召集人：张俊（中国科学技术大学，研究员）、段华贵（南开大学，教授）、Kaoru ONO（Research Institute for Mathematical Sciences (RIMS), Kyoto University，教授）

时间：2025.04.20—2025.04.26

 会议日程（Schedule）

报告摘要（Abstracts)

Topological and dynamical invariants in symplectic diffeomorphisms

Zhihong XIA（夏志宏）  大湾区大学

Abstract: We show that two dynamical invariants: flux and mean rotation vectors for symplectic diffeomorphisms, are in fact the same thing, under Poincare duality. Moreover, we show that a symplectic diffeomorphism that is isotopic to identity is Hamiltonian if and only if its mean rotation vector is zero.

粘性解奇性分析：历史与新观察

Wei CHENG（程伟）  南京大学

Abstract: 本报告将回顾近期我们与合作者对任意一对半凹函数和类Tonelli的Hamilton函数建立的Hamilton-Jacobi方程粘性解极大斜率曲线理论，以及相关粘性解奇性传播问题。根据以往关于Lax-Oleinik半群交换子与cut time function的分析，我们进一步给出一般半凹函数割迹的概念，并证明一般情形割迹为零测集。作为副产品，我们进一步给出关于凸函数几乎处处二次可微的Alexandrov定理一个有趣的初等证明。这是基于与Piermarco Cannarsa、洪家辉和魏文学近期的工作。报告中我们还将提及一些公开问题。 

An example of essential coexistence phenomenon in Hamiltonian dynamics

Jianyu CHEN（陈剑宇）  苏州大学

Abstract: The coexistence phenomenon of regular and hyperbolic behavior are numerically observed in a wide class of mechanical systems, however, a general theory is far out of reach. In this talk, we first review some results on the essential coexistence phenomena in volume-preserving category, whose constructions are based on the perturbation techniques in partially hyperbolic systems. Then we focus on a recent result by Huyi Hu, Yakov Pesin, Ke Zhang and myself, which provides a construction of essential coexistence in a 4-dimensional Hamiltonian system. The key ingredient is an explicit embedding the Katok map.

K-contact manifolds with minimal closed Reeb orbits

Hui LI（李慧）  苏州大学

Abstract: We use the Boothby-Wang fibration to construct certain simply connected K-contact manifolds and we give sufficient and necessary conditions on when such K-contact manifolds are homeomorphic to the odd dimensional spheres. If the symplectic base manifold of the fibration admits a Hamiltonian torus action, we show that on the total space of the fibration, other than the regular K-contact structures which have infinitely many closed Reeb orbits, there are K-contact structures whose closed Reeb orbits correspond exactly to the fixed points of the Hamiltonian torus action on the base manifold. Then we give a collection of examples of compact simply connected K-contact manifolds with minimal number of closed Reeb orbits which are not homeomorphic to the odd dimensional spheres, while having the real cohomology ring of the spheres. 

The sum of Hamiltonian manifolds

Hailong HER（赫海龙）  暨南大学

Abstract: Given a compact connected Lie group G, and two compact symplectic manifolds with Hamiltonian G-actions, (X+, ω+, μ+) and (X−, ω−, μ−), we study the operation of equivariant sum of these two Hamiltonian G-manifolds along a common codimension 2 Hamiltonian submanifold Z with the opposite equivariant Euler classes of the normal bundles. We verify that the symplectic reduction of the Hamiltonian sum agrees with the symplectic sum of the reduced symplectic manifolds. This is an equivariant version generalization of the the operation of the symplectic connected sum without group action, and is the foundation for studying the degeneration formula of Hamiltonian Gromov-Witten invariants. This is a joint work with Bohui Chen and Bai-Ling Wang.

Birkhoff sections for integrable system

Wentian KUANG（况闻天）  大湾区大学

Abstract: In this report, we introduce a method for directly constructing Birkhoff sections on energy surfaces of integrable Hamiltonian systems based on their dynamical behavior. We establish the necessary and sufficient conditions for various types of periodic orbits to serve as boundary orbits of a Birkhoff section. This construction applies to the boundaries of toric domains and energy surfaces of separable Hamiltonian systems, providing criteria for the existence or non-existence of different types of Birkhoff sections.

Intersection numbers and symmetric polynomials

Xiaobo LIU（刘小博）  北京大学

Abstract: Generating functions of intersection numbers on moduli spaces of curves provide geometric solutions to integrable systems. Notable examples are the Kontsevich-Witten tau function and Brezin-Gross-Witten tau function. In this talk I will first describe how to use Schur's Q-polynomials to obtain simple formulas for these functions. I will then discuss possible extensions for more general geometric models using Hall-Littlewood polynomials. This talk is based on joint works with Chenglang Yang. 

Mean complexity and Sarnak conjecture

Wen HUANG（黄文）  中国科学技术大学

Abstract: In this talk, we will review the mean complexity and the progress related to the Sarnak conjecture.In particular, we will discuss the logarithmic Sarnak conjecture and its equivalent forms, as well as our characterization by the ploynomial mean complexity.

Finite energy foliations for mechanical systems with two degrees of freedom

Pedro Salomão  南方科技大学

Abstract: I will discuss the dynamics of a two-degree-of-freedom mechanical system using pseudo-holomorphic curves tecniques. We consider a potential function on the plane with finitely many saddle points with the same critical value. As the energy increases across the critical value, a disk-like component of the Hill region gets connected to other components precisely at the saddles. Under certain convexity assumptions on the critical set, we obtain a weakly convex foliation in the region of the energy surface where the interesting dynamics takes place.  The binding of the foliation consists of the Lyapunov orbits near the rest points and a particular index-3 orbit. The transverse foliation forces the existence of periodic orbits, homoclinics, and heteroclinics to the Lyapunov orbits. We apply the results to the Henon-Heiles system, and some decoupled mechanical systems, including the frozen Hill’s lunar problem with centrifugal force, the Euler problem of two centers, and the potential of a chemical reaction. This talk is based on a joint work with Naiara de Paulo, Seongchan Kim, and Alexsandro Schneider.

Legendrian knots and quantum groups

Honghao GAO（高鸿灏）  清华大学

Abstract: Legendrian knots are fundamental objects in low dimensional contact topology. A robust invariant to study Legendrian knots is Chekanov’s dg algebra, built from Floer theoretic counting of disks. Quantum groups are deformations of the universal enveloping algebras of Lie algebras, which was first raised by Drinfeld and Jimbo to solve the quantum Yang-Baxter equations. In this talk, I will explain some work in progress with Roger Casals, Lenhard Ng, Linhui Shen, Daping Weng and Eric Zaslow trying to connect the two invariants. 

Reverse Lagrangian surgery on fillings

Yu PAN（潘宇）  天津大学

Abstract: A major theme in symplectic and contact topology is the study of Legendrian knots and exact Lagrangian surfaces. In the talk, we will talk about some flexibility results of immersed Lagrangian surfaces using augmentation, a Floer type invariant of Legendrian knots. In particular, for an immersed filling of a topological knot, one can do surgery to resolve a double point with the price of increasing the surface genus by 1. In the Lagrangian analog, one can do Lagrangian surgery on immersed Lagrangian fillings to treat a double point by a genus. In this talk, we will explore the possibility of reversing the Lagrangian surgery, i.e., compressing a genus into a double point. It turns out that not all Lagrangian surgery is reversible.

Characterizing standard autoequivalences via dynamical invariants

Yu-Wei FAN（范祐维）  清华大学

Abstract: The derived category of coherent sheaves on a smooth complex projective variety is equipped with standard autoequivalences, which originate from holomorphic automorphisms of the variety. The existence of autoequivalences beyond these standard ones reflects geometric properties of the underlying variety. In particular, spherical twists and P-twists, mirror to Dehn twists along Lagrangian spheres and complex projective planes, respectively, are expected to be non-standard, as they arise from symplectic rather than holomorphic constructions. In this talk, we will provide a rigorous proof of this expectation via certain dynamical invariants associated with autoequivalences. No prior familiarity with the aforementioned concepts will be assumed.

Turbulence, Lyapunov exponents, and SRB measures in infinite-dimensional dynamical systems

Kening LV（吕克宁）  四川大学

Abstract: In this talk, I will present several results related to Lyapunov exponents, SRB measures, entropy, and horseshoes in the context of infinite-dimensional dynamical systems. I will also discuss recent work on the ergodicity and statistical dynamics of the 2D Navier-Stokes equation, driven by both time-dependent deterministic and stochastic forces. Additionally, I will explore the connection between SRB measures and turbulence.

Common index jump theorem with applications to the multiplicity of closed geodesics

Huagui DUAN（段华贵）  南开大学

Abstract: In this talk, firstly we will introduce the Maslov-type theory and the generalized common index jump theorem. Then we will talk about its applications to the multiplicity of closed geodesics. 

Variants and generalizations of symplectic capacity and their applications

Rongrong JIN（晋榕榕）  中国民航大学

Abstract: In this talk, I will describe variants and generalizations of Hofer-Zehnder and Ekeland-Hofer symplectic capacities based on Hamiltonian boundary value problems, for example, generalized Hofer-Zehnder capacity, generalized Ekeland-Hofer capacity and coisotropic Ekeland-Hofer capacity, and their applications to Hamiltonian dynamics and billiard problems. These are joint works with Guangcun Lu.

Hamiltonian 2 group action on symplectic orbifolds and reduction

Bohui CHEN（陈柏辉）  四川大学

Abstract: Symplectic reduction on symplectic manifolds is defined for Hamiltonian group actions. It has important physics backgrounds and is an operation in symplectic world in the sene that a (regular) reduction preserves symplectic objects. On the other hand, the regular reduction of a symplectic manifold is a symplectic orbifold. It is a natural question to ask what objects should be included in symplectic "category " such that it is closed under symplectic reduction. It turns out symplectic reductions of Hamiltonian group actions on orbifolds are orbifolds. This seems answer the question. However, this is a fake answer since the natural Hamiltonian action on otbifolds is no longer groups, and the reduction will produce a symplectic 2-groupoid. (Note manifold/orbifold is treated as a 0/1 groupoid. ) I will explain this scenery in this talk. This is based on the joint work with Cheng-Yong Du and Fengyu Jiang.

C^0-closedness of Symp_0(X)

Weiwei WU（吴惟为）  浙江大学

Abstract: The C^0 topology of the symplectomorphism groups has lots of mysterious basic questions.  The famous symplectic rigidity theorem says that, given any symplectic manifold X, the symplectomorphism group Symp(X) are closed in Diff(X) with respect the the C^0 topology. It is also easy to check that Symp_h(X), the group of homologically trivial symplectomorphisms, is closed in Symp(X). The relation between Symp_h(X) and Symp_0(X) is more delicate. In this talk, we will present a proof of the closedness of Symp_0(X) in Symp_h(X) when X is a log Calabi-Yau surface of type D, in the sense of Li-Li-Wu.  For these symplectic manifolds, it was previously known that Symp_h(X) is a subset of Diff_0(X).  Our result implies that, there is a smooth isotopy of some symplectomorphism which cannot be C^0-approximated by a path of symplectomorphisms. This is an ongoing project with Marcelo Atallah and Cheuk-Yu Mak.

Non-contractible closed geodesics on the compact space forms

Yuchen WANG（王宇辰）  天津师范大学

Abstract: In this talk, we investigate the  multiplicity of non-contractible closed geodesics on Finsler manifolds that are diffeomorphic to compact space forms. By imposing natural geometric conditions analogous to those in the classical Riemannian setting, we establish multiplicity results for closed geodesics within prescribed nontrivial homotopy classes. Furthermore, for $C^\infty$-generic Finsler metrics, we demonstrate the existence of infinitely many distinct closed geodesics in every nontrivial homotopy class. 

The action function of Hamiltonian homeomorphisms on surfaces

Jian WANG（王俭）  南开大学

Abstract: In symplectic geometry, the action function is a classical object defined on the set of contractible fixed points of the time-one map of a Hamiltonian isotopy. Under a weaker boundedness condition, we can generalize the classical action function to the case of Hamiltonian homeomorphisms on surfaces. Through studying the properties of the generalized action function we can generalize several classical results from the smooth world to the C^0 world, e.g., the contractible fixed points set (and consequently the fixed points set) of a nontrivial Hamiltonian homeomorphism is not connected, the C^0 -Schwarz theorem, the continuity of the action functions, etc. Part of these works collaborated with Frédéric Le Roux.

Quantum cohomology/quantum K rings and cluster algebras

Yingchun ZHANG（张迎春）  上海交通大学

Abstract: I will introduce a relation between the quantum cohomology ring/quantum K ring of a quiver variety and the cluster algebra.  More explicitly, given a quiver with potential, there is an injective ring homomorphism from the cluster algebra to quantum cohomology/quantum K ring of the corresponding quiver variety. This relation has been proved for A and D type quivers. 

Non-resonant Hopf link near a Hamiltonian equilibrium

Lei LIU（刘磊）  山东大学

Abstract: In this talk, we will present explicit conditions on the Birkhoff-Gustvason normal form of a two-degree-of-freedom Hamiltonian system near an equilibrium point, under which there exists the existence of infinitely many periodic orbits on every sphere-like component of the energy surface near the equilibrium point. The equilibrium is supposed to be a nondegenerate minimum of the Hamiltonian. Every sphere-like component of the energy surface that sufficiently close to the equilibrium will contains at least two periodic orbits forming a Hopf link. The method in this study is to check a certain non-resonance condition (established by Hryniewitz, Momin and Salomao2015) for the Hopf link, and then infinitely many periodic orbits follow. This method does not need any global surface of section, as in the study of Hofer, Wysocki and Zehnder 1998. Finally, we apply our results to the spatial isosceles three-body problem, Hill's lunar problem and Henon-Heiles problem. 

Quantitative contact big fiber theorem

Jun ZHANG（张俊）  中国科学技术大学

Abstract: In this talk, we will present a proof of contact big fiber theorem, based on invariants read off from contact Hamiltonian Floer homology. The theorem concludes that any contact involutive map on a Liouville fillable contact manifold admits at least one fiber that is non-displaceable, under the condition that the Liouville filling has non-vanishing symplectic homology. Note that this result was also proved recently by Sun-Uljarević-Varolgunes, based on symplectic homology with compact support. Along with the proof, we will also propose a definition of partial contact quasi-state (and contact quasi-measure), serving as a contact analog to Entov-Polterovich’s quasi-state machinery in the symplectic setting. This talk is based on joint work with Igor Uljarević.

A Mathematical Formulation of SYZ conjecture

Hang YUAN（袁航）  北京雁西湖应用数学研究院

Abstract: The SYZ conjecture predicts that a Calabi-Yau manifold foliated by special Lagrangian fibrations has a dual torus fibration whose total space forms the mirror Calabi-Yau. We propose that the dual fibration should be a non-Archimedean torus fibration over the Novikov field. It is inspired by a toy model comparing the complex logarithm map and the non-Archimedean tropicalization map, both interpreted as torus fibrations over Euclidean space with a trivial integral affine structure. It is also motivated by the general need to define Floer homology and the Fukaya category over the Novikov field. Using (family) Floer theory for Lagrangian fibers, we establish the existence of a canonical dual fibration, unique up to isomorphism, which locally matches this toy model. The two fibrations, though defined on distinct spaces, induce exactly the same integral affine structures, in line with Arnold-Liouville and Kontsevich-Soibelman. If time permits, we will review basics of non-Archimedean geometry, present explicit examples (including singular fibers such as conifold transitions and A_n singularities), and discuss applications in symplectic geometry.