View the Project on GitHub hrzsea/AMSS-Topology-Seminar-2021Autumn
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Place: N802
Title: Topological Approaches to Weighted Data
Abstract: Topology and many other areas of pure mathematics have fruitful applications in data science nowadays. Topological data analysis has become a new-born research area. In this talk, we will discuss topological approaches to weighted data. The talk will consist of five sections. In section 1, we will give brief discussions on the question why Artificial Intelligence needs mathematics, as an introduction. Then we will mainly discuss mathematics and applications of weighted topological data analysis in sections 2 and 3. In Section 4, we will give short discussions on relevant topological approaches such as path homology and hypergraph homology. Finally, in section 5, we will revisit the question why Artificial Intelligence needs mathematics by displaying statistical information from google scholar on the research articles using topological data analysis.
Place: N802
Title: Orders of the canonical vector bundles over configuration spaces of finite graphs
Abstract: We prove that the order of the canonical vector bundle over the configuration space is $2$ for a general planar graph, and is $4$ for a nonplanar graph.
This is a joint work with Frederick R. Cohen.
Place: N802
Title: An extended rational dichotomy for certain open books
Abstract: We study the rational homotopy groups of open books in terms of those of their pages and bindings. Under a so-called homotopy order condition on monodromy we prove a rational dichotomy theorem for open books, as an extension of the classical dichotomy in rational homotopy theory. As a direct application, we show that the monodromies of Milnor’s open book decompositions of odd spheres are of infinite order in general. We also calculate the integral homotopy groups of open books when their monodromies are homotopic to the identity map.
This is a joint work with Stephen Theriault.
Place: N802
Title: Log-concave conjectures on cohomology of configuration spaces and flag manifolds
Abstract: A major advance in combinatorics in recent years is the Hodge theory of combinatorial geometry (matroids) established by Adiprasito-Huh-Katz, which in turn solved a series of long-standing open log-concave conjectures. Proudfoot and his collaborators make the log-concave conjecture to the equivariant case, propose some log-concave conjectures of cohomology representations. We discuss the case of configuration spaces and our similar conjectures for flag manifolds and Springer representations and (very little) progress at present.
Place: N802
Title: Nonexistence of symplectic structures on certain family of 4-manifolds
Abstract: 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 following new 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 nonminimal 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.
Place: N802
Title: Linear Independence in the Concordance Group and Knot Floer Invariants
Abstract: The first half of this talk is expository, introducing the knot concordance group and Heegaard knot Floer theory. We will survey a few invariants arising from the knot Floer complex. Then some families of knots will be given as examples to show how the invariants are used to prove linear independence. Finally, we will discuss concordance problems about algebraic knots.
Place: N802
Title: Realising sets of integers as mapping degree sets
Abstract: reading seminar about a paper by Neofytidis-Wang-Wang with the same title.
Place: N802
Title: Self covering and fibering manifolds over circle
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Place: 晨兴110
Title: Augmentations from Legendrian knots
Abstract: In this talk, I will present a cut-and-glue approach in the study of augmentations associated to Legendrian knots. Given a Legendrian knot, this induces from relative contact homology a constructible co-sheaf of DGAs over the real line, whose global co-section recovers the Chekanov-Eliashberg DGA. It follows that, the augmentations of the Chekanov-Eliashberg DGA form the augmentation variety with a sheaf property. As an application, this gives naturally a cell decomposition for the augmentation variety. Consequently, we obtain a new proof to the formula that the E-polynomial of the variety is computed by the ruling polynomial, a combinatorially defined Legendrian isotopy invariant. Time permitting, I will also mention a second application in my recent work concerning part of the geometric P=W conjecture in nonabelian Hodge theory.
Place: N502
Title: Algebraic topology of 24 dimensional string manifolds
Abstract: String can be viewed as higher version of spin, while the latter plays a fundamental role in Atiyah-Singer index theory. People try to develop parallel theory for string, the full story of which is still mystery. Geometry and topology of string manifolds then attract increasing attentions and interests, while the ones of dimensional $24$ are quite special among string manifolds. In this talk, we will discuss the algebraic-topological aspect of 24-dimensional string manifolds, following the work of Hirzebruch, Ochanine, Landweber-Stong, Mahowald-Hopkins, Chen-Han, and very recent work of mine joint with Fei Han. We will talk about string cobordism, various index-theoretic genera and some applications.
Place: N502
Title: A symplectic field theory and its applications (II)
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Place: N502
Title: Templates embedding and immersing in R3 （博士后开题报告）
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Place: N502
Title: Lagrangian surgery formulae and projective twists.
Abstract: I will explain the construction of Lagrangian Dehn twists along a given Lagrangian sphere S in symplectic geometry, then relate them to a symplectic operation called the Lagrangian surgery; the latter constructs Lagrangian embeddings of connected sums. The relation is two-fold: from a more geometric point of view, the Dehn twist of a given Lagrangian L can be regarded as a simultaneous surgery between L and the given Lagrangian sphere S; or, motivated by mirror symmetry, they are governed by a generalized surgery along clean intersections in a product symplectic manifold. The first perspective is usually difficult to work with an generalize, we will explain how the second perspective leads to a new view of the Huybrechts-Thomas conjecture regarding projective twists in Floer theories.
Place: N502
Title: G等变稳定同伦范畴的Balmer谱的计算 （博士后开题报告）
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Place: N902
Title: A symplectic field theory and its applications (I)
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Place: 腾讯会议：110 766 998
Title: Legendrian Dualities and Geometry of Submanifolds
Abstract: Singularity and degeneracy destroy the structure of manifolds and give rise to essential difficulties in researching the deteriorative manifolds. Thus, it is crucial to develop new methods for investigating the singular or degenerate manifolds. By characterizing the inner connection between pseudo spheres in semi-Euclidean space, the Legendrian dualities are effective methods developed by L. Chen and Izumiya in 2009 for studying the submanifolds in non-flat space. Specially, the Legendrian duality is applicable for studying the singular or degenerate submanifolds. In this talk, we will introduce the basic notions of Legendrian dualities. As their applications, using the singularity theory of mappings, we investigate the topological and geometrical properties of the (singular or degenerate) submanifolds immersed in non-flat space from the viewpoint of duality.