Differential geometry and symplectic topology seminar (Fall 2026)

Abstracts

• Jun Zhang
  • In this talk, we introduce a non-linear symplectic capacity for (4-dimensional) Euclidean domains X, known as the shape invariant of X, which describes the embeddability of Lagrangian tori into X. We will present a computational framework for this invariant using techniques from symplectic field theory (SFT). Furthermore, we explore its applications in uncovering new obstructions to symplectic embeddings and detecting Hamiltonian knottedness of Lagrangian tori. This talk is based on joint work with Richard Hind.
• Hao Fang

  • Gårding polynomials arose from joint work with Biao Ma on fully nonlinear elliptic equations, motivated more broadly by the Caffarelli–Nirenberg–Spruck theory. The underlying question is how polynomial structure can encode admissible regions and ellipticity. I will introduce Gårding polynomials through their positivity regions, describe their basic structural characterizations, and explain their connections with real stable polynomials, Gårding’s classical theory of hyperbolic polynomials, and the more recent theory of Lorentzian polynomials. In one variable, the theory admits an explicit description in terms of monotone root sequences (MRS), providing a useful comparison with real-rootedness.

  • I will discuss several intrinsic features of the theory, including recursive characterizations by derivatives, polarization, and a symbol-based theory of linear preservers. I will then turn to matroid applications. Gårding polynomial methods yield Rayleigh and ultra-log-concavity properties for several matroid generating functions, including naturally occurring non-homogeneous ones. I will present examples and non-examples and discuss a conjectural strengthening of Mason’s conjecture arising from the Gårding framework.

  • Finally, I will introduce ideal Gårding polynomials, a more rigid class motivated by stronger structural requirements in fully nonlinear PDE. Ideal Gårding polynomials exhibit additional convexity and concavity properties, admit a compatible polarization theory, and are closely related to localized Lorentzian structures. I will conclude with a brief discussion of how these ideas enter an ongoing program on a priori estimates for fully nonlinear elliptic equations.

  • This talk is based on joint work with Biao Ma.

• Francesco Lin
  • LeBrun conjectured that the Seiberg-Witten invariants of closed hyperbolic four-manifolds vanish. After discussing context and consequences, I will provide an overview of what is known, and then focus on the proof (joint work with Bruno Martelli) that the conjecture holds for the Davis manifold - arguably the simplest hyperbolic four-manifold.
• Dylan Cant
  • Eliashberg and Polterovich call a contact manifold orderable if non-negative contact isotopies induce a partial order on the universal cover of its contactomorphism group. The standard sphere $S^{2n-1}$ is not orderable, but the quotient $\mathbb{RP}^{2n-1}$ is. I will explain a Floer-theoretic approach to this phenomenon for a large class of contact manifolds: quotients of the ideal boundary of a convex-at-infinity symplectic manifold $W$ by a finite group acting on $W$, freely at infinity but with fixed points inside. Many of these quotients are orderable even though their covering spaces are not. The main tool is an equivariant contact Floer cohomology, a module over $k[[x]]$. Its key property is that the mapping cones of continuation maps are $x$-torsion and detect crossings with the discriminant. To make sense of these cones we work at chain level, defining the theory as an $\infty$-functor on an $\infty$-categorification of the Eliashberg-Polterovich relation. Equivariant PSS maps compare the theory with the Borel equivariant cohomology of $W$. When the action has a fixed point, a localization argument shows that the unit element is not eternal, and orderability follows. The same structure yields contact spectral invariants for every Reeb flow. When the symplectic cohomology of $W$ vanishes, the $k[[x]]$-module structure gives an integer-valued analogue of Givental’s nonlinear Maslov index. This is joint work with Eric Kilgore and Jun Zhang.
• Michael Zshornack
  • The infinite-index Zariski-dense subgroups of arithmetic groups exhibit several properties mirroring those of their ambient group despite not being arithmetic themselves. Motivated by various natural questions in geometric group theory, one seeks such groups possessing more interesting isomorphism types than what elementary constructions such as ping pong provide. I’ll discuss the case of surface groups in rank one, where the story also connects to a large body of work on the study of immersed surfaces in hyperbolic manifolds. This is based on joint work with Sara Edelman-Muñoz and joint work in progress with Sami Douba.