Differential geometry and symplectic topology seminar (Fall 2025)

Abstracts

• Julian Chaidez

  • Pseudo-Anosov flows, originally introduced by Thurston, are a broad class of hyperbolic flows on 3-manifolds that are Anosov except along a finite link of singular closed orbits. A longstanding conjecture states that any 3-manifold carries only finitely many transitive pseudo-Anosov flows, up to the appropriate type of equivalence. In this talk, I will explain a proof of this conjecture for a large class of pseudo-Anosov flows that are Reeb, in the appropriate sense. This extends recent work by Barthelme-Bowedn-Mann for Reeb Anosov flows. The proof involves a computation of cylindrical contact homology in this setting. This is joint work in preparation with Yijie Pan (USC).

• Peter Olver

  • I will present a new theorem proving convergence of normal form series for a broad range of Lie pseudo-group actions on submanifolds. This result generalizes the normal forms of Chern and Moser for hypersurfaces in CR geometry. The proof relies on combining the theory of involutive systems of partial differential equations with the method of equivariant moving frames. Joint work with Francis Valiquette and Masoud Sabzevari.

• Nicklas Day

  • In 1970, N. Tanaka gave a method for assigning a canonical frame to distribution with a constant Tanaka symbol. In 2009, B. Doubrov and I. Zelenko utilized a symplectification procedure to obtain a canonical frame for distributions independent of their Tanaka symbol, but an additional condition called “maximality of class” was required of the distribution. In a work with I. Zelenko, we recently proved that all rank 2 distributions with 5-dimensional cube are of maximal class at a generic point; this allowed us to assign a canonical frame at a generic point to every rank 2 distribution which is not Goursat. In the rank 2 case, I will give an interpretation of the symplectification procedure in terms of a classical construction called Cartan prolongation. Time permitting, I will also sketch what is known about a collection of fundamental invariants called harmonic invariants in the case of rank 2 distributions in dimensions 6 and 7 along with differential relations called syzygies satisfied by these invariants.

• Yuan Yao

  • The problem of symplectically packing k symplectic balls into a larger one has been solved in dimension four, i.e. there is now a combinatorial criteria of when this is possible. However, not much is known about symplectic packing problems in higher dimensions. We take a step in this direction in dimension six, by considering a “stabilized” packing problem, i.e. we consider symplectically packing a disjoint union of four dimensional balls times a closed Riemann surface into a bigger ball times the same Riemann surface. We show this is possible if and only if the corresponding four dimensional ball packing is possible. The proof is a mixture of geometric constructions, pseudo-holomorphic curves, and h-principles. This is based on work with Kyler Siegel.

• Nancy Mae Eagles

  • In 1997, Chekanov defined a DGA generated by Reeb chords of a Legendrian knot in R3, and whose differential counts rigid polygons between double points (chords) in the Lagrangian projection. His tool provided the first example of two Legendrian knots that are not Legendrian isotopic but have the same classical invariants. Despite this, the (stable tame isomorphism class) of the DGA is not a complete invariant. In this talk, I will recall the basic construction of the DGA and its homology and highlight some generalizations that have been cooked up over the years since its conception. Time permitting, I will discuss some recent work with Zijian Rong on another generalization that we hope can be described entirely in combinatorial terms.

• Sai Kee Yeung

  • We would like to explain a few results related to rigidity in studying moduli spaces of curves $M_g$ and locally Hermitian symmetric spaces $M$. The first on the classification of holomorphic mappings from $M_g$ to $M$. The second is related to a conjecture of Coleman-Oort on non-existence of a totally geodesic locally Hermitian symmetric subvarieties in the Torelli locus of a Siegel modular varieties. The third is on the upper bound of the slope of a Kodaira fibration surface, $c_1^2/c_2$, away from $3$.