Differential geometry and symplectic topology seminar (Spring 2026)

Abstracts

• Lina Liu
  • Orbifold (co)homology has been a hot topic in the last handful of years. A longstanding issue of defining Morse-Smale pairs (f,g) on an orbifold has now been resolved. This was done by applying the stabilization technique of Bao-Lawson onto the setting of orbifold Morse theory. This allows us to define Morse homology for effective compact orbifolds over integers.
• Honghao Jing
  • The Strominger–Yau–Zaslow approach to mirror symmetry for Fano toric varieties is well understood, with the mirror superpotential governed by Maslov index two holomorphic discs. For non-Fano manifolds, however, additional discs of non-positive Maslov index appear and fundamentally alter the SYZ construction. In this talk, I will use the non-Fano Hirzebruch surfaces F_3 and F_4 to illustrate that the resulting mirror can depend on the choice of perturbation used to regularize disc moduli spaces. For F_4, I will present a perturbation for which the mirror superpotential becomes an explicit infinite Laurent series, and explain how it is related to other superpotentials via a scattering diagram.
• Zijun Li
  • Legendrian links are a special class of links in R^3 that arise naturally in the study of particle motion. Consider a particle moving periodically along the real line. To analyze its motion, we can record its time x, velocity y, and displacement z and plot the tuples (x,y,z) in R^3. The resulting plot is a knot in R^3 satisfying the condition that the velocity coordinate is equal to the derivative of the displacement coordinate with respect to the time coordinate. Such a knot is called a Legendrian knot. If we consider n particles moving periodically on the real line, this construction produces a Legendrian link with n components. In this talk, I will discuss algebraic invariants of Legendrian links, immersed exact Lagrangian surfaces in R^4 with boundary on Legendrian links, and connections between them. Time permitting, I will also talk about how computing algebraic invariants of Legendrian links is related to gradient flow trees.
• Jun Zhang
  • In this talk, we will introduce a contact-geometric analogue of Hamiltonian Floer theory, based on the maximum principle established by Merry and Uljarević. This Floer theory can be regarded as an intermediate step leading to Viterbo’s symplectic homology, but it is built upon a general contact Hamiltonian dynamics that extends beyond classical Reeb dynamics. Various rigidity phenomena in contact geometry can be captured through quantitative invariants derived from this Floer theory. In particular, we will show how it can be used to detect a contact big fiber—a result parallel to the celebrated rigidity phenomenon discovered by Entov and Polterovich in symplectic geometry. We will also demonstrate how a Givental-type nonlinear Maslov index, constructed from this Floer theory, leads to orderability for a new class of contact manifolds. This talk is based on joint works with Djordjević, Uljarević, Cant, and Kilgore in two separate projects.
• Boyu Zhang
  • In this talk, I will present several results about isotopy problems in dimension 4. First, we give a classification of the isotopy classes of embeddings of $\Sigma$ in $\Sigma\times S^2$ that are geometrically dual to ${pt}\times S^2$, where $\Sigma$ is a closed oriented surface with a positive genus, and show that there exist infinitely many such embeddings that are homotopic to each other but mutually non-isotopic. This answers a question of Gabai. Second, we show that the space of symplectic forms on an irrational ruled surface homologous to a fixed symplectic form has infinitely many connected components. This gives the first such example among closed 4-manifolds and answers a question of McDuff-Salamon. We also show that symplectic forms on a closed 4-manifold with a fixed cohomology class do not admit the h-principle, which answers a question of Cieliebak-Eliashberg-Mishachev. The proofs are based on a generalization of the Dax invariant to embedded closed surfaces. This is joint work with Jianfeng Lin, Weiwei Wu, and Yi Xie.
• Tedi Draghici
  • I will present a mixture of old and new results regarding classes of almost Hermitian 4-manifolds whose Riemannian curvature tensor satisfies certain identities of Alfred Gray. Some old and new open questions will be presented as well. The talk is based on older works with Vestislav Apostolov and John Armstrong, but also on a very recent work with Ethan Addison and Mehdi Lejmi.
• Weimin Chen
  • A very old question in algebraic geometry asks what is the maximal number of nodes in a degree d hypersurface in CP^3. More generally, one asks what is the upper bound on the number of disjoint (-2)-curves in an algebraic surface. We consider the symplectic analog of this question and discuss some partial results. The symplectic version also offers a new perspective and a possible new approach involving Lagrangian geometry and Floer theory. This is work in progress.
• Cheuk Yu Mak
  • In joint work with Matessi-Ruddat-Zharkov, we prove the existence of Lagrangian torus fibrations on Calabi-Yau hypersurfaces in toric Fano manifolds given by a reflexive polytope. The result is motivated by the Strominger-Yau-Zaslow conjecture, which predicts the existence of these fibrations on Calabi-Yau manifolds near large complex structure limits. In this talk, I will outline the main setup of the construction. The idea is to replace the ordinary algebraic equation with a new one involving “ironing coefficients” and a convex (but not strictly convex!) potential, which breaks the manifold into local models. Over these models, we apply the Liouville flow technique in the style of Evans-Mauri.
• Feng Zhu

  • In this talk I will discuss a general theory of Patterson-Sullivan (PS) measures associated to coarse cocycles of convergence groups. Under some natural geometric assumptions on the coarse cocycle/s, we prove existence, uniqueness, ergodicity, and counting results.

    This framework includes, among other examples, PS measures on the Gromov boundary of a hyperbolic metric space, hitting measures of random walks on hyperbolic groups, as well as PS measures on flag manifolds associated to Anosov (or more generally transverse) subgroups of semisimple Lie groups.

    This is joint work with Pierre-Louis Blayac, Richard Canary, Tengren Zhang, and Andrew Zimmer.

• Caglar Uyanik
  • A celebrated theorem of Thurston asserts that a 3-manifold that fibers over the circle with fibers closed hyperbolic surfaces admit a hyperbolic metric if and only if the monodromy is pseudo-Anosov. The surface subgroup is exponentially distorted in the 3-manifold group. I will talk about how this distortion manifests itself by looking at the induced map on the boundary at infinities.
• Ely Kerman
  • Let $L(r, s)$ be a standard Lagrangian product torus in the 4-dimensional symplectic cylinder of capacity 1. Let $\phi$ be a Hamiltonian diffeomorphism of $\mathbb{R}^4$ that maps $L(r,s)$ back into the cylinder. In this talk I will describe some new constraints on the image $\phi(L(r,s))$ . These include some new Lagrangian intersection results and yield new computations of the shape invariant defined by Hind and Zhang. This is joint work with Richard Hind (https://arxiv.org/abs/2602.10291).
• Siu-Cheong Lau
  • Teleman conjectured that the mirror of a Hamiltonian action on a symplectic manifold is a holomorphic fibration. In this talk, I will explain this from the perspective of equivariant Lagrangian Floer theory and correspondence for symplectic quotients. Quantum corrections are captured by the Floer theoretical obstruction of Lagrangian correspondence, which can be computed by a proposed degeneration formula in the case G = U(1). This is a joint work with Nai-Chung Conan Leung and Yan-Lung Leon Li.
• Guangbo Xu
  • Gromov-Witten invariants in genus g(>0) overcount genus g holomorphic maps as certain configurations with ghost components have nontrivial contributions. The reduced Gromov-Witten invariants give a true count of smooth genus g holomorphic maps (including multiple covers), which is only defined in genus 1 by Zinger in symplectic geometry. Recently Ekholm-Shende defined their ``bare curve counts’’ in higher genus in Calabi-Yau threefolds using similar ideas. Both Zinger and Ekholm-Shende need to establish (via analysis) a delicate refinement of Gromov compactness. I will explain how to use a topological method to obtain reduced Gromov-Witten invariants in all genera for general closed symplectic manifolds, which is based on a new transversality result on stratified (virtual) manifolds.
• Xinle Clair Dai
  • Symmetric products of Riemann surfaces play a crucial role in symplectic geometry and low-dimensional topology. They are essential ingredients for defining Heegaard Floer homology and serve as important examples of Liouville manifolds when the surfaces are open. In this talk, I will discuss my work on the symplectic topology of these spaces through Liouville sectorial methods, along with examples and applications of this decomposition construction to homological mirror symmetry.