View the Project on GitHub hrzsea/AMSS-Topology-Seminar-2022Autumn
（return to homepage of AMSS Topology Seminar）
Place: 747-737-156
Title: The Stable Picard Group of A(n)
Abstract: In 1976, Adams calculated the Picard group of A(1) in “Uniqueness of BSO”, and it was not until 2017 that the conclusion of A(2) appeared. In the case of n greater than 2, it was initially impossible to determine whether its Picard group is finitely generated. In this talk, the Picard groups for all A(n) will be given. The method relies on reductions from a Hopf algebra to its proper Hopf subalgebras.
Place: 898 465 270
Title: Borel conjecture for tori
Abstract: The Borel conjecture considers the obstruction from homotopy equivalence to homeomorphism for aspherical manifolds. The torus is the first computed case of Borel conjecture with the idea of splitting used in the computation. In this talk I will review the basic results of surgery theory and prove the Borel conjecture for torus following Shanneson and Hsiang’s calculation. Moreover, the further Farrell’s splitting for structure set gives us further tools in the proof of more complicated case of Borel conjecture.
Place: 306-836-756
Title: Symplectic fillings of lens spaces and Seifert fibered spaces
Abstract: In this talk, we apply Menke’s JSJ decomposition for symplectic fillings to several families of contact 3-manifolds. Among other results, we complete the classification up to orientation-preserving diffeomorphism of strong symplectic fillings of lens spaces. For large families of contact structures on Seifert fibered spaces over S^2, we reduce the problem of classifying exact symplectic fillings to the same problem for universally tight or canonical contact structures. We show that exact symplectic fillings of contact manifolds obtained by surgery on certain Legendrian negative cables are the result of attaching a symplectic 2-handle to an exact symplectic filling of a lens space. This is joint work with Austin Christian.
Place: 腾讯会议：6965463074
Title: Suspension Homotopy of 4-manifolds And the Second 2-local Cohomotopy Sets
Abstract: In this talk we study the homotopy type of the (double) suspension of an orientable, closed, connected 4-manifold M, whose integral homology can have 2-torsion. Moreover, the decomposition results are applied to give a partial characterization of the second 2-local cohomotopy set of 4-manifolds.
Place: 腾讯会议：468-144-607
Title: Homological instability for the moduli space of smooth 4-manifolds
Abstract: The moduli space of a smooth manifold X is defined to be the classifying space of its diffeomorphism group. Understanding the cohomology group of this space is important because elements in this group one-to-one correspond to characteristic classes for smooth bundles with fiber X. A celebrated result of Harer states that homology groups of the moduli spaces of Riemann surfaces stabilize if one fixes a degree and increases the genus. Galatius and Randal-Williams established analogous homological stability for moduli spaces of manifolds of even dimension at least 6. In this talk, we will show that homological stability fails for the moduli space of any simply-connected closed smooth 4-manifold in any degree of homology. The central tool is a characteristic class constructed using Seiberg-Witten equations, which detects the subtle difference between the topological category and the smooth category of 4-manifolds. This is a joint work with Hokuto Konno.
Place: 701 801 036
Title: Linked periodic orbits of disc homeomorphisms
Abstract: In 1990, Gambaudo introduced the notion of linking of two invariant sets of a surface self-map. Most of the known results of linked periodic orbits are about an orbit linked with a fixed point. In this talk, based on the forcing relation of braids, we present a method for finding periodic orbits which are linked with a given periodic orbit of an orientation-preserving homeomorphism of the disc. New examples are provided.
Place: N533
Title: Legendrian knots and cluster algebras via augmentations
Abstract: Legendrian knots and their exact Lagrangian fillings are central objects to study in low dimensional contact and symplectic topology. In recent developments, cluster algebras have proven to be powerful tools to classify Lagrangian fillings. For a positive braid link, we introduce a cluster K2 structure on its augmentation variety. Using the perspective of Ekholm-Honda-Kalman theory, we prove that admissible exact Lagrangian fillings, a subset of decomposable ones, induce cluster seeds in the cluster K2 augmentation variety. We provide an algorithm to compute these cluster seeds. We also use the cluster Donaldson-Thomas transformation to produce infinitely many Lagrangian fillings. This is a joint work with L. Shen and D. Weng.
Place: N533
Title: Degree one maps between 4-manifolds with cyclic fundament groups
Abstract: I will present some results on the existence and finiteness of degree one maps between 4-manifolds with cyclic fundamental groups, as well as the relation between 1-domination and Euler characteristics. This is based on a joint work with Yang Su and Shicheng Wang.
Place: N533
Title: Rigidity and vanishing of elliptic genera of complex manifolds
Abstract:
Place: N533
Title: Equivariant cohomology, localization formula and moment graph
Abstract:
Place: 腾讯会议 295965402
Title: Triangulated persistence category in symplectic geometry
Abstract: In this talk, we will introduce a new algebraic structure called triangulated persistence category (TPC). A TPC combines the persistence module structure (from topological data analysis) and the classical triangulated structure so that a meaningful measurement, via cone decomposition, can be defined on the set of objects. Moreover, a TPC structure allows us to define non-trivial pseudo-metrics on its Grothendieck group, which is the first time that people can study a Grothendieck group in terms of the metric geometry. Finally, we will illustrate several unexpected properties of a TPC via its supporting example in symplectic geometry, the derived Fukaya category. In particular, we can distinguish classes in the Grothendieck group of a derived Fukaya category from a quantitative perspective. This is based on joint work with Paul Biran and Octav Cornea.
Place: N802
Title: Shadows of Hodge Theory (or: Never Underestimate Symmetry!)
Abstract: 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.
Place: N802
Title: Free circle actions on highly connected (2n+1)- manifolds
Abstract: A natural problem in topology is to determine which manifolds admit certain group actions. The problem we concern in this talk is to determine which highly connected (2n+1)-manifolds admit free circle actions. I will introduce some previous work and our progress on this problem.