Mini Workshop on Symplectic Geometry

December 13th, 2025  MCM110  

Invited Speakers

Organizers

Prof. Julian Chaidez (University of Southern California)

Title: Convex hypersurface theory from the dynamical viewpoint

Abstract: Any hypersurface in a contact manifold comes equipped with a natural dynamical system that governs the contact topology near the hypersurface, such as the property of being convex in the sense of Giroux. In this talk, I will discuss some new results in convex hypersurface theory proven using dynamical methods. This includes the existence of robust (local) obstructions to convexity and a complete characterization of convexity using recurrent dynamics. I will also discuss ongoing efforts to prove that convex hypersurfaces are generic in the C1-topology.

This talk will be partially based on joint work-in-progress with Yakov Eliashberg

(Stanford), Dishant Pancholi (IMS Chennai) and Michael Huang (USC).

Dr. Zhijing Wang (The University of Chicago)

Title: Complexity of Hofer's geometry in higher dimensional manifolds

Abstract: The group of Hamiltonian diffeomorphisms Ham(M, ω), equipped with the Hofer metric dH, is a central object in symplectic topology. A landmark result by Polterovich and Shelukhin established the profound geometric complexity of this group for surfaces and their products, showing that high powers are sparse in the metric space.

More recently, Álvarez-Gavela et al. demonstrated that the free group embeds quasi-isometrically into Ham of surfaces, revealing its large-scale non-commutativity.

In this talk, I will review these results and present a generalization to some higher dimensional symplectic manifolds, including surface bundles. We prove robust obstructions that prevent a Hamiltonian diffeomorphism from being a k-th power (for k≥2) or from being embedded in a flow. We also show that every asymptotic cone of (Ham(M, ω), dH) for our higher-dimensional manifolds contains an embedded free group.

Prof. Youlin Li (Shanghai Jiao Tong University)

Title: Algebraically overtwisted contact manifolds revisited

Abstract: We show that contact (+1) surgery along any Legendrian sphere in the standard contact (2n+1)-sphere whose Thurston-Bennequin invariant is not (−1)^{n(n−1)/2+1}produces an algebraically overtwisted contact manifold. In particular, for any Legendrian knot in the standard contact 3-sphere admitting an exact Lagrangian filling that is not a disk, contact (+1)-surgery produces an algebraically overtwisted but tight contact manifold. Analogous results hold for a class of ADC Legendrian spheres, including those in the standard contact (4n+1)-sphere where the Thurston-Bennequin invariants are always (−1)^{n+1}. This is joint work with Russell Avdek and Zhengyi Zhou.

Dr. Mohamad Rabah (MCM, CAS)

Title: Fukaya Algebra over \Z

Abstract: In their book, Fukaya-Oh-Ohta-Ono' 09, constructed an A_\infty-algebra structure on the singular cohomology of a Lagrangian submanifold over the Novikov ring with rational coefficients. Using the recent developments in Symplectic Topology, namely Bai-Xu' 22 realization of Fukaya-Ono' 97 proposal on Normally Complex polynomial perturbations and Abouzaid-McLean-Smith' 21 construction of Global Kuranishi charts, we show how to adapt such developments in the setting of Lagrangian Floer Theory, which leads us to an integral version of the above result. In this talk we will go over the necessary notions and background needed to state and prove our results, followed by a sketch of proofs.

Note:

There is no registration fee for this conference.

If you have any other questions, please send email to us (mcmoffice@math.ac.cn).