Hyperbolic Geometry and Quantum Invariants - Tian YANG 杨田, Texas A&M (2021-12-20)

There are two very different approaches to 3-dimensional topology, the hyperbolic geometry following the work of Thurston and the quantum invariants following the work of Jones and Witten. These two approaches are related by a sequence of problems called the Volume Conjectures. In this talk, I will explain these conjectures and present some recent joint works with Ka Ho Wong related to or benefited from this relationship.

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Past seminar info:

Norms on cohomology of non-compact hyperbolic 3-manifolds, harmonic forms and geometric convergence - Hans Xiaolong HAN 韩肖垄, Tsinghua (2021-12-13, part 2)

(This is part 2, and continues from where the previous talk left off)

We will talk about generalizations of an inequality of Brock-Dunfield to the non-compact case, with tools from Hodge theory for non-compact hyperbolic manifolds and recent developments in the theory of minimal surfaces. We also prove that their inequality is not sharp, using holomorphic quadratic differentials and recent ideas of Wolf and Wu on minimal geometric foliations. If time permits, we will talk about some results concerning the growth of L2 norm/Thurston norm for a sequence of closed hyperbolic 3-manifolds converging geometrically to a cusped manifold, using Dehn filling and minimal surface.

Norms on cohomology of non-compact hyperbolic 3-manifolds, harmonic forms and geometric convergence - Hans Xiaolong HAN 韩肖垄, Tsinghua (2021-12-06, part 1)

We will talk about generalizations of an inequality of Brock-Dunfield to the non-compact case, with tools from Hodge theory for non-compact hyperbolic manifolds and recent developments in the theory of minimal surfaces. We also prove that their inequality is not sharp, using holomorphic quadratic differentials and recent ideas of Wolf and Wu on minimal geometric foliations. If time permits, we will talk about some results concerning the growth of L2 norm/Thurston norm for a sequence of closed hyperbolic 3-manifolds converging geometrically to a cusped manifold, using Dehn filling and minimal surface.

Augmentations from Legendrian knots - Tao SU 苏桃, Tsinghua (2021-11-29)

In this talk, I will tell a story about Legendrian knots, with a focus on the associated Chekanov-Eliashberg DGAs and their augmentations. First, I will introduce the Chekanov-Eliashberg DGA. It’s a Legendrian isotopy invariant up to homotopy equivalence, which admits two equivalent descriptions: counting of pseudo-holomorphic disks, and combinatorics. Second, I will discuss the gluing property of the Chekanov-Eliashberg DGA, induced by cutting the Legendrian knot front diagram into elementary pieces. Finally, I will give an application of this gluing property: counting augmentations gives a state-sum Legendrian isotopy invariant, i.e. the ruling polynomial. Time permitting, I will also mention a second application in my recent work, concerning part of the geometric P=W conjecture.

How tight can a contact manifold be? - Zhengyi ZHOU 周正一, AMSS Chinese Academy of Science (2021-11-22)

The fundamental dichotomy of overtwisted v.s. tight in contact topology asserts that contact topology of overtwisted structures can be completely “understood” in a topological manner. On the other hand, the tight contact structures form a richer and more mysterious class. In this talk, I will explain how to use rational symplectic field theory to give a hierarchy on contact manifolds to measure their “tightness”. This is a joint work with Agustin Moreno.

Proofs of Mostow Rigidity Theorem - Qing LAN 蓝青, Tsinghua (2021-11-22)

In this talk I will sketch 2 proofs of Mostow rigidity, which essentially states that the geometry of a closed hyperbolic manifold of dimension greater than two is determined by the fundamental group. I will talk about a proof using ergodic theory and another proof using Gromov norm.

Extended graph manifolds, and Einstein metrics - Luca DI CERBO, University of Florida (2021-11-04)

In this talk, I will present some new topological obstructions for solving the Einstein equations (in Riemannian signature) on a large class of closed four-manifolds. I will conclude with some tantalizing open problems both in dimension four and in higher dimensions.

Trisection invariants of 4-manifolds from Hopf algebras - Xingshan CUI 崔星山, Purdue (2021-10-25)

The Kuperberg invariant is a topological invariant of closed 3-manifolds based on finite-dimensional Hopf algebras. Here we initiate the program of constructing 4-manifold invariants in the spirit of Kuperberg's 3-manifold invariant. We utilize a structure called a Hopf triplet, which consists of three Hopf algebras and a bilinear form on each pair subject to certain compatibility conditions. In our construction, we present 4-manifolds by their trisection diagrams, a four-dimensional analog of Heegaard diagrams. The main result is that every Hopf triplet yields a diffeomorphism invariant of closed 4-manifolds. In special cases, our invariant reduces to Crane-Yetter invariants and generalized dichromatic invariants, and conjecturally Kashaev's invariant. As a starting point, we assume that the Hopf algebras involved in the Hopf triplets are semisimple. Time permitting, we also sketch an ongoing effort to generalize the invariant using non-semisimple Hopf algebras. The generalized invariant is defined on 4-manifolds with a choice of spin^c structure. We expect the generalized invariant is more sensitive to extract information about 4-manifolds.

Let's stretch some hyperbolic surfaces! - Yi HUANG 黄意, Tsinghua (2021-10-18)

In an unpublished preprint, Thurston looked into the Lipschitz theory of hyperbolic surfaces and built from scratch a beautiful theory tying together stretch maps and the lengths of simple closed geodesics on hyperbolic surfaces. We hope to give a gentle introduction to this theory, and to introduce some modern explorations along this theme.

Turning smooth 4-manifolds into maps between spheres - Jianfeng LIN 林剑锋 (2021-10-04)

In the past 40 years, studying smooth structures on 4-manifolds has been an important topic in low dimensional topology. In this talk, I will talk about the celebrated Bauer-Furuta invariant of 4-manifolds. In particular, I will dicuss a technique called the ``finite dimensional approximation'' which is a general procedure that turns a nonlinear elliptic PDE into a map between two (finite dimensional) spheres. This allows us to use powerful tools from equivariant stable homotopy theory to attack hard problems in 4-dimensional topology. I will also talk about some recent applications of this invariant on exotic diffeomorphisms and exotic embeded surfaces in 4-manifolds.

Topological complexity of enumerative problem - Weiyan CHEN 陈伟彦 (2021-09-27)

Topological complexity measures how difficult is it to find solutions of a problem using an algorithm. It has been extensively studied in the context of topological robotics and superposition of algebraic functions. In this talk, I will propose a new research direction of determining topological complexity of enumerative problems in algeb