Expository Lecture (July 7-11)

Venue

Room 5303, the Fifth Teaching Building

Schedule

Titles & Abstracts

Dylan Cant (Université Paris-Saclay)

Title: Symplectic topology and Finsler Geometry

Abstract: To each compact smooth manifold $M$ one can associate to its cotangent bundle $T^{*}M$ a canonical symplectic structure (this is the crucial insight of Hamilton's formulation of classical mechanics). On the other hand, a Finsler metric on $M$ determines a unit codisk bundle $\Omega\subset T^{*}M$ which is a fiberwise convex set containing the zero section. In my lecture series, I will explain the surprising connection between the Finsler geometry and the symplectic invariants of $\Omega$. The main new symplectic invariant we will study is the socalled *parametric Gromov width,* which measures the maximal symplectic size that families of balls can attain inside of $\Omega$. For instance, with $M=T^{2}$, the lengths of meridians measured with respect to the Finsler metric bounds the maximal size of a loop of balls which travels along the longitudes. Such a result generalizes the famous *symplectic camel theorem* due to Y. Eliashberg and M. Gromov whose name comes from the biblical metaphor concerning a "camel passing through the eye of a needle". In the course of the lectures, we will cover a wide range of modern techniques in symplectic geometry, centering on the non-linear elliptic equations used by A. Floer to define symplectic invariants.

Charles Cifarelli (Stony Brook University)

Title: Explicit techniques in Kähler geometry

Abstract: We will survey techniques for the construction of Kähler metrics which can be represented by explicit formulas. The beginning point of the subject is the famous Calabi Ansatz, which reduces many important geometric equations to a simple ODE. We will introduce the theory of hamiltonian 2-forms, a generalization of the Calabi Ansatz which is designed in part to maintain this relationship with simple ODE solutions. We will discuss the Calabi Ansatz, its relevant generalization, and manifolds with special fibration structures. We will discuss as many applications to specific geometric problems as time permits, including extremal Kähler metrics, Ricci-flat metrics, and Kähler-Ricci solitons.

ManChun Lee (The Chinese University of Hong Kong)

Title: Overview of works on complete Kähler manifolds with positive curvature

Abstract: This mini-course will give an overview of results on the Kähler manifolds and Kähler-Ricci flow with non-negative curvature, in the complete noncompact settings, with applications to uniformization problems in complex geometry.

Conference (July 14-18)

Venue

Conference Room on 3rd Floor of Material Research Building (Section B&C), East Campus of USTC

Schedule

Titles & Abstracts

Zhongshan An (University of Michigan)

Title: Einstein manifolds with boundary

Abstract: We will talk about existence of Einstein metrics on manifolds with boundary, while prescribing the induced conformal metric and mean curvature of the boundary. In dimension 3, this becomes the existence of conformal embeddings of surfaces into constant sectional curvature space forms, with prescribed mean curvature. We will show existence of such conformal embeddings near generic Einstein background. We will also discuss the existence question in higher dimensions, where things become more subtle and a stability boundary condition is used to construct metrics with nonpositive Einstein constant.

*Simon Brendle (Columbia University in the City of New York)

Title: Systolic inequalities and the Horowitz-Myers conjecture

Abstract: Let $n$ be an integer with $3 \leq n \leq 7$, and let $g$ be a Riemannian metric on $B^2 \times T^{n-2}$ with scalar curvature at least $-n(n-1)$. We establish an inequality relating the systole of the boundary to the infimum of the mean curvature on the boundary. As a consequence, we obtain a new positive energy theorem where equality holds for the Horowitz-Myers metrics. This is joint work with Pei-Ken Hung.

Yifan Chen (UC Berkeley)

Title: when singular Kähler-Einstein metrics are Kähler currents

Abstract:  We show that a general class of singular Kähler metrics with Ricci curvature bounded below define Kähler currents. In particular the result applies to singular Kähler-Einstein metrics on klt pairs. This is a joint work with Shih-Kai Chiu, Max Hallgren, Gabor Szekelyhidi, Tat Dat To, and Freid Tong.

Aleksander Doan (University College London)

Title: Pseudo-holomorphic curves in positive symplectic manifolds

Abstract: This talk is about joint work with Alessio Cela on the fixed domain Gromov-Witten invariants of positive symplectic manifolds. A conjecture of Lian-Pandharipande asserts that in high degrees these invariants agree with geometric counts of curves for Fano manifolds. While the original conjecture was recently disproved by Beheshti et al. we prove that its symplectic analogue holds, when a complex structure is replaced by a generic almost complex structure. 

Lorenzo Foscolo (Sapienza Università di Roma)

Title: New minimal surfaces in hyperkähler 4-manifolds

Abstract: I will describe the construction of new minimal surfaces in hyperkähler 4-manifolds arising from the Gibbons–Hawking Ansatz, i.e. hyperkähler 4-manifolds that admit a triholomorphic circle action, and on certain K3 surfaces.

The minimal surfaces we produce are obtained via a gluing construction using well-known surfaces, the Scherk surface in flat space and the holomorphic cigar in the Taub-NUT space, as building blocks. The minimal surfaces obtained via this construction are not holomorphic with respect to any complex structure compatible with the metric, are not circle invariant, they can be parameterized by a harmonic map that satisfies a first-order Fueter-type PDE, and yet are unstable. This is joint work with Federico Trinca.

Mario Garcia-Fernandez (Instituto de Ciencias Matemáticas (ICMAT))

Title: Non-Kähler Hodge-Lefschetz theory and the Bianchi identity

Abstract: I will introduce a notion of variation of Hodge-Lefschetz structure for compact non-Kähler manifolds, which provides a generalization of the well-studied variations of polarised Hodge structure for projective varieties. A key ingredient for our construction is a "Bianchi identity" for hermitian metrics on the manifold, motivated by similar equations appearing in the string theory literature. Joint work with Raul Gonzalez Molina and Arpan Saha.

Siqi He (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)

Title:  Z/2 harmonic forms, harmonic maps into R-trees, and compactifications of character variety

Abstract: In this talk, we will explore the connection between the analytic compactification of the moduli space of flat SL(2,C) connections on closed, oriented 3-manifolds defined by Taubes, and the Morgan-Shalen compactification of the SL2(C) character variety. We will discuss how these two compactifications are related through harmonic maps to R-trees. Additionally, we will discuss several applications of this construction in the analytic aspects of gauge theory. This is joint work with R. Wentworth and B. Zhang.

Shouhei Honda (The University of Tokyo)

Title: From almost smooth spaces to RCD spaces

Abstract: We provide various characterizations for a given almost smooth space to be an RCD space, in terms of a local volume doubling and a local Poincaré inequality. Applications include a characterization of Einstein 4-orbifolds. This talks is based on a joint work with Song Sun (Zhejiang University).

Qiongling Li (Nankai University)

Title: Harmonic metrics of Higgs bundles over non-compact surfaces

Abstract: In this talk, I will discuss existence and uniqueness results for harmonic metrics solving the Hitchin equation on certain Higgs bundles over non-compact surfaces. The cases to be discussed include generically regular semisimple or nilpotent Higgs bundles and Hitchin section. This work is based on joint works with Takuro Mochizuki (Kyoto University) and joint work with Song Dai (Tianjin University).

Yang Li (University of Cambridge)

Title: Large mass of limit  G₂ & Calabi-Yau monopoles

Abstract: I will discuss some recent progress on the Donaldson Segal programme, and in particular how calibrated cycles (coassociative submanifolds, special Lagrangians) arise from the large mass limit of G₂ and Calabi Yau monopoles.

Yi Liu (Peking University)

Title: Hempel pairs and Turaev Viro invariants

Abstract: Hempel pairs are periodic surface bundles with profinitely isomorphic fundamental groups. In this talk, I will discuss whether Turaev--Viro invariants distinguish such pairs. I will explain some motivation of this work and discuss further questions.

Jiayin Pan (University of California, Santa Cruz)

Title: Ricci curvature and linear volume growth

Abstract: We will talk about old and new results on the topology of complete manifolds with nonnegative Ricci curvature and linear volume growth. The new results are based on a joint work with Dimitri Navarro and Xingyu Zhu.

Vivek Shende (UC Berkeley)

Title: Towards skein-valued counting of adiabatic associatives

Abstract: Donaldson has proposed that G2 manifolds can be profitably considered through adiabatic limits of their K3 fibrations over 3-manifolds; with Scaduto, he proposed that the associatives should converge to certain Morse flow graphs on the 3-manifold, and observed that the counting problem of said flow trees already exhibits the wall crossing behaviors identified by Joyce which obstruct a naive approach to invariantly counting associative submanifolds.

In this talk I will discuss the corresponding limiting structures for a (noncompact) G2 fibered in An surfaces.  The main observation is that the resulting graphs are in bijection with the flow graphs for a Lagrangian in the cotangent bundle to the base of the fibration, and correspondingly can be counted invariantly in the skein of said Lagrangian.  At least morally, this is an adiabatic limit of a Kaluza-Klein reduction.  This talk presents joint work in progress with Tobias Ekholm, Saman Eshafani, and Luya Wang. 

Zhe Sun (University of Science and Technology of China)

Title:  Exponential volumes of moduli spaces of hyperbolic surfaces

Abstract: Mirzakhani found a remarkable recursive formula for the volumes of the moduli spaces of the hyperbolic surfaces with geodesic boundary, and the recursive formula plays very important role in several areas of mathematics: topological recursion, random hyperbolic surfaces etc.

We consider some more general moduli spaces M_S(K,L) where the hyperbolic surfaces would have crown ends and horocycle decorations at each ideal points. But the volume of the space M_S(K,L) is infinite when S has the crown ends. To fix this problem, we introduce the exponential volume form given by the volume form multiplied by the exponent of a canonical function on M_S(K,L).

We show that the exponential volume is finite. And we prove the recursion formulas for the exponential volumes, generalising Mirzakhani's recursions for the volumes of moduli spaces of hyperbolic surfaces. We expect the exponential volumes are relevant to the open string theory. This is a joint work with Alexander Goncharov.

Wilderich Tuschmann (Karlsruher Institut für Technologie)

Title: Moduli Spaces of Riemannian Metric

Abstract: Focussing on the case of lower curvature bounds, I will present recent results and open questions about the global topological properties of moduli spaces of Riemannian metrics on manifolds, and also discuss metric approaches to obtain and study suitable compactifications of the former. 

Fang Wang (Shanghai Jiao Tong University)

Title: Rigidity Theorem for Poincare-Einstein Manifolds

Abstract: The rigidity problem for Poincare-Einstein manifold asks: when the conformal infinity of a Poincare-Einstein manifold (X, g) is the standard round sphere or Euclidean space, is (X,g) the standard hyperbolic space? In this talk, I will first introduce the classical rigidity theorem, under the condition that (X, g) is C^3 conformally compact. Then I will report some recent rigidity result for Poincare-Einstein manifold in the upper half-plane model, which takes the Euclidean space as the conformal infinity and whose adapted conformal metric has quadratic curvature decay at infinity. This is joint work with Sanghoon Lee (KIAS).

Xiaodong Wang (Michigan State University)

Title: Liouville Theorems on pseudohermitian manifolds with nonnegative Tanaka-Webster curvature

Abstract: I will discuss a geometric nonlinear PDE on Sasakian manifolds and present generalizations of the famous Jerison-Lee classification theorem to curved setting, in both compact and noncompact cases. This is based on joint work with Giovanni Catino, Dario Monticelli and Alberto Roncoroni.

Yunhui Wu (Tsinghua University)

Title: Spectral gaps on thick part of moduli spaces

Abstract: We study spectral gaps of closed hyperbolic surfaces for large genus. And we show that for any fixed $k\geq 1$, as the genus goes to infinity, the maximum of the $k$-th spectral gap over any thick part of the moduli space of closed Riemann surfaces approaches to 1/4. This is joint work with Haohao Zhang (Tsinghua University).

Mingchen Xia (University of Science and Technology of China)

Title: Partial Okounkov bodies

Abstract: Given a big line bundle L on a projective manifold, Lazarsfeld–Mustată and Kaveh–Khovanskii introduced a method of constructing convex bodies associated with L. These convex bodies are known as Okounkov bodies. When L is endowed with a singular positive Hermitian metric h, I will explain how to construct smaller convex bodies from the data (L,h). These convex bodies play important roles in the study of the singularities of h. I will also explain some applications in toric geometry.

Yulun Xu (University of Toronto)

Title: The uniqueness of Poincar\'e type extremal K\"ahler metric

Abstract: Let D be a smooth divisor on a closed K\"ahler manifold X. Suppose that Aut_0(D)={Id}. We prove that the Poincar\'e type extremal K\"ahler metric with a cusp singularity at D is unique up to a holomorphic transformation on X that preserves D. This generalizes Berman-Berndtsson's work on the uniqueness of extremal K\"ahler metrics from closed manifolds to some complete and noncompact manifolds.

Chengjian Yao (ShanghaiTech University)

Title: Some existence and uniqueness results for the Kahler-Yang-Mills equations

Abstract: The Kahler-Yang-Mills equations were introduced as a tool to study pair of complex structures (on the base manifold and a complex vector bundle) based on moment map considerations. Following the line of developments in Kahler geometry, we studied the existence and uniqueness problem of solutions to the coupled system. In particular, we have established relatively general existence result in the dimensional reduction case of gravitating vortices, and uniqueness of solutions in any dimension. The talk is based on joint works with Alvarez-Consul, Garcia-Fernandez, Garcia-Prada and Pingali in recent years.