Geometry and related topics セミナー

Date

2014年11月18日 (火)15:00-16:30

Room

Room 206, RIMS, Kyoto University

Speaker

Manabu Akaho (Tokyo Metropolitan University)

Title

Symplectic displacement energy for exact Lagrangian immersions

Summary We give an inequality of the displacement energy for exact Lagrangian immersions and the symplectic area of punctured holomorphic discs. Our approach is based on Floer homology for Lagrangian immersions and Chekanov's homotopy technique of continuations. Moreover, we discuss our inequality and the Hofer--Zehnder capacity.
Organizer Kaoru Ono

Date

2014年11月18日 (火)13:00-14:30

Room

Room 206, RIMS, Kyoto University

Speaker

Dingyu Yang (Institut de Mathématiques de Jussieu)

Title

Deriving maximality and 4 applications of level-1 structures from FOOO Kuranishi structures

Summary In this talk, I will sketch elements of polyfold--Kuranishi correspondence which perfectly identifies Fukaya-Oh-Ohta-Ono's Kuranishi structures with Hofer-Wysocki-Zehnder's polyfold Fredholm structures up to germs in two geometric ways intertwining associated perturbation theories. These two aforementioned structures are abstract geometric structures appearing in the foundational theory of moduli spaces in Lagrangian Floer theory and symplectic field theory respectively. Polyfold Fredholm theory generalizes the notion of smoothness, and underpins the ideal of making domain reparametrization smooth and equipping smooth structures to general moduli spaces in symplectic geometry while making almost no extra choice. As a finite dimensional approach, a Kuranishi structure is a global way to consistently fit together finite dimensional reductions constructed from local neighborhoods of moduli spaces, and provides a framework to globally perturb moduli spaces in spite of possible dimension jumps when switching to nearby local models. I defined a notion of Kuranishi structures by adding maximality and topological matching conditions to FOOO's Kuranishi structures; from this, one naturally obtains a Hausdorff level-1 good coordinate system, which is a coherent system of structured tubular neighborhoods built upon an ordered finite cover and serves as a device to overcome dimension jumps (in 4 applications below). I also defined a notion of a (global) germ via common refinements possibly increasing local dimensions, which is applicable to any version of Kuranishi structures. I will show how to functorially obtain these two conditions from any representative (and thus all of its shrinkings) within each germ of FOOO's Kuranishi structures, and thus level-1 structures become available to use to show the following: (i) A germ of FOOO's Kuranishi structures is an equivalence class. (ii) We can define and compose maps between germs (where level-1 structures define left inverses to chart embeddings), and obtain a category of FOOO Kuranishi structure germs. (iii) We can lift local perturbations using level-1 structures to globally perturb a Kuranishi structure, and can lift small such global perturbation through level-1 refinement to show that perturbation theory descends to germs; and combining (i) to (iii), we do not need to compare different perturbations constructed using choices for truncated moduli space of different finite energy levels and can do a single perturbation for the entire moduli space, just as in polyfold theory. (iv) Level-1 structures provide sc-smooth retract models for gluing across dimension-jumping coordinate changes and thus convert FOOO Kuranishi structures to polyfold Fredholm structures, and a further polyfold--Kuranishi correspondence for analysis involved in constructing these geometric structures would merge methods and results from these two fields and make construction of Legendrian symplectic field theory easier and as a common extension.
Organizer Kaoru Ono

Date

2014年11月17日 (月) 10:30-12:00

Room

Room 110, RIMS, Kyoto University

Speaker

Naichung Conan Leung (Chinese University of Hong Kong)

Title

Witten deformation, A_infinity Morse category and scattering

Date

2014年11月17日 (月) 13:30-15:00

Room

Room 110, RIMS, Kyoto University

Speaker

Naichung Conan Leung (Chinese University of Hong Kong)

Title

Donaldson-Thomas invariants for Calabi-Yau fourfolds

Organizer Kaoru Ono

Date

2013年7月18日 (木) 15:00-16:30

Room

Room 006, RIMS, Kyoto University

Speaker

Sheel Ganatra (Stanford University)

Title

Symplectic cohomology and duality for the wrapped Fukaya category

Abstract To an exact symplectic manifold M, one can two important Floer-theoretic invariants: symplectic cohomology SH^*(M) and the wrapped Fukaya category W(M). We will explain how, when M contains enough Lagrangians, the natural geometric open-closed string maps between the Hochschild homology of W(M), symplectic cohomology, and the Hochschild cohomology of W(M) are all isomorphisms. The induced isomorphism between Hochschild homology and cohomology is an instance of a new self-duality for the wrapped Fukaya category, a non-compact version of a Calabi-Yau structure.
Organizer Kaoru Ono

Date

2013年7月11日(木)15:00--16:30

Room

Room 006, RIMS, Kyoto University

Speaker

Bai-Ling Wang (Australian National University)
微分トポロジーセミナーと共同開催

Title

K-theory virtual fundamental classes for moduli spaces of stable maps

Abstract In this talk, I will propose K-theory virtual fundamental classes for moduli spaces of stable map, based on joint work in progress with Bohui Chen and Jianxun Hu in an attempt to define quantum K-theory for general symplectic manifolds. 
Organizer K. Ono

Date

2013年5月30日(木)15:30−17:00 

Room

Room 006, RIMS, Kyoto University

Speaker

Andrei Pajitnov (Universite Nantes)

Title

Twisted Novikov homology, jump loci and non-abelian Hodge theory

Abstract We begin with a brief introduction to the Novikov homology and circle-valued Morse theory. Then I will explain our joint work with Toshitake Kohno about the applications of the Novikov homology to the jump loci in the homology with local coefficients, in particular on compact Kaehler manifolds.
Organizer K. Ono

Date

April 2 (Tue), 2013, 11:00−12:00, 13:30−14:30, 15:00-16:00

Room

Room 204, RIMS, Kyoto University

Speaker

Naichung Conan Leung (Chinese University of Hong Kong)

Title

SYZ mirror transformation

Organizer K. Ono

セミナートップへ