O Seminário Simplético Conjunto é uma iniciativa de pesquisadores do IMPA, PUC-RJ, UFF e UFRJ, com encontros mensais rotativos. Cada encontro conta com duas palestras, ministradas por pesquisadores locais ou convidados, em temas relacionados à geometria simplética (num sentido amplo). Alunos são particularmente encorajados a participar. The Joint Symplectic Seminar is organized by IMPA, PUC-RJ, UFF and UFRJ, and consists of monthly meetings that alternate among these institutions. Each meeting features two talks on topics related to symplectic geometry (in a broad sense). Students are particularly encouraged to attend the seminars.

Encontros 2017

# Sexta 8/dez @ UFF, Auditório Pós-graduação, Bloco H Gragoatá

This is joint work with Adriano Regis (UFRPE) and Cesar Castilho (UFPE).  In the first part of the talk we review Jacobi's construction of confocal quadrics coordinates on an octant of the triaxial ellipsoid, and how he parametrized the whole ellipsoid except for two segments connecting the umbilics. Jacobi used those coordinates to show that the geodesics are integrable. This problem, as we know is the first example of a completely integrable two degrees of freedom system without geometric symmetries. Jacobi also showed how to find local isothermal coordinates. In the second part of the talk, I will show how to make these coordinates global, and used them to derive the equations for point vortices. We show that the vortex pair (two opposite vortices) system is an nonintegrable extension of Jacobi's geodesics. Themes for future work will be presented.

I will discuss a method, based on the notion of ``Euler-like'' vector fields, to obtain splitting theorems in several geometric settings (including the classical Frobenius theorem and Weinstein's splitting theorem for Poisson structures). The method actually proves more general normal forms around certain submanifolds (called ``transversals'') as well as their equivariant versions. This is joint work with H. Lima and E. Meinrenken.

# Terça-feira 7/nov @ UFRJ, Sala C-116.

A classical construction due to Paul Biran allows to lift a Lagrangian submanifold $L$ from a Donaldson $Y$ divisor to a Lagrangian $L'$ in an ambient symplectic manifold $X$. Biran shows that if the minimal Chern number of $Y$ is greater than 1, then the count of Maslov index 2 holomorphic disks with boundary on the lifted Lagrangian $L'$ is equivalent to the similar count of disks with boundary on $L$ plus one extra disk. We study this enumerative geometry problem in the case when the minimal Chern number of $Y$ is 1. This reveals several new, previously unexplored connections it has with relative closed-string Gromov-Witten theory of the pair $(X,Y)$. We explore applications, in particular, we use that to distinghish (up to action of $\mathrm{Symp}(X)$) lifts of previously known Lagrangians. This is joint work with Luis Diogo, Dmitry Tonkonog and Weiwei Wu.

An extremal metric, as defined by Calabi, is a canonical Kahler metric: it minimizes the curvature within a given Kahler class. According to the Yau-Tian-Donaldson conjecture, polarized Kahler manifolds admitting an extremal metric should correspond to stable manifolds in a Geometric Invariant Theory sense. In this talk, we will explain that a projective extremal Kahler manifold is asymptotically Chow stable. This fact was conjectured by Apostolov and Huang, and its proof relies on quantization techniques. We will explain various implications, such that unicity or splitting results for extremal metrics. Joint work with Yuji Sano ( Fukuoka University)

# Sexta 22/set @ PUC, Sala 856L.

De acordo com Gromov, "Loewner made an amazing discovery around 1949". A descoberta à qual Gromov se refere, devido ao matemático alemão Charles Loewner, é a seguinte desigualdade sistólica: "em qualquer 2-toro riemanniano, o comprimento da curva não-contrátil mais curta é menor ou igual que a raiz quadrada da área vezes uma constante universal". Loewner ainda dá a constante explicitamente, e mostra que a geometria fica determinada quando a desigualdade é igualdade. Esse foi o início do que se entende hoje por geometria sistólica, uma área bastante divulgada por matemáticos como Thom, Berger, Gromov, entre outros. Recentemente ficou claro que desigualdades sistólicas têm interpretação natural no contexto de geometria de contato (Alvarez Paiva-Balacheff) e que esse ponto de vista permite atingir resultados novos não-triviais. Eu gostaria, nesta palestra informal, de discutir alguns problemas básicos e também alguns resultados, como a confirmação da conjectura (Babenko-Balacheff) de que a geometria redonda na 2-esfera maximiza razões sistólicas localmente, e a refutação da conjectura (Hutchings) de que razões sistólicas de formas de contato são limitadas superiormente. Além disso, se o tempo permitir, apresentarei uma cota superior ótima para a razão sistólica de 2-esferas riemannianas de revolução. Todos esses enunciados estão intimamente ligados com uma importante conjectura devido a Viterbo. Esses resultados são em colaboração com Abbondandolo, Bramham e Salomão.

(This is joint work with Jason Lotay) I shall describe my joint work with Jason Lotay concerning existence and classification results for G2-instantons on noncompact manifolds. That work investigates the particular case of $\mathbb{R}^4 \times S^3$, with it's two explicitly known distinct G2-holonomy metrics, exhibiting the different existence/behavior of G2-instantons. I will also give an explicit example of sequences of G2-instantons where bubbling and removable singularity phenomena occur in the limit. If time permits, I will state some quite accessible (I hope) open problems.

# Segunda 28/08 @ IMPA, Sala Auditorio 1.

Embedded contact homology is an invariant of contact 3-manifolds. It has proven to be useful in studying the Reeb dynamics of contact manifolds as well as studying embeddings of symplectic 4-manifolds they bound. However, it is difficult to compute in general. In this talk, we discuss how to compute the ECH of toric contact manifolds and some applications to embedding problems.

It is well known that for adjoint orbits of the unitary and orthogonal compact Lie groups we have (Liouville) integrability for collective Hamiltonians systems. These integrable systems are called Gelfand-Tsetlin integrable systems and they were introduced in 1983 by V. Guillemin and S. Sternberg. Since then, these systems have been widely studied due to their relations with representation theory, geometric quantization and more recently topics related to mirror symmetry (Potential functions via Toric degeneration). In this talk, I will present a Lax formulation for Gelfand-Tsetlin integrable systems, the main purpose is to show how this formulation allows us to describe the conserved quantities of Gelfand-Tsetlin integrable systems as spectral invariants associated to a suitable Lax matrix.

# Sexta 23/jun @ UFF, Sala 407 - bloco H, Campus Gragoatá, Niterói.

A Gelfand Pair is a pair $(G,K)$, where $G$ is a Lie group and $K$ is a spherical subgroup. These pairs are relevant in Geometric Representation Theory. Any symmetric pair $(G,K)$ is a Gelfand pair, but there are many other examples. When $G$ is reductive, Gelfand pairs were classified by Krammer, Mikityuk and Brion. In this talk I will explain how to approach Gelfand pairs from the perspective of symplectic geometry and I will explain a connection with symplectic realizations of Poisson manifolds, which leads to a characterization of Gelfand pairs that are symmetric spaces of maximal rank, a result that does not seem to be known.

Almost-toric manifolds are symplectic four-manifolds that are equipped with a singular Lagrangian fibration that has the local structure of a completely integrable system with toric and focus-focus singularities, the same types of singularities as in the integrable system on the phase space of the spherical pendulum that is defined by the energy and angular momentum. The K3 surface, a much-studied algebraic surface, can be endowed with both holomorphic and special-Lagrangian fibrations, the latter of which gives it the structure of an almost-toric manifold. After introducing almost-toric manifolds and how to read aspects of their topology from two-dimensional diagrams analogous to moment map images of toric manifolds, I will explain some of the topology of the K3 surface and explain a few open questions about almost-toric structures on the K3 surface.

# Terca 16/maio @ UFRJ, Sala C-116.

The algebra of symmetric polynomials on the dual of a Lie algebra carries a natural Poisson structure given by the Kirillov-Kostant-Souriau bracket. The celebrated Duflo homomorphism quantises this Poisson algebra to the enveloping algebra, in such a way that the restriction to the G-invariant subspaces is a ring isomorphism. In this talk I will first review the Duflo map and its relation to equivariant cohomology. I will then introduce a refinement of equivariant cohomology based on the affine Kac-Moody vertex algebra. Interestingly this construction only works at the critical level. I will show that the refined equivariant cohomology of a point coincides with the Feigin-Frenkel center generated by Segal-Sugawara vectors and discuss a jet analogue of the Duflo homomorphism.

Abordamos ferramentas topológicas na Teoria do Índice de Conley, com o propósito de explorar complexos de cadeia $(C,\Delta)$ gerados pelos pontos críticos de funções de Morse. Descrevemos um algoritmo de varredura que, a partir da matriz de conexão $\Delta$, nos permite codificar as informações dadas pela sequência espectral, produzindo um sistema de geradores para os seus módulos e determinando suas diferenciais. Finalmente, obtemos resultados dinâmicos explorando as informações algébrica da sequ&ehatncia espectral via este algoritmo. Além disso, mostramos que este processo corresponde a propriedades de continuação do fluxo associado.

# Terca 25/abr @ PUC, Sala 856 L.

Na primeira parte da palestra vou recordar a definicao de capacidade simplectica, de espaco Hermitiano simetrico e as principais propriedades. Na segunda parte, depois de recordar a definicao de positive Jordan triple system, vou calcular o Gromov width dos espacos Hermitianos simetricos e mais em geral limites inferiores e superiores por uma qualquer capacidade simplectica. Os resultados e os calculos sao parte de um trabalho conjunto com A. Loi, F. Zuddas (J. Symplectic Geom. 2016).

Joint work with Eva Miranda (UPC) and Francisco Presas (ICMAT). This talk addresses the focus-focus contribution to real geometric quantisation (with real polarisations given by integrable systems), with the quantisation of K3 surfaces serving as model examples and motivation. Near a singular Bohr-Sommerfeld focus-focus fibre of an integrable system in dimension four, the zeroth and first cohomology groups computing geometric quantisation are trivial, but the second cohomology group is not; the latter is actually infinite dimensional. As a consequence, the real geometric quantisation of semitoric integrable systems and almost toric manifolds are computed.

# Segunda 27/3 @ IMPA, Auditório 3.

In my talk I will describe a rather particular construction of Poisson structures arising from $r$-matrices. While the construction itself is well- known it is not completely obvious how (non-)generic such Poisson structures really are. I will point out some hard obstructions for their existence. On the other hand, they enjoy many nice features as e.g. a simple way to quantize them via twists. I will outline a simplified construction of a Drinfel'd twist and corresponding universal deformation formulas based on the Fedosov construction of star products. The talk is based on joint results with Chiara Esposito, Jonas Schnitzer and Thomas Weber.

In this talk I will prove that the mean Euler characteristic of a Gorenstein toric contact manifold is equal to half the normalized volume of the corresponding toric diagram. I will also give some immediate applications of this result. This is joint work with Leonardo Macarini.