Seminário Simplético do Rio - 2022

- Seminários de 2022: -

# Sex 11/Nov, 10:30hs-17hs: 
Mini-workshop on Poisson Geometry and Related Topics
Presencial @UFF, sala 407 do bloco H no Campus Gragoatá.

Schedule

10:30 -- 11:30 Alejandro Cabrera (UFRJ)

11:30 -- 11:45 Coffee break

11:45 -- 12:45 Ivan Struchiner (USP)

12:45 -- 14:30 Lunch break 

14:30 -- 15:30 Pedro Frejlich (UFRGS)

15:30 -- 16:00 Coffee break

16:00 -- 17:00 Ioan Marcut (Radboud Universiteit Nijmegen)

Titles and Abstracts

• Title: Overview on quantization of Poisson brackets (Cabrera)

Abstract: In this talk, we shall revisit the idea of quantization of Poisson brackets by so-called star products. We shall establish a direct connection between this quantization notion and the underlying Lie theory for the Poisson brackets. Finally, we shall review recent results of the speaker and collaborators, a work in progress involving integrability, and open problems.

• Title: TBA (Struchiner)

• Title: Hierarchies and Dirac geometry (Frejlich)

Abstract: A well-known result of Magri and Morosi says that out of a symplectic and a Poisson structure which commute, one obtains a whole "hierarchy" (= sequence) of commuting Poisson structures by successive application of the induced Nijenhuis tensor. The literature which has since developed focused mostly on the Nijenhuis condition to generalize this result in the setting of Dirac structures. Our approach is rather that the notion of compatibility is god-given ("concurrence"), and we investigate the following question: given concurring Dirac structures, when do their iterated tangent and cotangent products concur? We conjecture the answer to be positive in general, and prove intermediary results which underpin our opinion. This suffices to produce new hierarchies, and to strengthen the classical  result of Magri and Morosi.

• Title: Poisson structures with compact support (Marcut)

Abstract: When compared to symplectic structures, Poisson structures are much more flexible. However, it is still poorly understood whether this flexibility is obstructed by some hidden invariants. I will talk about the Poisson Extension Problem, which I learned many years ago from Pedro Frejlich. Can any Poisson germ be extended to a global Poisson structure? Together with Gil Cavalcanti, we build explicitly such extensions, even with compact support, for certain classes of Poisson structures. I will discuss these constructions and some interesting consequences. This talk is based on the preprint: "Poisson structures with compact support", Gil R. Cavalcanti, Ioan Marcut, arXiv:2209.14016.

# Sex 28/Out, 14:30hs:  2 palestras
Presencial @IMPA, sala a confirmar

Global Dynamics and Stability by Conley-Zehnder Index theory
Dan Offin (Queen's University, Canada)

Abstract:  I review some recent results on global stability analysis of periodic orbits in Hamiltonian systems, using the Maslov and Conley-Zehnder indices associated to these orbits. An earlier result on the hyperbolicity of minimizing brake orbits on energy surfaces is described. The Henon-Hieles potential illustrates some of the applications of this technique in low dimensional systems. Hyperbolic behaviour on energy surfaces, and stability transitions of a family of periodic solutions limiting onto a homoclinic to saddle center equilibrium are explained using the Conley-Zehnder index.

Higher Koszul brackets on the cotangent complex
Hans-Christian Herbig (UFRJ)

Abstract:  A Let $(A,\{\ ,\ \}$ be an affine Poisson algebra over a field $k$ of characteristic zero. Then the module of Kähler differentials $\Omega^1_{A|k}$ together with the ring $A$ becomes a Lie-Rineart algebra with respect to the Koszul bracket. If $A$ is singular it is not projective as an $A$-module. The cotangent complex $\mathbb L_{A|k}$ can be seen as a replacement of $\Omega^1_{A|k}$, it is however not always a resolution. We lift the Koszul bracket from $\Omega^1_{A|k}$ to $\mathbb L_{A|k}$ as an $L_\infty$-Lie algebroid. The main tool is a $P_\infty$-algebra structure on a resolvent $R$ of $A$ that is constructed by homological perturbation theory.

# Ter 13/Set, 14hs:  1 palestra introdutória + 2 palestras de pesquisa
Presencial @UFRJ (Fundão), CT Bloco C, sala C116

(0): Introdução à palestra (1) [30min]
Renato Vianna (UFRJ)

(1): Lagrangian toric fibrations on smoothing of algebraic cones
Santiago Achig Andrango (IMPA)

Abstract:  Altmann used the Minkowski decomposition of a lattice polytope Q to describe the versal deformation of a toric Gorenstein singularity Y induced by Q. In this paper, we introduce the notion of an admissible decomposition of a lattice polytope Q, for which we prove that a deformation of Y, as described by Altmann, is smooth and admits a Lagrangian toric fibration with singularities. This singular fibration admits a convex base diagram representation with cuts, as the almost toric manifolds defined by Symington.  The convex base diagram obtained is the dual cone of the cone with Q at height 1. We will discuss some consequences of our result.

(2): Multiplicative Ehresmann connections
Ioan Marcut (Nijmegen)

Abstract:  In this talk, I will discuss the theory of multiplicative Ehresmann connections for Lie groupoid submersions covering the identity, as well as their infinitesimal counterparts. I will discuss interesting classes of Lie groupoids (including proper ones) and Lie algebroids, for which multiplicative Ehresmann connections always exist, also discuss obstructions to their existence. Finally, I will motivate this theory by the construction of local models and linearization results in Poisson geometry. This is joint work with Rui Loja Fernandes: arXiv:2204.08507.

# Ter 5/Abr, 17hs: Maxim Zabzine (Uppsala)
Presencial @IMPA: SALA 236

Shift equations for equivariant volumes

Abstract:  Motivated by the study of the Donaldson-Thomas theory on toric CY 3-folds I will reconsider application of Duistermaat-Heckman formula for non-compact toric Kahler manifolds. I will review the calculation of equivariant volumes for non-compact manifolds and explain some underlying issues. I will derive the set of finite-difference equations obeyed by equivariant volumes and their quantum versions. The talk is based on my joint paper with Nikita Nekrasov and Nicolo Piazzalunga (with appendix by Michele Vergne). 

Seminários de 2020/21:  accessar no link do canto superior direito ou aquí [2020], [2021]