Links of Gorenstein toric isolated singularities are good toric contact manifolds with zero first Chern class. In this talk I will present some results relating contact and singularity invariants in this particular toric context. Namely,
(i) I will explain why the contact mean Euler characteristic is equal to the Euler characteristic of any crepant toric smooth resolution of the singularity (joint work with Leonardo Macarini).
(ii) I will discuss applications of contact invariants of Lens spaces that arise as links of Gorenstein cyclic quotient singularities (joint work with Leonardo Macarini and Miguel Moreira).
Louis Bonthrone: J-holomorphic curves from J-anti-invariant forms in dimension 4
Since the 1980's there has been a well known folklore theorem which says that for a generic Riemannian metric on a 4-manifold the zero set of a self-dual harmonic 2-form is a finite number of embedded circles. We prove that in the almost complex setting the corresponding result holds without a genericity assumption. That is, we show the zero locus of a closed J-anti-invariant 2-form is a J-holomorphic curve in the canonical class. Also it is shown that the closed J-anti-invariant 2-form is determined, up to scaling, by the zero divisor. Some applications to the geometry of 4-manifolds will be discussed. This is based on joint work with Weiyi Zhang.
Yalong Cao: Gopakumar-Vafa type invariants for Calabi-Yau 4-folds
As an analogy of Gopakumar-Vafa conjecture for CY 3-folds, Klemm-Pandharipande proposed GV type invariants on CY 4-folds using GW theory and conjectured their integrality. In this talk, we propose a sheaf theoretical interpretation to these invariants using Donaldson-Thomas theory on CY 4-folds. This is a joint work with Davesh Maulik and Yukinobu Toda.
Jonny Evans: Bounding symplectic embeddings of rational homology balls in surfaces of general type
This is joint work with Ivan Smith. Developing ideas of T. Khodorovskiy and J. Rana, we prove that if the Milnor fibre of a Wahl singularity embeds symplectically in a (canonically polarised) surface of general type with $b^+>1$ then the length of the singularity is bounded above by $4K^2+7$, where the length is the number of components of the exceptional divisor of the minimal resolution and $K^2$ is the square of the canonical class. This implies the corresponding length bound for singularities of stable surfaces of general type, which is an improvement on the current bound known to algebraic geometers ($400(K^2)^4$, due to Y. Lee). Our proof uses Seiberg-Witten theory and holomorphic curves.
Jianxun Hu: Gromov-Witten invariant and symplectic rationally connectedness
In this talk, I will first talk about how to use the Gromov-Witten invariant to study the birational properties of symplectic manifolds. Then I will talk about the symplectic rationally connectedness of del Pezzo manifolds of degree $\geq 5$.
A central problem in modern Riemannian geometry is to determine which compact smooth manifolds admit Einstein metrics, and to then completely understand the moduli space of such metrics when they exist. For arbitrary 4-manifolds, the problem remains mysterious and daunting, but when the 4-manifold also admits a symplectic structure, our current understanding becomes much more satisfactory. I will describe what we currently do and do not know about the problem in this context, and how our current paradigms are guided by ideas from algebraic geometry. I will also compare and contrast these results with the wildly different situation that occurs in the higher dimensions.
Yankı Lekili: Mirror symmetry for punctured surfaces and Auslander orders
We consider partially wrapped Fukaya categories of punctured surfaces with stops at their boundary. We prove equivalences between such categories and derived categories of modules over the Auslander order on certain nodal stacky curves. As an application, we deduce equivalences between derived categories of coherent sheaves (resp. perfect complexes) on such nodal stacky curves and the wrapped (resp. compact) Fukaya categories of punctured surfaces of arbitrary genus and arbitrary non-zero number of boundary components. This is joint work with Polishchuk.
Tian-Jun Li: Geography of symplectic fillings in dimension 4
We show that, given any contact 3-manifold, there is a lower bound of 2 Euler + 3 signature for all its minimal symplectic fillings. This result generalizes the similar bound of Stipsicz for Stein fillings. It is proved by constructing Donaldson caps and apply the notion of maximal surfaces developed by Weiyi Zhang and myself. We also introduce the Kodaira dimension of contact 3-manifolds and show that contact 3-manifolds with distinct Kodaria dimensions behave differently when it comes to the geography of various kinds of fillings. This talk is based on joint works with Cheuk Yu Mak, and partly with Koichi Yasui.
Cheuk Yu Mak: Spherical Lagrangian submanifolds and spherical functors
In this talk, we discuss Dehn twists along Lagrangian submanifolds which are quotients of spheres. We explain the auto-equivalences of the Fukaya category induced by these Dehn twists in terms of twist along spherical functors in the sense of Anno and Logvinenko. This is a joint work with Weiwei Wu.
Mark McLean: The Cohomological McKay Correspondence and Symplectic Cohomology
Suppose that we have a finite quotient singularity $C^n/G$ admitting a crepant resolution $Y$ (i.e. a resolution with $c_1 = 0$). The cohomological McKay correspondence says that the cohomology of Y has a basis given by irreducible representations of $G$ (or conjugacy classes of $G$). Such a result was proven by Batyrev when the coefficient field F of the cohomology group is Q. We give an alternative proof of the cohomological McKay correspondence in some cases by computing symplectic cohomology of $Y$ in two different ways. This proof also extends the result to all fields F whose characteristic does not divide $|G|$ and it gives us the corresponding basis of conjugacy classes in $H^*(Y)$. We conjecture that there is an extension to certain non-crepant resolutions. This is joint work with Alex Ritter.
Milena Pabiniak: Toric degenerations in symplectic geometry
A toric degeneration is a construction from algebraic geometry which allows us to "degenerate" a given projective manifold $M$ to some (symplectic) toric variety $X_0$, i.e. form a flat family over $C$ with a generic fiber $M$ and a special fiber $X_0$. As observed by Harada and Kaveh, if M is symplectic then there exists an open dense subset of $M$ and an open dense subset of $X_0$ which are symplectomorphic. This allows us to study symplectic invariants of $M$ by studying $X_0$ which, as a toric variety, is usually much better understood. In certain nice situations the whole $M$ is symplectomorphic to $X_0$ and the degeneration provides a symplectomorphism.
In this talk I will discuss two applications of toric degenerations in symplectic geometry.
1. To find lower bounds for the Gromov width of $M$, concentrating mostly on the case when $M$ is a coadjoint orbit (based on projects with I. Halacheva, and with X. Fang and P. Littelmann).
2. To study questions of the form: given two (symplectic) Bott manifolds and an isomorphism $F$ between their integral cohomology rings, sending $[\omega_1]$ to $[\omega_2]$, is there a diffeomorphism inducing $F$? (Based on a project with S. Tolman.)
Dmitri Panov: Symplectic Fano 6-manifolds with Hamiltonian S^1-action are simply connected
A compact symplectic manifold is called a symplectic Fano manifold if its first Chern class is equal to the cohomology class of its symplectic form. Thanks to work of Gromov-Taubes-McDuff it is known that there exist only 10 such manifolds in dimension 4 and they are all simply connected. On the other hand, starting from dimension 12 such manifolds can have infinite fundamental group. The talk is bases on a joint work with Nick Lindsay. We study the case of dimension 6 and prove that symplectic Fano 6-manifolds with Hamiltonian $S^1$-action are simply connected.
Daniel Pomerleano: Symplectic cohomology and mirror symmetry
I will explain how, under suitable hypotheses, one can construct a flat degeneration from the symplectic cohomology of log Calabi-Yau varieties to the Stanley-Reisner ring on the dual intersection complex of a compactifying divisor. I will explain how this result connects to classical mirror constructions of Batyrev and Hori-Iqbal-Vafa as well as ongoing work of Gross and Siebert.
Ivan Smith: A tale of two plumbings
We discuss the symplectic topology of the Stein manifolds obtained by plumbing two 3-spheres along a circle, using these examples to illustrate various open questions.
Richard Thomas: Vafa-Witten invariants for projective surfaces
I'll describe joint work with Yuuji Tanaka defining Vafa-Witten invariants of projective surfaces. Depending on how the time goes, I'll discuss extensions to the semistable case, relations to degeneracy loci and Carlsson-Okounkov operators, and how to refine the theory.
Zhiyu Tian: Crepant resolution conjecture, Chow motive, and hyperkahler manifolds
The crepant resolution conjecture relates the GW theory of a stack with that of a crepant resolution of its coarse moduli. I will make some remarks about lifting this to Chow motives in the global quotient stack case and discuss some applications to the study of the Chow ring of some hyperkahler manifolds. This is based on joint work with Lie Fu and with Lie Fu-Charles Vial.
Zuoqin Wang: Inverse spectral results on toric manifolds
Let $M$ be a compact manifold and $G$ a compact Lie group acting on $M$. Let $P$ be a $G$-equivariant pseudo-differential operator on $M$ which is elliptic and self-adjoint. In this talk I will explain how the symmetry induces extra structure in the spectrum of $P$, and how symplectic reduction can be used in studying this equivariant spectrum. Then I will discuss some applications of this idea in the study of the inverse spectral problem for Schrodinger operator when $M$ is a toric manifold. I will also discuss related inverse spectral results for Toeplitz operators. This is based on joint works with Victor Guillemin and Alejandro Uribe.