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Place: N702
Title: Algebraic Model of Manifolds
Abstract: The surgery theory of dimension at least 5 stayed in the central position of classification of manifolds in the last century. In 1970s, Ranicki invented the cosheaf theory of derived quadratic forms to give an algebraic discription of the existence of high dimensional manifolds over a fixed homotopy type and the homotopy manifold classifications. On the other hand, we could understand a homotopy type by passage to rational and p-adic information. Quillen and Sullivan individually developed the rational homotopy theories. Sullivan sketched the approach to algebraize simply connected rational manifolds, i.e., a free dga over Q together with a Poincare duality and several rational Pontryagin classes, and Zhixu Su completed the proof in her thesis. About two decades ago, Mandell proved that E-inifinity algebras over Fp-bar canb be the algebraic models of p-adic nilpotent homotopy types. Five years ago, Rivera and Zeinalian showed that the cobar construction of coalgebras could give algebraic models of integral homotopy types. We want to generalize Sullivan’s rational discussion of manifolds to give algebraic models of manifolds. The project is still in progress but I will talk about our recent work of 2-adic discussion of Ranicki’s theory and one application of this powerful theory to understand Galois actions on smooth complex varieties.
Place: N702
Title: Slope detections and the L-space Conjecture for toroidal 3-manifolds
Abstract: The L-space conjecture connects topological (taut foliations), analytic (Floer invariants), and algebraic (orderability of groups) properties of 3-manifolds, which has been a very popular topic in low dimensional topology. In this talk, we will introduce the notion of slope detections, and show results to demonstrate how slope detection is powerful notion to study the L-space conjecture for toroidal 3-manifolds.
This is joint work with Steve Boyer and Cameron Gordon.
Place: N702
Title: On the loop spaces of certain polyhedral products
Abstract: A polyhedral product X of given topological spaces is a special subspace of their cartesian product, which is determined by a simplicial complex K. As a special case when K is a disjoint union of its vertices, X is a wedge sum. Moreover, every moment-angle manifold is homotopy equivalent to the homotopy fiber F of the inclusion of such a polyhedral product into the corresponding cartesian product. In this talk we show some recent progress on the homotopy and homology the loop spaces of general X and F, together with some relationships with simplicial groups and Hopf algebras.
Place: N702
Title: From Stanley-Reisner Ring to Topological Face Ring
Abstract: We introduce how the Stanley-Reisner ring of a simplicial complex can be understood from algebraic topology viewpoints via some spaces arising in toric topology. This reveals some interesting connections between commutative algebras and algebraic topology. Then we explain how the topological interpretation of Stanley-Reisner rings leads to the definition of topological face ring of a nice manifold with corners (and more generally a space with abstract faces).
Place: MCM110
Title: String topology, cyclic homology, and the Fukaya A-infinity algebra Abstract: First, I will describe a new chain model of the (based and free) loop space of a path-connected topological space X, defined using the fundamental groupoid of X, which can be viewed as a generalization of classical theorems of J. F. Adams and K-T Chen. Then I will combine this model with a Jones’ type theorem on cyclic homology and S^1 equivariant homology, as well as K. Irie’s work on string topology, to describe chain level string topology operations in the S^1-equivariant setting, in particular, chain level string bracket (cyclic loop bracket). Finally, I will use this chain model to lift the Fukaya A-infinity algebra of a Lagrangian submanifold L to a Maurer-Cartan element in the dg Lie algebra of cyclic invariant chains on the free loop space of L, and discuss applications in symplectic topology.
Place: N702
Title: 广义构型空间在Borsuk-Ulam定理推广上的应用
Abstract: 令 M 为向量空间或者射影空间的子集。报告人及其合作者定义了 M 的广义构型空间，该空间由 M 中元素的 n-元组构成，要求其中任意 k 个元素线性无关。 作为广义构型空间的应用，报告人推广了经典的Borsuk-Ulam 定理。报告人证明了，对于某些连续映射，存在着另外一对非对径点在映射下的像相同。这给出了经典Borsuk-Ulam 定理与Yang-Bourgin 定理的更强形式，这其中利用的主要工具为Fadell定义的取值在理想的上同调指标。实际上，通过利用这个指标，我们同样得到了一些条件，来判断哪些点在球面到欧氏空间映射下的像相同。
Place: N702
Title: Rationalization theories for non-nilpotent spaces and groups
Abstract: We give an overview of five rationalization theories for non-nilpotent spaces (Bousfield–Kan’s Q-completion; Sullivan’s rationalization; Bousfield’s homology rationalization; Casacuberta– Peschke’sΩ-rationalization; G´omez-Tato–Halperin–Tanr´e’s π_1- fiberwise rationalization) that extend the classical rationalization of simply connected spaces. We also give an overview of the corresponding rationalization theories for groups (Q-completion; HQ-localization; Baumslag rationalization) that extend the classical Malcev completion. In addition, we will tell about our results on the rationalization of the wedge of two circles for all these types of rationalisations.
Place: N702
Title: The earthquake metric
Abstract: Earthquakes are natural generalisations of Fenchel-Nielsen twists deformations on Teichmueller space, and Thurston’s remarkable earthquake theorem asserts that any hyperbolic metric on a given closed surface can be deformed to any other by a unique (left) earthquake. This was famously employed by Kerckhoff in his proof of the Nielsen realisation problem, which quickly cemented their importance in Teichmüller theory. Geometrically speaking, however, (long) Earthquake paths are far from being “twist efficient” - indeed, Mirzakhani shows that earthquake flows on Teichmüller space are measure conjugate to the horocyclic flow. Motivated by wishing to understand how one might efficiently “earthquake” between hyperbolic structures, we initiate the first systematic study of the earthquake metric—a Finsler metric first introduced in Thurston’s “Minimal stretch maps between hyperbolic surfaces” preprint, and discover surprising connections to both the Thurston metric and the Weil-Petersson metric. This is work in collaboration with K. Ohshika, H. Pan and A. Papadopoulos.
Place: N702
Title: Recent developments on random hyperbolic surfaces of large genus
Abstract: In this talk, we report several very recent asymptotic results on certain classical geometric quantities viewed as random variables on the moduli space of Riemann surfaces for large genus (and many cusps). This field was initiated by M. Mirzakhani in 2013. This talk is based on several joint works with Hugo Parlier, Xin Nie, Yang Shen and Yuhao Xue.
Place: 781-789-068
Title: Unstable homotopy groups of indecomposable A_3^2-complexes
Abstract: In this talk, I will introduce the methods for calculating the unstable homotopy groups of finite CW-complexes which builds on works of the relative James construction initiated by B. Gray in 1973 and further developed by Yang, Mukai and Wu recently. Then I will introduce our works of using these methods to calculate the unstable homotopy groups of indecomposable A_n^2-complexes, which are elementary indecompsable homotopy type classified by by Chang in the 1950s.
Place: N802
Title: Meromorphic differentials and Teichmüller space
Abstract: 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 where one replaces holomorphic differentials by meromorphic differentials. I will explain how this fits into an even more general story involving spaces of stability conditions and cluster varieties.
Place: N802
Title: General Blue-Shift Phenomenon and Generalized Relations of Roots and Coefficients of a Polynomial （II）
Abstract: same as that in Part I
Place: N802
Title: Higher dimensional Heegaard Floer homology and Hecke algebras
Abstract: Higher dimensional Heegaard Floer homology (HDHF) is a higher dimensional analogue of Heegaard Floer homology in dimension three. It’s partly used to study contact topology in higher dimensions. In a special case, it’s related to symplectic Khovanov homology. In this talk, we discuss HDHF of cotangent fibers of the cotangent bundle of an oriented surface and show that it is isomorphic to various Hecke algebras. This is a joint work with Ko Honda and Tianyu Yuan.
Place: N802
Title: General Blue-Shift Phenomenon and Generalized Relations of Roots and Coefficients of a Polynomial （I）
Abstract: In chromatic homotopy theory, there is a well-known conjecture called blue-shift phenomenon (BSP). Recently, Balmer-Sanders and Barthel-Hausmann-Naumann-Nikolaus-Noel-Stapleton showed that a new BSP is closely related to the Zariski topology of Balmer spectrum of the category of compact genuine A-spectra for a finite abelian group A. To unify these two BSP to one framework, we propose a general blue-shift phenomenon (GBSP) in this paper and have a new idea to explain it in a more conceptual way. To carry out our idea, we use the roots of p^j-series of formal group law of a complex oriented spectrum E in the homotopy group of the generalized Tate spectrum of E originally due to the seminal paper of Hopkins-Kuhn-Ravenel. This motivates us to go further to study the relation of roots and coefficients of a polynomial in a commutative ring R. And we propose a notion called n-tuple of a polynomial in R to obtain generalized relations of roots and coefficients of this polynomial in R. These generalized relations have a broad application prospect in reducing the relations of R, especially they play an extremely important role in explaining GBSP. By taking this brand-new approach, we successfully achieve our idea of the explanation of GBSP for some abelian cases, and obtain that the generalized Tate construction lowers Bousfield class along with many Tate vanishing results. This strengthens and extends previous theorems of Balmer-Sanders and Ando-Morava-Sadofsky. Though our approach could only recover Barthel-Hausmann-Naumann-Nikolaus-Noel-Stapleton, it seems more accessible to deal with GBSP for non-abelian cases. Besides, our approach greatly simplifies the original proof of Bonventre-Guillou-Stapleton (arXiv:2204.03797), which showed that its applications are not restricted to GBSP. Thus our approach deserves more applications and further study.