Venue

Lecture Hall (Building 1-Section B-3rd Floor)

USTC Shanghai Institute for Advanced Studies

Schedule(August 3-7)

TBA

Titles & Abstracts

Hanwool Bae (Junnam National University)

Title: Rabinowitz Fukaya category as a Tate $A_\infty$ category and Calabi--Yau structures

Abstract: The notion of Tate vector space enables one to formulate self-duality for infinite-dimensional vector spaces. It was shown by Cieliebak-Hingston-Oancea that the Rabinowitz Floer homology of a Liouville manifold admits a 2-dimensional TQFT structure and, as a consequence, satisfies a Poincar\'{e} duality in the sense of Tate vector spaces.

In this talk, I will introduce the notion of an $A_\infty$-category enriched in Tate vector spaces. or more generally in ultrasolid modules, which I will call a Tate $A_\infty$-category. I will then explain that the Rabinowitz Fukaya category of a Liouville manifold carries the structure of Tate $A_\infty$-category, and admits a categorical Poincar\'{e} duality, namely, a strong proper Calabi--Yau structure. This is based on joint work with Yuan Gao.

Huai-Liang Chang (Wuhan University)

Title: Mixed-Spin-P field theory of \(X_{3,3}\subset\mathbb{P}^2\times\mathbb{P}^2\)

Abstract: This survey introduces Mixed-Spin-P (MSP) field theory, the most effective approach to organizing phase transitions and all genus mirror predictions among all Gromov-Witten-type theories on compact Calabi-Yau threefolds. MSP theory mixes Calabi-Yau and Landau-Ginzburg phases, and supplies comprehensive algebraic geometric explanations for BCOV theory, holomorphic anomaly equations, Yamaguchi-Yau functional equations, high genus mirror symmetry, and analyticity of all-genus curve-counting potentials Fg’s. The theory naturally provides proofs of various conjectures for diverse phases of CY, including FJRW invariants or, more generally, hybrid Landau-Ginzburg models. I will review the story of MSP theory and sketch its recent developments, including quintic-FJRW phases and also the Calabi-Yau hypersurface \(X_{3,3}\subset\mathbb{P}^2\times\mathbb{P}^2\).

Shuho Kanda (University of Tokyo) 

Title: Holomorphic polynomial crystallographic actions of nilpotent groups

Abstract: Hasegawa conjectured that every simply connected nilpotent Lie group endowed with a left-invariant complex structure is biholomorphic to \mathbb{C}^n, and this natural problem was recently settled by him. In this talk, we discuss a refinement of this result from the viewpoint of polynomial geometry: can one choose such a biholomorphism so that the induced group law on \mathbb{C}^n is polynomial?

In the case where the complex structure is nilpotent, we give an affirmative answer by constructing such a biholomorphism explicitly in exponential coordinates. As a consequence, every lattice in such a Lie group admits a free, properly discontinuous, and cocompact action on \mathbb{C}^n by holomorphic polynomial automorphisms. We interpret this as a holomorphic analogue of polynomial crystallographic actions, that is, free, properly discontinuous, and cocompact actions on \mathbb{R}^n by polynomial diffeomorphisms, as introduced by Dekimpe, Igodt, and Lee.

Tatsuki Kuwagaki (Kyoto University)

Title: Sheaves and Fukaya category over the Novikov ring

Abstract: Over the past decade, the relationship between the Fukaya categories of exact Lagrangian submanifolds and microlocal sheaf theory has been understood, and has been used as an important technique for studying symplectic topology and mirror symmetry. In this talk, I would like to review recent progress in our understanding of how such correspondences can be generalized to non-exact Lagrangian submanifolds, as well as their applications to homological mirror symmetry. 

Jungsoo Kang (Seoul National University)
Title: Floer homology for prequantization bundles

Abstract:A prequantization bundle is a circle bundle that admits a contact/symplectic structure. I will first present a quantum version of the Gysin sequence for prequantization bundles, which relates the Rabinowitz Floer homology of the total space to the quantum homology of the base. We then consider a Lagrangian submanifold in the base and its Legendrian lift in the total space, whose singular homologies are related via the transfer map. I will present a quantum version of this relation. This is based on joint work with Hanwool Bae, Joonghyun Bae, and Sungho Kim.

Naichung Conan Leung (Chinese University of Hong Kong)

Title: 3d mirror symmetry and mirror symmetry

Abstract: We will explain relationship between 3d mirror symmetry for Rozansky-Witten theory and 2d mirror symmetry for string theory.

Mingyang Li (Stony Brook University)

Title: Gravitational instantons and harmonic maps

Abstract: It is known from general relativity that axisymmetric stationary black holes can be reduced to axisymmetric harmonic maps into the hyperbolic plane H^2, while in the Riemannian setting, 4d Ricci-flat metrics with torus symmetry can also be locally reduced to such harmonic maps satisfying a tameness condition. We study such harmonic maps and application includes a construction of infinitely many new complete, asymptotically flat, Ricci-flat 4-manifolds with arbitrarily large second Betti number b_2. Joint work with Song Sun.

Natsuo Miyatake (Tohoku University)

Title: Equilibrium Metrics and Complete Harmonic Metrics

Abstract: An equilibrium metric on a positive line bundle $L$ over a compact Riemann surface $X$ is a special singular Hermitian metric whose curvature current induces an equilibrium measure. It is defined from data consisting of a continuous Hermitian metric $e^{-\phi}h_\ast$ on $L$ and a nonpolar compact subset $K\subseteq X$, using the envelope of subharmonic weight functions bounded above by $\phi$ on $K$. The approximation of equilibrium metrics by sequences of singular metrics has been studied in a variety of contexts, including complex geometry, potential theory, dynamical systems, and probability theory.

In this talk, I will introduce two functions, called entropy and free energy, associated with a semipositive singular Hermitian metric on the canonical bundle of a Riemann surface. These quantities are defined using an extension of the notion of complete harmonic metrics on cyclic Higgs bundles, whose existence and uniqueness were established by Li--Mochizuki, to complete Hermitian metrics associated with a semipositive singular metric on the canonical bundle that is not necessarily induced by a holomorphic $r$-differential. I will give several estimates for these functions and discuss my ongoing research aimed at quantitatively establishing the principles of entropy increase and free energy decrease in various situations where an equilibrium metric is approximated by a sequence of singular metrics.

Ngoc Cuong Nguyen (KAIST)

Title: Fine properties of functions in complex Sobolev Spaces

Abstract: We study comprehensively local properties of functions in complex Sobolev spaces on a bounded open subset of C^n. The main tool is the corresponding functional capacity for the space which is inspired by the global one due to Vigny [2007]. An inequality between this capacity and the Bedford-Taylor capacity for plurisubharmonic functions is proved, which is sharp as far as the exponents are concerned. Moreover, it is shown that the functional capacity is a Choquet capacity. 

*Valentino Tosatti (New York University)

Title: Holomorphic tubular neighborhoods

Abstract: A classical result in differential topology shows that every submanifold of an ambient manifold admits a tubularneighborhood diffeomorphic to a neighborhood of the zero section of its normal bundle. The analogous result in the category of complex manifolds does not hold in general, already for plane algebraic curves of degree at least 2, and deciding in which cases it holds can be a very difficult question. In 1975, Ogus studied the case of certain elliptic curves embedded in the blowup of the plane at 9 points, which have topologically trivial (but not holomorphically torsion) normal bundle. Arnol'd showed in 1976 that almost all of these (in the measure-theoretic sense) admit a holomorphic tubular neighborhood, but not a single one was known which does not admit such a neighborhood.  I will discuss recent work with Simion Filip where we construct many such examples, answering Ogus's question.

Weiwei Wu (Zhejiang University)

Title: Ball Swapping and Configuration Spaces

Abstract:Ball swapping gives a geometric way to realize monodromy coming from motions in algebraic configuration spaces. I will explain this relation in the setting of Milnor fibers, briefly reviewing the known uniqueness results for the A-type Milnor fiber and then discussing the D-type case. The goal is to understand how configuration-space topology produces symplectic mapping classes via ball-swapping constructions. This is ongoing joint work with Jialiang Chen.

Qizheng Yin (Peking University)

Title: Lagrangian fibrations and hyper-Kähler varieties: an analogy

Abstract:We discuss several open questions concerning the topology and algebraic geometry of (holomorphic) Lagrangian fibrations and hyper-Kähler varieties, and draw an analogy between the relative and absolute settings. We also explain how this analogy serves as a guiding principle for solving some of these questions. Based on series of joint work with Younghan Bae, Davesh Maulik, Junliang Shen, and Ruxuan Zhang.

Zhengyi Zhou (MCM, CAS)

Title: On symplectic CPn

Abstract: We show that the existence of a pseudo-holomorphic line passing through two generic points on a symplectic manifold X, phrased using Gromov-Witten invariants, implies that X is homotopy equivalent to CPn with identical first Chern class and small quantum cohomology. We then deduce from this result and its generalizations some rigidity results regarding symplectic hyperplanes in CPn+1 as well as symplectic fillings. The proof is based on Rabinowitz Floer homology.