Seminar notes.
Weiyan CHEN 陈伟彦, Honghao GAO 高鸿灏, Yi JIANG 江怡, Jianfeng LIN 林剑锋, and I organise the YMSC Topology seminar group at the Yau Mathematical Sciences Center (Tsinghua University).
A symplectic excision is a symplectomorphism between a manifold and the complement of a closed subset. We focus on the construction of symplectic excisions by Hamiltonian vector fields and give some criteria on the existence and non-existence of such kinds of excisions. The talk is based on arXiv:2101.03534.
The non-abelian Hodge theory identifies moduli spaces of representations with moduli spaces of Higgs bundles through solutions to Hitchin's selfduality equations. On the one hand, this enables one to relate geometric structures on surfaces with algebraic geometry, and on the other hand, one obtains interesting hyper-Kähler metrics on the solution spaces. In my talk, I will explain how to construct new hyper-Kähler metrics from certain singular solutions to Hitchin's self-duality equations. The main ingredients are graftings of projective structures, twistor spaces, and Deligne's notion of \(\lambda\)-connections.
In my talk, I will report on recent joint work with I. Biswas, S. Dumitrescu and S. Heller showing the existence of holomorphic maps from a compact Riemann surface of genus \(g>1\) into a quotient of \(\mathrm{SL}(2,\mathbb{C})\) modulo a cocompact lattice which is generically injective. This gives an affirmative answer to a question raised by Huckleberry and Winkelmann and by Ghys. The proof uses ideas from harmonic maps into the hyperbolic \(3\)-space, WKB analysis, and the grafting of real projective structures.
The classification of manifolds in various categories is a classical problem in topology. It has been widely investigated by applying techniques from geometric topology in the last century. However, the known results tell us very little information about the homotopy of manifolds. In the last ten years, there have been attempts to study the homotopy properties of manifolds by using techniques in unstable homotopy theory. In this talk, we will discuss the loop decomposition method in this topic and review the known results and our recent work.
In this talk, we show that the length of a shortest closed geodesic on a Riemannian manifold of dimension \(4\) with diameter \(D\), volume \(v\), and \(|Ric|<3\) can be bounded by a function of \(v\) and \(D\). In particular, this function can be explicitly computed if the manifold is Einstein. The proof of this result depends on a structural theorem proven by J. Cheeger and A. Naber. This is joint work with N. Wu.
We consider a closed oriented surface of genus at least \(2\). To describe curves on it, one natural idea is to choose once for all a collection of curves as a reference system and to hope that any other curve can be determined by its intersection numbers with reference curves. For simple curves, using the work of Dehn and Thurston, it is possible to find such a reference system consisting of finitely many simple curves. The situation becomes more complicated when curves have self-intersections. In particular, for any non negative integer \(k\), it is possible to find a pair of curves having the same intersection number with every curve with \(k\) self-intersections. Such a pair of curves are called \(k\)-equivalent curves. In this talk, I will discuss the general picture of a pair of \(k\)-equivalent curves and the relation between \(k\)-equivalence relations for different \(k\)'s. This is a joint-work with Hugo Parlier.
Branching random walks (BRW) on groups consist of two independent processes on the Cayley graphs: branching and movement. Start with a particle on a favorite location of the graph. According to a given offspring distribution, the particles at the time n split into a random set of particles with mean \(r \ge 1\), each of which then moves independently with a fixed step distribution to the next locations. It is well-known that if the offspring mean \(r\) is less than the spectral radius of the underlying random walk, then BRW is transient: the particles are eventually free on any finite set of locations. The particles trace a random subgraph which accumulates to a random subset called limit set in a boundary of the graph. In this talk, we consider BRW on relatively hyperbolic groups and study the limit set of the trace at the Bowditch and Floyd boundaries. In particular, the Hausdorff dimension of the limit set will be computed. This is based on a joint work with Mathieu Dussaule and Longmin Wang.
Counting embedded curves on a hyperbolic surface as a function of their length has been much studied by Mirzakhani and others. I will discuss analogous questions about counting incompressible surfaces in a hyperbolic 3-manifold, with the key difference that now the surfaces themselves have more intrinsic topology. As there are only finitely many incompressible surfaces of bounded Euler characteristic up to isotopy in a hyperbolic 3-manifold, it makes sense to ask how the number of isotopy classes grows as a function of the Euler characteristic. Using Haken’s normal surface theory and facts about branched surfaces, we can characterize not just the rate of growth but show it is (essentially) a quasi-polynomial. Moreover, our method allows for explicit computations in reasonably complicated examples. This is joint work with Stavros Garoufalidis and Hyam Rubinstein. Based on https://arxiv.org/abs/2007.10053
In the late 1980s, Nigel Hitchin and Michael Wolf independently discovered a parametrization of the Teichmüller space of a compact surface by holomorphic quadratic differentials. In this talk, I will describe a generalization of their result. I will explain how, by replacing holomorphic differentials by meromorphic differentials, one is naturally led to consider an object called the enhanced Teichmüller space. The latter is an extension of the classical Teichmüller space which is important in mathematical physics and the theory of cluster algebras.
Legendrian links play a central role in low dimensional contact topology. A rigid theory uses invariants constructed via algebraic tools to distinguish Legendrian links. The most influential and powerful invariant is the Chekanov-Eliashberg differential graded algebra, which set apart the first non-classical Legendrian pair and stimulated many subsequent developments. The functor of points for the dga forms a moduli space which acquires algebraic structures and can be used to distinguish exact Lagrangian fillings. Such fillings are difficult to construct and to study, whereas the only known complete classification is the unique filling for Legendrian unknot. A folklore belief was that exact Lagrangian fillings might be scarce. In this talk, I will report a joint work with Roger Casals, where we applied techniques from contact topology, microlocal sheaf theory and cluster algebras to find the first examples of Legendrian links with infinitely many Lagrangian fillings.
In this talk, we will discuss the behavior of the separating systole for random hyperbolic surfaces with respect to the Weil-Petersson measure of the moduli space. We show that its length is approximately \(2\log(g)-4\log(\log(g))\) and it separates out a one-holed torus for generic surfaces. Some other geometric quantities are also considered. This talk is based on joint works with Xin Nie, Hugo Parlier and Yunhui Wu.
(joint work with V. Delecroix, E. Goujard and P. Zograf)
It is common in mathematics to study decompositions of compound objects into primitive blocks. For example, the Erdos-Kac Theorem describes the decomposition of a random large integer number into prime factors. There are theorems describing the decomposition of a random permutation of a large number of elements into disjoint cycles.
I will present our formula for the asymptotic count of square-tiled surfaces of any fixed genus g tiled with at most N squares as N tends to infinity. This count allows, in particular, to compute Masur-Veech volumes of the moduli spaces of quadratic differentials. A deep large genus asymptotic analysis of this formula performed by Aggarwal and the uniform large genus asymptotics of intersection numbers of psi-classes on the moduli spaces of complex curves proved by Aggarwal allowed us to describe the decomposition of a random square-tiled surface of large genus into maximal horizontal cylinders. Our results imply, in particular, that with a probability which tends to 1, as genus grows, all "corners" of a random square-tiled surface live on the same horizontal and on the same vertical critical leave.
Maryam Mirzakhani has ingeniously computed frequencies of simple closed multi-geodesics of any topological type on a hyperbolic surface. Developing the results of Mirzakhani we give a detailed portrait of a random hyperbolic multi-geodesics (random multicure) on a Riemann surface of large genus.
A long standing question in the study of exotic behavior in dimension four is whether exotic behavior is “stable". For example, in thinking about the four-dimensional h-cobordism theorem, Wall proved that simply connected, exotic four-manifolds always become smoothly equivalent after applying a suitable stabilization operation enough times. Similarly, Hosokawa-Kawauchi and Baykur-Sunukjian showed that exotic surfaces become smoothly equivalent after stabilizing the surfaces some number of times. The question remains, how many stabilizations are necessary, and is one always enough? By considering certain satellite operations, we provide a negative answer to this question in the case of exotic surfaces with boundary. (This draws on joint work with Hayden, Kang, and Park).
Scalar curvature is interesting not only in analysis, geometry and topology but also in physics. For example, the positive mass theorem, which was proved by Schoen and Yau in 1979, is equivalent to the result that the three-dimension torus carries no Riemannian metric with positive scalar curvature (PSC metric). A widely open conjecture says that a closed aspherical manifold does not admit a PSC metric. I will show that the connected sum of a closed manifold and some exotic aspherical manifolds carries no PSC metric. The enlargeable length-structure and some of Prof. Tom Farrell and his coauthors' work will be used in the talk.
Z2 harmonic 1-forms was introduced by Taubes as the boundary appearing in the compactification of the moduli space of flat SL(2,C) connections. Although from gauge theory aspect, Z2 harmonic 1-forms should exist widely, it is highly challenging to explicitly construct examples of them. Besides the curvature condition, there seems to have more obstruction to the existence of Z2 harmonic 1-forms. In this talk, we will discuss a method to construct examples of Z2 harmonic 1-forms using symmetry. Moreover, we will also discuss the connection between Z2 harmonic 1-forms and special Lagrangian geometry and present a non-existence result.
The original version of Sullivan's rational surgery realization theorem provides necessary and sufficient conditions for a prescribed rational cohomology ring to be realized by a simply-connected smooth closed manifold. We will present a version of the theorem for almost complex manifolds. It has been shown there exist closed smooth manifolds \(M^n\) of Betti number \(b_i=0\) except \(b_0=b_{n/2}=b_n=1\) in certain dimensions \(n> 16\), which realize the rational cohomology ring \(\mathbb{Q}[x]/\langle x\rangle^3\) beyond the well-known projective planes of dimension 4, 8, 16. By the obstructions from the signature equation and the Riemann-Roch integrality conditions among Chern numbers, one can show that none of these manifolds with sum of Betti number three in dimension \(n>4\) can admit almost complex structure. More generally, any \(4k\) (\(k > 1\)) dimensional closed almost complex manifold with Betti number \(b_i = 0\) except \(i=0,n/2, n\) must have even signature and even Euler characteristic, one can characterize all the realizable rational cohomology rings by a set of congruence relations among the signature and Euler characteristic.
This talk aims to advertise a pattern/phenomenon that has emerged in many different mathematical areas during the past decades but is not currently well-understood. I will begin with a broad overview of the Kahler packages (Poincare duality, Hard Lefschetz, and Hodge-Riemann relations) that appear in geometry, algebra, and combinatorics, from the classics of Lefschetz to the recent work of this year's Fields medalist June Huh, in a down-to-earth way. Then I will discuss two new Kahler packages we discovered that are equivariant and have no geometric origin. The equivariant log-concavity in representation theory hints at our discoveries. This talk will be non-technical and accessible to the general audience: nothing will be assumed other than elementary linear algebra. Partly based on joint work with Rui Xiong.
In the 1980s, Neumann and Zagier introduced a symplectic vector space associated to an ideal triangulation of a cusped 3-manifold, such as a knot complement. We give an interpretation for this symplectic structure in terms of the topology of the 3-manifold, via intersections of certain curves on a Heegaard surface. We also give an algorithm to construct curves forming a symplectic basis for this vector space. This approach gives a description of hyperbolic structures on a knot complement via Ptolemy equations, which can be used to calculate the A-polynomial. This talk involves joint work with Jessica Purcell and Joshua Howie.
In this talk, I will give a brief introduction to discrete curvature notions and their motivations from Riemannian Geometry. To name a few (which arose and became popular in the last 10~20 years), there are Ollivier Ricci curvature, Bakry-Emery curvature, and Entropic Ricci curvature.
I will help you visualize curvature values in small and simple graphs via this interactive graph curvature calculator Graph Curvature (ncl.ac.uk) created by my colleagues from Newcastle upon Tyne. (And of course, you are welcome to try it beforehand). I will also tell you some stories of our discoveries by playing around this app.
Hyperbolic surfaces and flat surfaces look very different, but they're linked by a remarkable correspondence. I'll show you two versions of it: a geometric "collapsing" process that flattens hyperbolic surfaces, and a representation-theoretic "abelianization" process that diagonalizes \(\mathrm{SL}(2,\mathbb{R})\) local systems.
Let \(X\) be a compact hyperbolic surface. We can see that there is a constant \(C(X)\) such that the intersection number of the closed geodesics is bounded above by \(C(X)\) times the product of their lengths. Consider the optimum constant \(C(X)\). In this talk, we describe its asymptotic behavior in terms of systole, length of a shortest closed geodesic on \(X\).
In this talk, I will describe a joint work with Bena Tshishiku on Nielsen Realization problem for 3-manifolds, in particular, about the twist subgroup. The twist subgroup is a normal finite abelian subgroup of the mapping class group of 3-manifold, generated by the sphere twist. The proof mainly uses the geometric sphere theorem/torus theorem and geometrization.
For a translation surface, the associated saddle connection graph has saddle connections as vertices, and edges connecting pairs of non-crossing saddle connections. This can be viewed as an induced subgraph of the arc graph of the surface. In this talk, I will discuss both the fine and coarse geometry of the saddle connection graph. We show that the isometry type is rigid: any isomorphism between two such graphs is induced by an affine diffeomorphism between the underlying translation surfaces. However, the situation is completely different when one considers the quasi-isometry type: all saddle connection graphs form a single quasi-isometry class. We will also discuss the Gromov boundary in terms of foliations. This is based on joint work with Valentina Disarlo, Huiping Pan, and Anja Randecker.
Let W be a closed smooth n-manifold and W' a manifold which is homeomorphic but not diffeomorphic to W. In this talk I will discuss the extent to which W' supports the same symmetries as W when W is a n-torus or a hyperbolic manifold, and W' is the connected sum of W with an exotic n-sphere. As a sample of results, I will indicate how to classify all finite cyclic groups that act freely and smoothly on an exotic n-torus. For hyperbolic manifolds W, I will show how to produce examples of W' which admit no nontrivial smooth action of a finite group, while Isom(W) is arbitarily large. This is joint work with Bena Tshishiku.
In this talk, we consider volumes of hyperbolic 3-manifolds and construct a new distance on the Teichmüller space of a closed surface of genus >1. We will compare the new distance with other known distances: Teichmüller distance, Weil-Petersson distance. If time permits, I would also like to talk about several questions about the new distance. This talk is based on the preprint https://arxiv.org/abs/2108.06059.
We consider the space of all complete hyperbolic surfaces with basepoint equipped with the pointed Gromov-Hausdorff topology. In this talk, I will begin by motivating this topology and reviewing basic surface hyperbolic geometry. Then, I will describe certain deformations on a hyperbolic surface and concrete geometric constructions which are used to show that the space is globally path-connected and is locally weakly connected at points whose underlying surfaces are either the hyperbolic plane or hyperbolic surfaces of the first kind.
The moduli space of decorated twisted G-local systems on a marked surface, originally introduced by Fock--Goncharov, is known to have a natural cluster K_2 structure. In particular, we have a canonically defined cluster algebra A and an upper cluster algebra U inside its field of rational functions. In order to investigate the structure of the function ring of that moduli space, we introduce the Wilson lines valued in the simply-connected group G, which are “framed versions” of those studied by myself and Hironori Oya. We see that the function ring of the moduli space is generated by the matrix coefficients of Wilson iines, and some of them are cluster monomials. As an application, we prove that both A and U coincide with the function ring. Time permitting, I will also mention some relations to the skein theory. This talk is based on a joint work with Hironori Oya and Linhui Shen.
I will start the talk with basics about Higman--Thompson groups and then introduce its braided version and ribbon version.I will build a geometric model for the ribbon Higman--Thompson groups, namely as a nice subgroup for the mapping class group of a disk minus a Cantor set. We use this model to prove that the ribbon Higman--Thompson groups satisfy homological stability. This can be treated as an extension of Szymik--Wahl's work on homological stability for the Higman--Thompson groups to the surface setting.This is a joint work with Rachel Skipper.
William Thurston proposed regarding the map induced from two circle packings with the same tangency pattern as a discrete holomorphic function. A discrete analogue of the Riemann mapping is deduced from Koebe-Andreev-Thurston theorem. One question is how to extend this theory to Riemann surfaces and relate classical conformal structures to discrete conformal structures. Since circles are preserved under complex projective transformations, we consider circle packings on surfaces with complex projective structures. Kojima, Mizushima and Tan conjectured that for a given combinatorics the deformation space of circle packings is diffeomorphic to the Teichmüller space. In this talk, we explain how discrete Laplacian is used to prove the conjecture for the torus case and its connection to Weil-Petersson geometry.
Inspired by Deligne's use of the simplicial theory of hypercoverings in defining mixed Hodge structures we replace the index category \(\triangle\) by the symmetric simplicial category \(\triangle S\) and study (a class of) \(\triangle_{inj}S\)-hypercoverings, which we call spaces admitting symmetric (semi)simplicial filtration - this special class happens to have a structure of a module over a graded commutative monoid of the form \(Sym M\) for some space \(M\). For \(\triangle S\)-hypercoverings we construct a spectral sequence, somewhat like the Čech-to-derived category spectral sequence. The advantage of working on \(\triangle S\) is that all of the combinatorial complexities that come with working on \(\triangle\) are bypassed, giving simpler, unified proof of results like the computation of (in some cases, stable) singular cohomology (with \(\mathbb{Q}\) coefficients) and étale cohomology (with \(\mathbb{Q}_{\ell}\) coefficients) of the moduli space of degree \(n\) maps \(C\to\mathbb{P}^r\), \(C\) a smooth projective curve of genus \(g\), of unordered configuration spaces, of the moduli space of smooth sections of a fixed \(\mathfrak{g}_d^r\) that is \(m\)-very ample for some \(m\) etc. In the special case when a \(\triangle_{inj} S\)-object \(X_{\bullet}\) admits a symmetric semisimplicial filtration by \(M\), the derived indecomposables of \(H^*(X_{\bullet})\) as a \(H^*(Sym M)\)-module (in the sense of Galatius-Kupers-Randal-Williams) give the cohomology of the space of \(M\)-indecomposables.
A manifold which is like a projective plane is a simply-connected closed smooth manifold whose homology equals three copies of Z. In this talk I will discuss our computation of the mapping class group of these manifolds, as well as some applications in geometry. This is a joint work with WANG Wei from Shanghai Ocean University.
Let Symp(X) be the group of symplectomorphisms on a symplectic 4-manifold X. It is a classical problem in symplectic topology to study the homotopy type of Symp(X) and to compare it with the group of all diffeomorphisms on X. This problem is closely related to the existence of symplectic structures on smooth families of 4-manifolds. In this talk, we will discuss the proof of following results: (1) For any X that contains a smoothly embedded 2-sphere with self-intersection -1 or -2, there exists a loop of self-diffeomorphisms on X that is not homotopic to a loop of symplectomorphisms. (2) Consider a family of 4-manifolds obtained by resolving an ADE singularity using a hyperkahler family of complex structures, this family never support a family symplectic structure in a constant cohomology class. (3) For any non-minimal symplectic 4-manifold whose positive second-betti number does not equal to 3, the space of symplectic form is not simply connected. The key ingredient in the proofs is a new gluing formula for the family Seiberg-Witten invariant.
Asaeda, Przytycki and Sikora defined a generalization of Khovanov homology for links in thickened compact surfaces. In this talk I will show that the Asaeda-Przytycki-Sikora homology detects the unlink and torus links in the thickened torus. This is joint work with Boyu Zhang.
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.
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.
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.
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.
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.
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.
Seminar slides.
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.
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.
There's a major conference in honour of Chern happening this week, click here for some info. There's slightly more information (including the talk schedule) on wechat.
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.
Seminar notes.
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 algebraic geometry. As an example, I will talk about our recent theorem on the topological complexity of finding flex points on smooth cubic plane curves. This talk is based on joint work with Zheyan Wan 万喆彦. I will try to make the talk accessible to undergraduate students who have taken a course in algebraic topology.