Differential geometry and symplectic topology seminar (Fall 2024)

Abstracts

• Xiaolong Li

  • The second kind of curvature operator has received recent attention following the resolution of Nishikawa’s conjecture by Cao-Gursky-Tran, Nienhaus-Petersen-Wink, and myself. These works have shown that a closed Riemannian manifold with 3-nonnegative curvature operator of the second kind is either flat or diffeomorphic to a spherical space form. In this talk, I will briefly introduce the curvature operator of the second kind and talk about the proof of Nishikawa’s conjecture. Then I will talk about my recent work on new sphere theorems under negative lower bounds of the curvature operator of the second kind.

• Rohil Prasad

  • Punctured holomorphic curves have been a key tool in symplectic topology and dynamics since their introduction by Hofer in 1993. I’ll describe a new compactness theorem for sequences of punctured holomorphic curves with actions converging to 0 and with bounded topology, without the traditional assumption of bounded Hofer energy. In the limit, one obtains a family of compact invariant subsets of the underlying Hamiltonian flow. Such sequences turn out to be abundant in low-dimensional symplectic dynamics. Dynamical applications include (1) a generalization of the Le Calvez-Yoccoz theorem to higher-genus surfaces and three-manifolds and (2) a generalization of a recent theorem by Fish-Hofer on four-dimensional autonomous Hamiltonian flows. This talk is based on arXiv preprints 2401.14445 (joint with Dan Cristofaro-Gardiner) and 2405.01106.

• Soham Chanda

  • We introduce a class of surgery operations on Lagrangian submanifolds which switch between two different fillings of Bohr-Sommerfeld Legendrians. We will also discuss the effect of such a surgery operation on the disk potential in the setting of monotone Lagrangians. As an application we will prove that such surgery operations yield exotic monotone Lagrangian tori in projective spaces of dimension 3 and higher.

• Jie Min

  • Curves on surfaces have been very effective in understanding topology of 3- and 4-manifolds, for example Heegaard diagrams, Lefschetz fibrations and trisection/multisection diagrams. I will talk about an analogous way to understand contact and symplectic topology in dimension 3 and 4, called the contact cut graph. We show that each path in the contact cut graph corresponds to a Weinstein domain, from which we define a new invariant for Weinstein domains called the Weinstein L-invariant. We also give some examples of L=0 and L arbitrarily large Weinstein domains. This is joint work with N. Castro, G. Islambouli, S. Sakalli, L. Starkston and A. Wu. (arXiv:2408.05340)

• Shengzhen Ning

  • Since Thurston’s first example of a non-Kahler symplectic 4-manifold, people have gradually realized that the world of Kahler manifolds only occupies a tiny part of the symplectic world. However one can still ask, for a Kahler manifold, can we find some other symplectic forms different from the Kahler form and not compatible with any complex structures? Following the work by Cascini and Panov, I will talk about a new example which is the blowup of Enriques surface (arXiv:2407.10217).

• Siyang Liu

  • In this talk, I’ll explain our recent work studying the symplectic topology of complements of complexified hyperplane arrangements. When hyperplanes intersect transversely, we proved that symplectic handles can be found essentially from the intersection points of the hyperplanes. If time permits, I’ll also discuss the strategy of extending the result to general hyperplane arrangements. This is based on the recent joint work with Sukjoo Lee, Yin Li and Cheuk Yu Mak and work in preparation with Sheel Ganatra, Wenyuan Li and Peng Zhou.

• Lenhard Ng

  • Lagrangian surfaces and Legendrian links play a crucial role in the modern study of the topology of symplectic 4-manifolds and contact 3-manifolds. Given two Legendrian links in a contact 3-manifold, one can ask whether there is an exact Lagrangian cobordism between them: this is a type of Lagrangian surface bounded by the two links in the symplectization of the contact manifold. In this talk, we’ll discuss what’s known about this surprisingly subtle question, from constructions of cobordisms to obstructions from topology and Floer theory.

• Habib Alizadeh

  • We first shortly introduce the Polterovich-Shelukhin Lagrangian spectral estimators [PS23] defined on the space of Hamiltonian functions on a product of two spheres. One objective of the talk will be to prove C^0-continuity of PS-spectral estimators for which we shall use fragmentation. Continuity of these spectral estimators have interesting applications among which is the proof of the long-standing simplicity conjecture which was first proved in [CGHS24]. In the second part of the talk, we explain how one can use pseudo-holomorphic curves in dimension four to prove fragmentation type results and we show its application to continuity of PS-spectral estimators. Finally, we mention some applications to the Hofer geometry of four-dimensional polydisks.

• Charles Daly

  • In this talk we’ll provide some background in the deformation theory of representations of hyperbolic 3-manifolds M into PSO(3,1) < PGL(4,R). We review some of the main results and address Weil’s Infinitesimal Rigidity Theorem and how (twisted) cohomology groups appear in this theory. We’ll explain how a certain group action makes an appearance in the context of when M is a once-punctured torus bundle and how to relate it to the notion of `infinitesimal projective rigidity rel cusp(s).

• Zhengyi Zhou

  • We will present two results in complex geometry: (1) A Kahler compactification of C^n with a smooth divisor complement must be P^n, which confirms a conjecture of Brenton and Morrow(1978) under the Kahler assumption; (2) Any complete asymptotically conical Calabi-Yau metric on C^3 with a smooth link must be flat, confirming a modified version of Tian’s conjecture regarding the recognition of the flat metric among Calabi-Yau metrics in dimension 3. Both proofs rely on relating the minimal discrepancy number of a Fano cone singularity to its Reeb dynamics of the conic contact form. This is a joint work with Chi Li.