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.