Seminário Simplético do Rio
O Seminário Simplético Conjunto é uma iniciativa de pesquisadores do IMPA, PUC-RJ, UFF e UFRJ. As palestras envolvem pesquisadores locais ou convidados, em temas relacionados à geometria simplética e de Poisson. Alunos são particularmente encorajados a participar.
The Joint Symplectic Seminar is organized by IMPA, PUC-RJ, UFF and UFRJ. The talks involve topics related to symplectic and Poisson geometry. Students are particularly encouraged to attend the seminars.
Em 2024, o seminário será presencial e itinerante, com frequência aproximada de 1 vez por mes.
Para receber anúncios e os links, entrar em contacto com:
contact: semsimp (arr) impa (pont) br
- Seminários de 2024: -
# Sex 14/Jun, 14hs: 2 palestras
UFRJ Fundão, sala C119 - IM
[14-15hs]
Sergey Galkin (PUC - Rio)
On Counter-Examples to the Gamma Conjecture I
Abstract: Gamma conjectures form a family of conjectures and theorems at the crossroads of symplectic topology with number theory, classical analysis of ODEs, and 2D SCFT/string theory. They are closely related to Dubrovin’s conjectures, mirror symmetry, loop spaces, supersymmetric central charge of branes, and the disc partition function. A common element among them is the appearance of the Gamma class, or a Chern character twisted by the Gamma class (known as Iritani’s map), and its relation to asymptotic classes built from Gromov-Witten invariants or periods.
Gamma conjectures for Fano manifolds were precisely formulated and heuristically justified by myself, Vasily Golyshev, and Hiroshi Iritani in a series of works starting from 2008: 1604.04652 (experiments and zeta-conjectures), 1404.6407 (conjectures O, Gamma I, and II), 1404.7388 (conifold point and “lower bound conjecture”), 1508.00719 (mirror and loop space heuristics), 2307.15938 (conjecture Gamma III, based on Sanda-Shamoto). Additionally, Abouzaid-Ganatra-Iritani-Sheridan heuristically justified Gamma conjectures via tropical geometry.
Conjecture O is used to define Gamma conjecture I, which predicts a series of equalities of real transcendental numbers. Conjecture O and the "lower bound conjecture" concern the relative location on the complex plane of eigenvalues of quantum multiplication by the first Chern class of a Fano manifold. These have been verified for many types of computable examples, including large finite and various countable families. However, these more precise conjectures remained unverified even for toric Fano manifolds, where mirror symmetric Landau-Ginzburg models (Floer potentials) are quite explicit, and both mirror symmetry and Gamma conjecture II are well-known.
Recently, Jianxun Hu, Huazhong Ke, Changzheng Li, and Zhitong Su made significant progress by reducing the analysis of critical values of mirror superpotentials in toric cases to optimization problems in non-linear programming. This enabled them to systematically verify the aforementioned conjectures in a large pool of known toric Fano manifolds. They found counter-examples to the lower bound conjecture (2405.16987) and, more importantly, to Conjecture O and Gamma conjecture I (2405.16979). The latter is estimated to fail in 1-2% of cases. For the original counter-examples, we were able to identify the respective asymptotic classes and relate them to exceptional objects arising from a step in the minimal model program. We have made some steps toward modifying Gamma conjecture I, but so far, we lack an understanding of how and why various heuristics, such as the loop space heuristics, fail in the aforementioned counter-examples.
This talk is based on joint works with Iritani, particularly on recent work with Hu, Ke, Li, and Su (2405.16979), and earlier works with Golyshev.
[15:30-16:30hs]
Leonardo Macarini (IMPA)
Periodic orbits of non-degenerate lacunary contact forms on prequantizations
Abstract: A non-degenerate contact form is lacunary if the indexes of every contractible periodic Reeb orbit have the same parity. To the best of my knowledge, every contact form with finitely many periodic orbits known so far is non-degenerate and lacunary. I will show that every non-degenerate lacunary contact form on a suitable prequantization of a closed symplectic manifold $B$ has precisely $r_B$ contractible closed orbits, where $r_B=\dim H_*(B;{\mathbb Q})$. Examples of such prequantizations include the standard contact sphere and the unit cosphere bundle of a compact rank one symmetric space (CROSS). I will also consider some prequantizations of orbifolds, like lens spaces and the unit cosphere bundle of lens spaces. This is joint work with Miguel Abreu.
# Ter 26/Mar, 14:30hs: 2 palestras
IMPA, sala a determinar
[14:30-15:30hs]
Vortex sheets and diffeomorphism groupoids
Boris Khesin (Toronto)
Abstract: We discuss ramifications of Arnold’s group-theoretic approach to ideal hydrodynamics as the geodesic flow for a right-invariant metric on the group of volume-preserving diffeomorphisms. It turns out that many equations of mathematical physics, such as the motion of vortex sheets or fluids with moving boundary, have Lie groupoid, rather than Lie group, symmetries. We present their geodesic setting, which also allows one to describe multiphase fluids, homogenized vortex sheets and Brenier’s generalized flows. This is a joint work with Anton Izosimov.
[16-17hs]
Geometrical aspects of magnetic flows
Valerio Assenza (IMPA)
Abstract: Magnetic flows are the toy models for the motion of a charged particle over a Riemannian manifold under the influence of a magnetic force. These dynamics appear in several physical and mathematical contexts and they had been investigating with different approaches in the last four decades. From a geometrical point of view, to every magnetic system we associate an operator called magnetic curvature. Such an operator encodes the classical Riemannian curvature together with terms of perturbation due to the magnetic interaction, and it carries relevant properties in terms of magnetic dynamics. For instance, in this talk we bring into the discussion of magnetic flows a theorem by E.Hopf (refined by Green): if a geodesic flow is without conjugate points, then the total scalar curvature is non positive and equal to zero if and only if the metric is flat. The generalization of this result leads naturally to investigate the flatness in a magnetic sense. This is part of a joint work with James Marshall Reber and Ivo Terek.