Differential geometry and symplectic topology seminar (Spring 2025)

Abstracts

• Guangbo Xu

  • Symmetry and transversality are two conflicting features as symmetric objects are not generic. This causes the failure of generic transversality on orbifolds and makes the foundation of symplectic geometry very hard, as the moduli spaces or holomorphic curves are modelled on orbifolds. An accompanying feature is that the symplectic invariants (such as Gromov-Witten invariants) are in general rational numbers, even though they morally ``count’’ the numbers of holomorphic curves. In late 1990s Fukaya-Ono proposes a method to obtain integer-valued Gromov-Witten invariants by using specific kinds of perturbations. Building on their vision, we have developed a rigorous framework grounded in a refined notion of transversality on orbifolds. As an application, we can define integer-valued Gromov-Witten invariants and more related invariants.

• Nick Miller

  • Seminal work of Thurston shows that one can construct a plethora of finite volume, hyperbolic 3-manifolds using mapping tori of surface homeomorphisms. In fact, Thurston shows that such a mapping torus is hyperbolizable precisely when the corresponding bundle is atoroidal which is equivalent to the associated mapping class being pseudo-Anosov.

    Motivated by the desire to push this construction into dimension 4, it has been an open question as to whether one can similarly construct atoroidal surface bundles over surfaces. This was finally resolved this year by Kent and Leininger, who produced infinitely many such bundles. It is a folklore conjecture that, despite the analogy, such manifolds should not be hyperbolizable and one hope was to prove this by showing the non-vanishing of a certain obstruction to hyperbolicity called the signature. In this talk we will show that, on the contrary, the signature of these bundles vanishes and hence they retain the potential to be hyperbolic 4-manifolds. This is joint work with Jean-Francois Lafont and Lorenzo Ruffoni.

• Yu-Shen Lin

  • Special Lagrangians are an important class of minimal submanifolds, introduced by Harvey-Lawson. However, there are very few examples of them. It is well-known folklore conjecture that holomorphic curves in Calabi-Yau manifolds with collapsing special Lagrangian fibrations converge to tropical curves. Such conjecture motivates the development of tropical geometry. In this talk, we will consider the special Lagrangians in compact Calabi-Yau manifolds with collapsing holomorphic fibrations. Under this setting, one might expect that the special Lagrangians also converge to some graphs in the base. Indeed, we will construct special Lagrangians from certain geodesic intervals or loops in this talk. Such construction is a tip of the iceberg of the conjecture of Donaldson-Scaduto. This is a joint work with Shih-Kai Chiu.

• Shengzhen Ning

  • A classical theorem of Liu-Ohta-Ono asserts that any symplectic 4-manifold with negative pairing between the symplectic form and canonical class must be rational or ruled. This result is a symplectic reminiscence of the more classical characterization of complex surfaces with Kodaira dimension −∞. In this talk, we will discuss the generalization of Liu-Ohta-Ono’s theorem to the relative setting by considering symplectic divisors whose adjoint class has negative pairing with canonical class. In parallel to the theorem of Fujita-Miyanishi-Sugie-Russell in the algebraic context, we show that the complement of such divisors is foliated by a certain family of unicuspidal rational curves, thereby admitting the affine-ruled structure. This is based on joint work with Tian-Jun Li.