aut(L) there are no known examples; and when G is a Lie subgroup the notion is far from selective enough. In this talk I’ll introduce the concept of a quantum state *localized at Y *, where Y is a coadjoint orbit of a subgroup H of G. I’ll explain how such states often exist and are unique when Y has lagrangian preimage in X, and how this can be regarded as a solving, in a number of cases, A. Weinstein’s “fundamental quantization problem” of attaching state vectors to lagrangian submanifolds.

Notes: http://math.berkeley.edu/~libland/gone-fishing-2014/ziegler.pdf

Eckhard Meinrenken: Lu’s Lie algebroid.

Jiang-Hua Lu proved that for any Poisson action H × M → M, the action Lie algebroid M × 𝔥 and the cotangent Lie algebroid T_π^*M form a matched pair: That is, their direct sum is a Lie algebroid. I will explain a simple construction of Lu’s Lie algebroid, and some generalizations and applications.

Michael Gekhtman: Cluster structures on Drinfeld doubles.

I will report on the progress in the ongoing joint project with M. Shapiro and A. Vainshtein aimed at constructing cluster structures compatible with Poisson-Lie structures on a double of a simple complex group arising in a Belavin-Drinfeld classification.

Notes: http://math.berkeley.edu/~libland/gone-fishing-2014/gekhtman.pdf

Rui Loja Fernandes: Pic(𝔤^*) = OutAut(𝔤) .

In this talk I will discuss some fundamental properties of the Picard group and Picard Lie algebra of a Poisson manifold. I will use them to proof a conjecture of Bursztyn and Weinstein stating that Pic(𝔤^*) = OutAut(𝔤), when 𝔤 is semisimple of compact type. This is joint work with Henrique Bursztyn.

Li-Sheng Tseng: A Symplectic Product for Differential Forms.

I will describe a new product for differential forms on symplectic manifolds that in general involves derivatives and is non-associative. The product interestingly leads to an A-infinity structure for forms. This is joint work with C.-J. Tsai and S.-T. Yau.

Ivan Contreras: Integration of Poisson manifolds and canonical relations.

In arXiv:1401.7319 we define a way to integrate Poisson manifolds, compatible with the usual integration via symplectic groupoids, from the phase space construction in the Poisson sigma model (PSM). This new object is described in an extended symplectic categroid of weak symplectic Banach manifolds as objects, and canonical relations as morphisms. In this talk, we will briefly describe such construction, in the infinite dimensional context, stressing the advantages with respect to the finite dimensional integration. If time allows, we will mention an application to the study of Topological Field Theories with boundary. This work is joint with Alberto Cattaneo.

Notes: http://math.berkeley.edu/~libland/gone-fishing-2014/contreras.pdf

Markus Pflaum: Differentiable stratified groupoids and inertia spaces.

The inertia space of a Lie groupoid encodes interesting topological, geometric, and analytic information about the original Lie groupoid. In general, the inertia space is not the quotient space of a Lie groupoid but a singular groupoid. In the talk, this phenomenon will be explained, and the connection with cyclic homology theory of proper Lie groupoids will be indicated.

José Antonio Vallejo: Averaging of Dirac structures.

We describe an averaging procedure on a Dirac manifold, with respect to a class of compatible actions of a compact Lie group. Applications to invariant realizations of Poisson structures around symplectic leaves and coupling Dirac structures will be described, along with a discussion on the existence of obstructions to this procedure.

Notes: http://math.berkeley.edu/~libland/gone-fishing-2014/vallejo.pdf

Associated publications: http://dx.doi.org/10.1063/1.4817863, and http://dx.doi.org/10.3842/SIGMA.2014.096.

Songhao Li: Toric log symplectic manifolds.

A log symplectic structure on a 2n-manifold is a Poisson structure that is degenerate linearly on a collection of normal intersecting hypersurfaces. Generalizing the result of Delzant, we characterize log symplectic manifolds with a Lagrangian action of the n-torus. Furthermore, generalizing the notion of Hamiltonian, we classify Hamiltonian toric log symplectic manifolds. This is joint work with M Gualtieri, A Pelayo and T Ratiu.

Notes: http://math.berkeley.edu/~libland/gone-fishing-2014/songhaoli.pdf

Mathieu Stienon: A Hopf algebra associated to a Lie pair.

The quotient L∕A[-1] of a pair A ⊂ L of Lie algebroids is a Lie algebra object in the derived category D^b(C) of the category C of left modules over the universal enveloping algebra of A — the Lie bracket is the Atiyah class of L∕A. We describe the universal enveloping algebra of L∕A[-1] and we prove that it is a Hopf algebra object in D^b(C).

Preprint: http://arxiv.org/abs/1409.6803

Associated publication: http://dx.doi.org/10.1016/j.crma.2014.09.010.

Henrique Bursztyn: On the integration of double structures.

I will discuss integration results relating V B-algebroids to V B-groupoids, and double Lie algebroids to LA-groupoids, mentioning examples and applications. This is based on joint work with M. del Hoyo and A. Cabrera.

Rajan Mehta: Representing Representations up to Homotopy.

Slides: http://math.berkeley.edu/~libland/gone-fishing-2014/mehta.pdf