数学研究所
数学科学全国重点实验室

华罗庚青年数学论坛综合报告

时间：8月·11日

地点：晨兴楼110

1. 9:30-10:30

Guy Boyde (Utrecht University)

题目: Planar loops and the homology of Temperley-Lieb algebras
摘要: Temperley-Lieb algebras are certain finite-dimensional algebras coming originally from statistical physics and knot theory. Around 2019, they became one of the first examples of homological stability for algebras, when Boyd and Hepworth showed that in low dimensions the homology vanishes. I will explain what we mean by homology of an algebra, and then explain what we know about this problem. We're now able to give some complete calculations of the homology, which has a surprisingly rich structure. This is joint work in progress with Rachael Boyd, Oscar Randal-Williams, and Robin Sroka. Prerequisites will be minimal.

2. 10:50-11:50

Yunze Lu (UCSD)

题目：Calculations in nonabelian equivariant cohomology

摘要：Computing the coefficients of equivariant generalized cohomology theories is a central problem in equivariant homotopy theory. While much of the classical work has focused on abelian groups, there is growing interest in nonabelian groups due to their increasing relevance in recent developments. In this talk, I will present some of my work in this direction. Examples include the RO(G)-graded Eilenberg–MacLane cohomology of a point with constant coefficients for dihedral groups of order 2p and for the quaternion group Q8, as well as the coefficient ring of Σ3-equivariant complex cobordism. If time permits, I will also discuss some ongoing questions related to compact Lie groups.

3. 13:30-14:30

Wenyuan Li (USC)

题目: C^0-rigidity of Legendrian submanifolds
摘要: One instance of rigidity phenomenon in contact topology is that the contactomorphism group is closed in the diffeomorphism group under the C^0-topology. We can thus define contact homeomorphisms as points in the closure of the contactomorphism group in the homeomorphism group. Dimitroglou Rizell and Sullivan showed that the images of closed Legendrians under contact homeomorphisms are still closed Legendrians if smooth. We will show that for closed Legendrians in cosphere bundles, vanishing of the Maslov class and certain Floer theory invariants is preserved under contact homeomorphisms, and when the conformal factors of contactomorphisms are uniformly bounded, the full Maslov data and Floer theory invariants are preserved. This is joint work in preparation with Tomohiro Asano, Yuichi Ike and Chris Kuo.

4. 14:50-15:50

Yuhan Sun (Imperial College London)

题目: Rigid fibers in symplectic and contact manifolds
摘要: Given a Hamiltonian integrable system on a closed symplectic manifold, Entov-Polterovich proved it has at least one rigid fiber. We will first survey several ideas of the proof and recent enhancements. Then we will present a rigid fiber theorem for contact manifolds. Based on joint works with Mak, Uljarevic and Varolgunes.

5. 16:10-17:10

Luya Wang (IAS)

题目：Spinal open books and symplectic fillings with exotic fibers
摘要：An evolving theme in symplectic topology has been the classification of symplectic manifolds, where pseudoholomorphic curves are often useful for establishing uniqueness and finiteness results. In the closed case, this traces back to Gromov and McDuff. In a series of works, Wendl used punctured pseudoholomorphic foliations to classify symplectic fillings of contact three-manifolds supported by planar open books, turning it into a problem about monodromy factorizations. In a joint work with Hyunki Min and Agniva Roy, we build on the recent works of Lisi--Van-Horn-Morris--Wendl in using spinal open books to further study the classification problem of symplectic fillings of higher genus open books. In particular, we provide the local model of the mysterious "exotic fibers" in a generalized version of Lefschetz fibrations, which captures a natural type of singularity at infinity. We will give some applications to classifying symplectic fillings via this new phenomenon.