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Place: N818
Title: Generation of 3-manifolds through surgery triads
Abstract: Surgery triads is an important concept in various versions of Floer homology theories of 3-manifolds. If three 3-manifolds form a surgery triad, then the Floer homology of these three manifolds are usually related by an exact triangle. A strategy of proving certain isomorphism theorems for Floer homology is to use exact triangles and naturality. In order to apply this strategy, we often need finite generation results for a class of 3-manifolds through surgery triads. For example, a folklore theorem asserts that all closed oriented connected 3-manifolds are generated by the 3-sphere through surgery triads.In this talk, we will review the above strategy and discuss the proof of the above folklore theorem and its generalization to a class of sutured manifolds.
Place: MCM110
Title: Integral measure equivalence versus quasiisometry for some right-angled Artin groups
Abstract: Two finitely generated groups G and H are quasi-isometric (QI), if they admit a topological coupling, i.e. an action of G times H on a locally compact topological space such that each factor acts properly and cocompactly. This topological definition of quasi-isometry was given by Gromov, and at the same time he proposed a measure theoretic analogue of this definition, called the measure equivalence, which is closely related to the notion of orbit equivalence in ergodic theory. Despite the similarity in the definition of measure equivalence and quasi-isometry, their relationship is rather mysterious and not well-understood. We will start with a discussion of an interesting similarity between some QI invariants and ME invariants for general cubical groups. Then we look at the case of right-angled Artin groups. We show that if H is a countable group with bounded torsion which is integrable measure equivalence to a right-angled Artin group G with finite outer automorphism group, then H is finitely generated, and H and G are quasi-isometric. This allows us to deduce integrable measure equivalence rigidity results from the relevant quasi-isometric rigidity results for a large class of right-angled Artin groups. This is perhaps the first instance a rigidity result in the ME side is obtained via establishing quasi-isometry. This is joint work with Camille Horbez.
Place: N818
Title: Homology of random subgroups of mapping class groups
Abstract:
Place: N818
Title: Quasi-conformal homeomorphisms of Morse boundaries
Abstract: Boundaries of hyperbolic groups have played an important role in geometric group theory. By restricting to Morse geodesic rays, one can define an analogous boundary for more general groups. Inspired by a theorem of Paulin, we investigate precise conditions for when a homeomorphism between the Morse boundaries of two groups is induced by a quasi-isometry of the groups themselves. In this talk, I will discuss the quasi-conformal homeomorphisms of Morse boundaries.
Place: N818
Title: From nonhyperbolic 3-manifold to premodular tensor category
Abstract: In this talk，I will introduce a connection between nonhyperbolic 3-manifolds and premodular tensor categories. I will talk about a method to use Chern-Simons invariants and Reidemeister torsions to construct modular data.
Place: N818
Title: Free Circle Action on Certain Simply Connected 7-manifolds
Abstract: Manifold classification is a fundamental problem in topology. In higher dimensions, complete classifications are often intractable, motivating the study of manifolds with special structures or significant connections to other areas of mathematics.This talk concerns manifolds admitting a free smooth circle action. I will begin with an overview of existing classification results and applications, then present for which nonnegative integers k, l and for which homotopy 7−sphere Σ the manifold (k S^{2}× S^{5})#(l S^{3}× S^{4})#Σ admits a free smooth circle action.
Place: N818
Title: Immersed Lagrangian Floer theory, Lagrangain compositions, and bounding cochains on surfaces
Abstract: In this talk, I will first give the background of immersed Lagrangian Floer theory based on character varieties. Then I will focus on the relation between immersed quilted Lagrangian Floer theory and the usual immersed Lagrangian Floer theory when they are related by a Lagrangian correspondence, in the topological aspect. I will use many examples to explore this relation. If time permitted, I will give some examples about bounding cochains.
Place: N702
Title: Chern numbers on positive vector bundles and combinatorics
Abstract: 经由Griffiths(1960s)，Fulton-Lazarsfeld(1980s)和Demailly-Peternell-Schneider(1990s)等的基本工作我们知道正向量丛的Chern数满足很强的限制条件。他们的工作也遗留了几个基本的问题。在这个报告中我们将回顾这些相关工作并讲讲最近的一些进展。
Place: N818
Title: The homotopy theory of connected sums with projective spaces
Abstract: Let M be a simply-connected 2n-dimensional Poincare Duality complex. We describe a homotopy decomposition of the based loops on the connected sum of M and a 2n-dimensional complex projective space that holds integrally if n is even and localized away from 2 if n is odd. This generalizes results of Duan that relied on geometric methods; our proof is purely homotopy theoretic. This is joint work with Ruizhi Huang.
Place: N818
Title: Properties of Poincare Duality complexes - from rational to integral
Abstract: A provocative theorem of Halperin and Lemaire states that if M is a simply-connected Poincare Duality complex of dimension n and the rational cohomology of M is not generated as an algebra by a single element, then the based loops on M retracts off the based loops on its (n-1)-skeleton. This implies that the attaching map for the n-cell of M kills off homotopy groups but does not introduce any new ones. It is rare that a cell attachment has this property and it is surprising that it holds in such generality for Poincare Duality complexes.In this talk we show that this theorem secretly originates in integral homotopy theory. We give conditions for when the based loops on M integrally retracts off the based loops on its (n-1)-skeleton, and use these to recover the rational result.
Place: N818
Title: The homotopy theory of Poincare Duality complexes
Abstract: A Poincare Duality complex is a CW-complex M whose cohomology satisfies Poincare Duality, meaning it is a topological generalization of an oriented manifold. The goal is to gain insight into the homotopy theory of M by determining properties of its based loop space. An approach will be outlined that has had success for several interesting families of Poincare Duality complexes.
Place: N818
Title: Symplectic fillings of unit cotangent bundles of spheres
Abstract: This talk is concerned with symplectic fillings of contact manifolds. After briefly giving an overview of results on the topology of symplectic fillings, I will present joint work with Myeonggi Kwon, where we give some topological restrictions on symplectic fillings of unit cotangent bundles of spheres. In particular, we show a uniqueness result on diffeomorphism types of them, under certain condition, for the case where the base manifold is a 3-sphere. Additionally, I will explain an application of our result to exact symplectic cobordisms.
Place: N818
Title: Real Bott periodicity at higher chromatic heights
Abstract: Real Bott periodicity shows that the homotopy groups of Real topological K-theory is 8-periodic. From chromatic homotopy theory perspective, Real topological K-theory is a height 1 theory, and for each natural number n, there are height n periodic cohomology theories. Important examples includes 192 periodic topological modular forms at height 2 and Hill—Hopkins—Ravenel’s 256 periodic Kervarie detection spectrum at height 4. In this talk, we give a general formula of the periodicity for all heights. This is joint work in progress with Zhipeng Duan, Mike Hill, Yutao Liu, Danny XiaoLin Shi, Guozhen Wang, and Zhouli Xu.
Place: N818
Title: Knot invariants in thickened surfaces derived from chord index
Abstract: An effective method to construct a knot invariant is to sum up certain weighted crossings (chord index) in a knot diagram. In this talk, we demonstrate a way to define chord indices in thickened surfaces. Specifically, for a given closed orientated surface , we introduce a chord index homomorphism from a subgroup of H1(, Z) to the group of chord indices of a knot K in I. Some knot invariants can be derived from this homomorphism.