Ching-Hao Chang (Academia Sinica)
Title:
The isotopy problems of nodal symplectic spheres and J-holomorphic spheres in rational manifolds
Abstract:
The isotopy problem for symplectic submanifolds in a symplectic manifold is always a very interesting topic in symplectic topology. One can use various techniques to study the isopoty problems for different categories of symplectic submanifolds in different symplecitc manifolds.
For the compact symplectic 4-manifold case, one of the approaches to attack the isotopy problem for the symplectic surfaces in a compact symplectic 4-manifold (M,w) is to study the isotopies for J-holomorphic curves in (M,w) where J is an w-tamed almost complex structure on M.
This approach is first introduced by M. Gromov. He (1985) proved that any symplectic sphere of degree 1 in CP2 is symplectically isotopic to an algebraic line. J. Barraud (1999) extended Gromov's work to show that any symplectic sphere of degree d in CP2 with only positive ordinary double point singularities is symplectically isotopic to an algebraic curve. In fact, We can imitate Barraud's work and further extend the result to the nodal symplectic spheres in rational manifolds by studying the nodal J-holomorphic spheres in them.
In this talk, we will briefly introduce the idea and take a walk in the moduli space of J-holomorphic curves.
Huai-Liang Chang (Hong Kong University of Science and Technology)
Title:
Mixed-Spin-P field moduli and its virtual fundamental class
Abstract:
Mixed-Spin-P field (MSP field) is a moduli space which contain both objects in GW theory of a projective hypersurface and also FJRW theory of the corresponding cone singularity. It is designed for studying CY-LG correspondence and also enumeration of higher genus GW/FJRW invariants. In this talk we will survey its motivation and current progress.
Bohui Chen (Sichuan University)
Title:
Quantization of Kirwan morphism
Abstract:
In this talk, I will explain the quantum version of Kirwan morphism using moduli spaces of symplectic vortex and generalized symplectic vortex equations.
Gabriel C. Drummond-Cole (IBS Center for Geometry and Physics)
Title:
Chain level string topology operations
Abstract:
String topology on a manifold is related to symplectic invariants on the cotangent bundle of that manifold. So far, this relationship has been expressed rigorously as an isomorphism of certain algebraic structures at the level of homology. It is anticipated that the isomorphism is the shadow of a more fundamental equivalence at the level of chain complexes. I will discuss a chain level string topology structure that could be seen as a candidate for one side of this potential equivalence.
Huijun Fan (Peking University)
Title:
Analytic theory of gauged linear sigma model
Abstract:
GLSM was proposed by Witten to understand the mirror symmetry, including the LG/CY correspondence conjecture. Recently, there are much work on constructing the invariants theory both from algebraic geometry and from analysis. In algebraic geometry, there are the efforts by Fan-Jarvis-Ruan, Chang-Li-Li-Liu and other groups. The quantum invariants in "narrow"case have been built up (under some stability conditions). But the construction of all quantum invariants including "narrow" and "broad" cases has not been completed, and it is a challanging work. This is even a challenging problem for algebraic geometers to construct the FJRW invariants in "broad case". In this talk, I will report the recent progress of the work by Tyler, Ruan and me (since 2012) to construct the invariants by studying the module problem of the gauged Witten equation. In detail, I will report the context of our first paper in analysis:
1) Basics of Gauged Witten equations;
2) Regularity and compactness theorems;
3) Linear theory;
4) Boundary theory;
5) Asymptotic behavior;
6) Center manifold theorm;
7) Slice theorem;
8) Fredholm theory of the fiber moduli space.
Bohan Fang (Peking University)
Title:
Global mirror curve of a toric Calabi-Yau 3-fold
Abstract:
I will describe the construction of a family of mirror curves over the secondary toric variety of a toric Calabi-Yau 3-fold. The recent progress of all genus mirror symmetry (BKMP remodeling conjecture) allows us to interpret A-model open-closed Gromov-Witten invariants in terms of the topological recursion on the mirror curve. Thus this construction automatically has implications on the modularity of GW invariants and the crepant resolution conjecture.
Viktor Ginzburg (University of California, Santa Cruz)
Title:
Non-contractible periodic orbits in Hamiltonian dynamics on closed symplectic manifolds
Abstract:
In this talk, we focus on Hamiltonian diffeomorphisms of closed symplectic manifolds with non-contractible periodic orbits. We will discuss several conjectures and results concerning the existence of such orbits and also some relevant aspects of Floer theory. We will show that, in a variety of settings, the presence of one non-contractible periodic orbit implies the existence of infinitely many such orbits in a specific collection of free homotopy classes. The talk is based on a joint work with Basak Gurel.
Chung-I Ho (National Tsing Hua University)
Title:
Minimal genus problems in 4-manifolds.
Abstract:
In this talk, I will explain some recent progress in estimating the minimal genus of
embedded surfaces in smooth 4-manifolds. In particular, I will give lower bounds which
only depends on the homology type rather than the smooth structure.
Yohsuke Imagi (Kavli IPMU)
Title:
Simple singualrity of special Lagrangian submanifolds
Abstract:
Special Lagrangian submanifolds are volume-minimizing Lagrangian submanifolds of Calabi-Yau manifolds.
McLean established in 1990's a local theory on the moduli space of compact special Lagrangian submanifolds,
but its global structure has not been well-understood; in other words we have not a good understanding of singularity of special Lagrangian submanifolds.
Joyce developed in 2000's a fairly good theory on a simple class of singularity, and I have recently made further progress in that direction. I will talk about it.
Hiroshi Iritani (Kyoto University)
Title:
Global mirror symmetry for toric stacks and its application
Abstract:
I will describe a partially compactified
Landau-Ginzburg model which is mirror to the big
quantum cohomology of a toric Deligne-Mumford
stack. I will also describe its application to
the functoriality of quantum cohomology.
Suguru Ishikawa (Kyoto University, RIMS)
Title:
Spectral invariants of distance functions
Abstract:
Entov and Polterovich introduced the notion of a (super)heavy
subset of a symplectic manifold by using spectral invariants of
Floer homology.
Superheavy set cannot displaced by symplectic isotopy,
and heavy set cannot be displaced by Hamiltonian isotopy.
Important fact is that (super)heavyness is preserved by product,
which enabled them a lot of examples of non-displaceable subsets.
We found new kind of superheavy set by direct calculations of
spectral invariants of distance-like Hamiltonians.
We showed if convex open subsets in Euclidian space are disjointly
embedded in a spherically negative monotone closed symplectic manifold,
their compliment is superheavy.
The key of the proof is estimates of the Conley-Zehnder index of
periodic orbits of distance-like Hamiltonian.
Byeongho Lee (National Institute for Mathematical Sciences)
Title:
Orbifolding Frobenius manifolds
Abstract:
Given a Frobenius manifold and a finite group acting on it, it has been speculated that there may be a general procedure to produce a new FM, called orbifolding. Examples of orbifolding at the level of Frobenius algebras appeared in the works of Fantechi-Goettsche, and Kaufmann. In fact, Kaufmann's description of orbifolding A series singularities to get D series essentially solves the problem at the level of FM, since their Frobenius structures can be obtained from their undeformed superpotentials using standard procedures. But we will take a different view point and look directly at their FM structures, so that we can generalize to more general types of FM's that do not have simple descriptions such as superpotentials.
Ziming Nikolas Ma (National Taiwan University)
Title:
Scattering in the SYZ programme
Abstract:
We consider semi-flat torus bundles X_0 and X_0^ over a common base B_0,
which is semi-flat limit of a mirror pair of Calabi-Yau manifold X and X^.
We prove that the process of solving Maurer-Cartan equation in L_{X_0}*,
the fiberwise Fourier transform of the Kodaira-Spencer dgLa on X_0^, has semi-classical limit as the
scattering process introduced by Kontsevich-Soibelman,
which is known to govern the deformation from X_0^ to X^.
This is a part of joint work in progress with K.-W. Chan and Conan Leung to
carry out Fukaya's program on understanding quantum corrections.
Yong-Geun Oh (IBS Center for Geometry and Physics)
Title:
Topological extension of Calabi invariants and its application
Abstract:
In this talk, I will explain how we can extend the classical
Calabi homomorphism on the area-preserving diffeomorphism group of
two-disc to the group of Hamiltonian homeomorphisms. I will also mention
implication of this extension result on the structure of area-preserving
homeomorphism group of the two-disc.
Miles Reid (University of Warwick)
Title:
Ice cream and orbifold Riemann-Roch
Abstract:
Ice cream expresses periodicity: if you have income of 3/7 per day, you
get an ice cream on Wed, Fri and Sun of each week. The same can be
expressed as Dedekind sums, that are defined as trigonometric sums, but
better thought of as the Fourier coefficients of a periodic function.
An orbifold is an algebraic variety X locally modelled on quotient
singularities such as CC^n/G where G is a finite subgroup of GL(n,CC).
For my purposes, G is a diagonal cyclic group ZZ/r acting on CC^n with
only isolated fixed points. Orbifold RR asks for the Euler
characteristic chi(X,F) of a sheaf F modelled on the character sheaves
of G. For a global quotient X = Y/G, orbifold RR is the same thing as
the holomorphic Lefschetz fixed point formula of Atiyah, Singer and
Segal, expressing the character of G acting on the cohomology of an
equivariant sheaf as a local property at the fixed points, that in turn
works out as Dedekind sums.
We use these ideas to give a very simple and practical formula for the
Hilbert series of projectively Gorenstein varieties, which includes, for
example, the log del Pezzo surfaces, QQ-Fano 3-folds or Calabi--Yau
orbifolds that are popular in string theory.
Fumihiko Sanda (Kyoto University)
Title:
Computations of quantum cohomology from Fukaya categories
Abstract:
In this talk, we explain how to compute quantum cohomology from Fukaya categories. As an application, we compute quantum cohomology of a blow-up of the P^2×P^1 along a elliptic curve.
Chris Wendl (University College London)
Title:
Tight contact structures on connected sums need not be contact connected sums
Abstract:
In dimension three, convex surface theory implies that every tight contact structure on a connected sum M # N can be constructed as a connected sum of tight contact structures on M and N. I will explain some examples showing that this is not true in any dimension greater than three. The proof is based on a recent higher-dimensional version of a classic result of Eliashberg about the symplectic fillings of contact manifolds obtained by subcritical surgery. This is joint work with Paolo Ghiggini and Klaus Niederkrüger.
Toru Yoshiyasu (Kyoto University)
Title:
On Lagrangian submanifolds in the complex projective spaces
Abstract:
In 2013, Y. Eliashberg and E. Murphy established the h-principle for exact Lagrangian embeddings with a concave Legendrian boundary. In this talk, I will explain that their h-principle holds for exact Lagrangian embeddings with a concave Legendrian boundary into 6-dimensional compact symplectic manifolds. I will also show its application to the existence of Lagrangian submanifolds in the complex projective spaces.
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