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.