Do dia 6 a 8 de novembro de 2019 acontecerá o IV Mini Workshop em Geometria Simplética, no Instituto de Matemática e Estatística (sala 407, Bloco H), no Campus Gragoatá da Universidade Federal Fluminense (UFF). O objetivo do evento é reunir especialistas em geometria simplética e áreas próximas, incluindo teoria de foliação, geometria algébrica, física matemática e geometria de Poisson.
Haverá um minicurso que será ministrado por Yoshihiko Mitsumatsu (Chuo University, Japan) e palestras com especialistas da área, como Francesco Bonechi (Università degli Studi di Firenze, Italy), Margaret Symington (Mercer University, USA), Jethro van Ekeren (UFF, Brazil), Misha Verbitsky (IMPA, Brazil), Renato Vianna (UFRJ, Brazil) e Maxim Zabzine (Uppsala University, Sweden).
6 de novembro de 2019 (quarta-feira)
14:00 - 14:50: Mitsumatsu
15:00 - 15:50: Mitsumatsu
15:50 - 16:15: Coffee break
16:15 - 17:15: Symington
7 de novembro de 2019 (quinta-feira)
10:00 - 11:00: Vianna
11:15 - 12:15: Verbitsky
12:15 - 14:00: Lunch
14:00 - 14:50: Mitsumatsu
15:00 - 15:50: Mitsumatsu
15:50 - 16:15: Coffee break
16:15 - 17:15: Bonechi
8 de novembro de 2019 (sexta-feira)
10:00 - 11:00: van Ekeren
11:15 - 12:15: Zabzine
Título: Lefschetz fibrations on Milnor fibres of cusp and simple elliptic singularities
Abstract: Recently we found there exist Lefschetz fibration structures on Milnor fibres of cusp or simple elliptic singularities in three complex variables, whose regular fibre is closed 2-torus. This also implies that there exists a symplectic structure on such a Milnor fibre whose end is symplectically periodic.
We start with the notion of convexity in symplectic structures as well as the (strong) pseudo convexity in complex variables, then proceed to the topologial flexibility of the symplectic convexity. How the cohomological condition is related with the modification of convex symplectic structures into end-periodic ones is discussed.
As a motivation and as an application of such end-periodic symplectic structures, leafwise symplectic structures on Lawson's foliation on the 5-sphere (a regular Poisson structure) and on similar foliations are shown to exist.
The construction of Lefschetz fibrations is roughly sketched. Local modifications of isolated critical points are discussed on the way. Such local study might enables us to attack further global existence results in future.
As applications to 4-dim (symplectic) topology, by gluing two Milnor fibres together with Lefschetz fibrationn structures, we obtain closed symplectic 4-manifolds and in fact elliptic surfaces over CP^1. Some of them are diffeomorphic to the K3 surface. A relation of these phenomena with the singularity theory, in particular the strange duality of Arnold, is also discussed.
The results are largely form the joint work with Naohiko KASUYA, Hiroki KODAMA, and Atsyhide MORI.
Francesco Bonechi (Università degli Studi di Firenze, Italy)
Title: Quantization of symplectic groupoids from multiplicative integrable models
Abstract: I will present a class of non trivial examples where Weinstein's dream of quantizing Poisson manifolds through the quantization of the symplectic groupoid can be concretely realized. The construction uses singular polarizations, for instance those given by integrable models that are compatible with the groupoid structure. We call such models multiplicative. The main source of examples comes from Poisson-Nijenhuis geometry. I will discuss in detail the example of Bruhat-Poisson structure on complex projective spaces.
Margaret Symington (Mercer University, USA)
Title: Integral affine surfaces: cylinders to spheres
Abstract: A key feature of an almost toric manifold (a symplectic four-manifold equipped with a singular Lagrangian fibration having singularities of elliptic and focus-focus type) is the singular integral affine structure induced on the base of the fibration. Building on an understanding of the integral affine structures induced by semitoric systems and inspired by the algebraic and combinatorial descriptions of integral affine structures on on S^2 (coming from toric degenerations and mirror symmetry), I will explain an approach to describing integral affine structures that is suitable in the more general symplectic context. I will give some examples of structures on cylinders and explain how they may help in understanding such structures on S^2.
Jethro van Ekeren (UFF, Brazil)
Title: Quantisation of Poisson Arc Spaces and Statistical Models
Abstract: Differential Poisson Algebras and Vertex Algebras provide a context in which to discuss quantisations and classical limits of 2 dimensional field theories. In this talk I will describe a homological criterion to decide freeness of classical limits of such field theories as well as recent results on classical limits of the Ising model and its generalisations and open problems. I will also discuss interesting links with modular forms and partitions and with recent work of physicists on Schur indices of 4 dimensional superconformal field theories. (Joint work with R. Heluani.)
Misha Verbitsky (IMPA, Brazil)
Title: Closed Reeb orbits on Sasakian manifolds
Abstract: Sasakian manifolds are related to contact ones in the same way as Kahler manifolds are related to symplectic. Sasakian manifold can be defined as a quotient of a Kahler manifold by R acting by non-trivial Kahler homotheties, in the same was as one defines a contact manifold as a quotient of a symplectic manifold by homotheties. For each Sasakian manifold Q there is a circle S^1 acting on Q by Sasakian automorphisms, in such a way that the quotient Q/S^1 is a projective orbifold. We prove that the number of closed Reeb orbits on Q is bounded from below by the sum of Betti numbers of Q/S^1. This implies that a Sasakian manifold of real dimension 2n+1 has at least n closed Reeb orbits. This is a joint work with Liviu Ornea. The proof is based on counting elliptic curves on a certain non-Kahler complex manifold which is naturally associated to a Sasakian manifold.
Renato Vianna (UFRJ, Brazil)
Title: Applications of almost toric fibrations
Abstract: We will survey a series of results in 4 dimensional symplectic topology in the last 6 years that rely on almost toric fibrations developed by Symington. We will present a subset of the following list of results: existence of infinitely many (Hamiltonian isotopy classes) of monotone Lagrangian tori in Del Pezzo surfaces (comments about generalisations to higher dimensions [joint with L. Diogo, D. Tonkonog, W. Wu]); computations and other results of isotopy shapes/star-shapes for (almost) toric fibres and relationship with embedding of Weinstein neighbourhoods [joint with E. Shelukhin, D. Tonkonog]; ball packing results in the complement of exotic Lagrangian tori in CP^2 [joint with W. Lee, Y-G. Oh]; recover results about volume filling embeddings of ellipsoids into the ball and other toric domains [joint with R. Cassals]; classification of almost toric fibres of CP^2 [joint with E. Shelukhin, D. Tonkonog]. We may also mention how to visualise Lagrangians fibering over tropical curves on almost toric fibrations, which was independently discovered by G. Mikhalkin / D. Matessi / J. Hicks.
Maxim Zabzine (Uppsala University, Sweden)
Title: On generalised Kahler potential
Abstract: I will discuss the problem of generalised Kahler potential. I will outline the past progress and open problems.
Para mais informações sobre o evento, clique AQUI.
Título: Modelo fatorial quantílico: uma abordagem bayesiana para redução de dimensão sob a ótica de quantis
Data: 25 de outubro de 2019 (sexta-feira)
Local: Sala C116, CT
Kelly Cristina Mota Gonçalves – IM-UFRJ (Presidente)
Helio dos Santos Migon - IM-UFRJ
Mariane Branco Alves - IM-UFRJ
Larissa de Carvalho Alves - ENCE
Gustavo da Silva Ferreira - ENCE
João Batista de Morais Pereira
Título: "Processos Gaussianos e Multifidelidade".
Data: 30 de outubro de 2019 (quarta-feira)
Local: C-116, IM/UFRJ, CT, BL. C
Orientador(es): Fábio Antonio Tavares Ramos
Fábio Antonio Tavares Ramos (Presidente) - IM-UFRJ
Bernardo Freitas Paulo da Costa - IM-UFRJ
Hugo Tremonte de Carvalho - DME-UFRJ
Carlos Tomei - PUC-Rio
Luca Roberto Augusto Moriconi - IF-UFRJ
Título: Uma Análise da Equação de Muñoz-Delgado
Data: 31 de outubro de 2019 (quinta-feira)
Local: C-116, IM/UFRJ, CT, BL. C
Orientador(es): Rolci de Almeida Cipolatti
Rolci de Almeida Cipolatti (Presidente) - IM-UFRJ
Ricardo Martins da Silva Rosa - IM-UFRJ
Adán José Corcho Fernández - IM-UFRJ
Juan Bautista Límaco Ferrel - UFF
Patrícia Nunes da Silva - UERJ
Ademir Fernando Pazoto (Suplente) - IM-UFRJ
Dinamérico Pereira Pombo Júnior (Suplente) - UFF
Minicurso: Perturbations of matrix pencils and polynomials
Ministrado por: Andrii Dmytryshyn
Data: 24, 29 e 31 de outubro
Horário: 10 às 12
Local: Sala Eliana Aude (localizada nas dependências do NCE)
Resumo: Complete eigenstructures of matrices, matrix polynomials and pencils are reflected in the corresponding canonical forms, e.g., Jordan and Kronecker canonical forms. These canonical forms are well known and studied with various purposes but the reductions to these forms are unstable operations: both the corresponding canonical forms and the reduction transformations depend discontinuously on the entries of an original matrix or matrix pencil. Therefore, V.I. Arnold introduced a normal form, with the minimal number of independent parameters, to which an arbitrary family of matrices A' close to a given matrix A can be reduced by similarity transformations smoothly depending on the entries of A'. He called such a normal form a miniversal deformation of A. Now the notion of miniversal deformations has been extended to various matrices and matrix pencils. Miniversal deformations can help us to construct stratifications, i.e., closure hierarchies, of orbits and bundles. These stratifications are the graphs that show which complete eigenstructures the matrices, matrix pencils or polynomials may have in an arbitrarily small neighbourhood of a given matrix, matrix pencil or polynomial. In particular, the stratifications show how the eigenvalues may coalesce or split apart, appear or disappear. The course will start with the classical results by V.I. Arnold on the perturbations of matrices and cover their recent extensions to the structured matrix systems. The second part of the course will be dedicated to the geometrical aspects of the perturbation theory.
About Andrii Dmytryshyn: Andrii research interests are in the fields of matrix analysis and computational mathematics. Before starting as an Associate Senior Lecturer at Örebro University, Andrii worked and studied in Umeå (Sweden), Bordeaux (France), Padua (Italy), and Kyiv (Ukraine).
Andrii received the SIAM Student Paper Prize 2015 (which is one of the major prizes awarded by the Society for Industrial and Applied Mathematics). He was selected as one of the top 7 candidates for the Householder Prize XX (The Householder Prize is an award for the best dissertation in numerical linear algebra in a 3-year period). Andrii was also an LAA Early Career Speaker at ILAS 2017 and invited and plenary speaker at several other conferences.
Andrii teaches courses on numerical methods for differential equations, applied mathematics, mathematical control theory, matrix computations, and case studies in computational mathematics.