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Title: Progress on the study of Gauged linear Sigma models
Abstract: Gauged linear sigma model (GLSM) proposed by Witten nearly thirty years ago was used to explain the mirror symmetry phenomena. GLSM relates to many important conjectures, like Landau-Ginzburg A model mirror to Landau-Ginzburg B model, Calabi-Yau model to Landau-Ginzburg model correspondence. GLSM has also intimate relation with GIT and can also be formulated in category theory. This report gives a short survey on the recent progress on this topic.
Title: Instanton homology and knot detection on thickened surfaces
Abstract: In this talk, we will characterize all knots in product sutured manifolds that have minimal sutured instanton homology. As an application, we will show that the Asaeda-Przytycki-Sikora homology of a knot K in a thickened genus zero surface has rank 2 if and only if K is isotopic to an embedded knot in a slice of the surface. A variant of Asaeda-Przytycki-Sikora will be discussed. This is joint work with Zhenkun Li and Boyu Zhang.
Title: Dynamics of composite symplectic Dehn twists
Abstract: It is classically known in Nielson-Thurston theory that the mapping class group of a hyperbolic surface is generated by Dehn twists and most elements are pseudo Anosov. Pseudo Anosov elements are interesting dynamical objects. They are featured by positive topological entropy and two invariant singular foliations expanded or contracted by the dynamics. We explore a generalization of these ideas to symplectic mapping class groups. With the symplectic Dehn twists along Lagrangian spheres introduced by Arnold and Seidel, we show in various settings that the compositions of such twists has features of pseudo Anosov elements, such as positive topological entropy, invariant stable and unstable laminitions, exponential growth of Floer homology group, etc. This is a joint work with Wenmin Gong and Zhijing Wang.
Title: Symplectic Torelli groups of log Calabi-Yau surfaces
Abstract: The symplectic Torelli group of a symplectic manifold is formed by homotopy classes of homologically trivial symplectomorphisms. For log Calabi-Yau surfaces, such symplectomorphisms have the trivial smooth isotopy class, hence the symplectic Torelli group exactly captures the exotic symplectomorphisms. A question of Donaldson asks whether Lagrangian Dehn twists generate symplectic Torelli groups. In this talk, we explain how to compute the symplectic Torelli group for all but three symplectic surfaces which support log Calabi-Yau structures. This leads to an affirmative answer to Donaldson’s question and proves that all toric surfaces have trivial symplectic Torelli groups. As a by-product, we show the existence of a Hamiltonian Z/2-action on a symplectic four manifold which does not extend to an S^1-action. This is joint work with Tian-Jun Li and Jun Li.
Title: Noncommutative Gröbner basis and Ext groups
Abstract: The Gröbner basis is a powerful tool in commutative algebra. We can use it to do many calculations such as computing the presentations of the kernel and cokernel of a map between finitely presented modules over a commutative algebra. However, many important algebras including the Steenrod algebra in algebraic topology are not commutative. We make a noncommutative generalization of the Gröbner basis which can be applied to the Steenrod algebra A. This leads to highly efficient calculations in the category of A-modules including the computation of E2 pages of Adams spectral sequences.
Title: On Schubert’s Problem of Characteristics
Abstract: Hilbert’s 15th problem called for a rigorous foundation of Schubert calculus, in which a long standing and challenging part is the problem of characteristics. In the course of securing the foundation of algebraic geometry, Van der Waerden and Andre Weil attributed the problem to the intersection theory of flag manifolds. This talk surveys the background, content, and solution of the problem of characteristics. Our main results are a unified formula for the characteristics, and a system description for the intersection rings of flag manifolds. We illustrate the effectiveness of the formula and the algorithm by explicit examples.
Title: 拓扑和数论间的一些联系
Abstract: 我们将讨论由研究流形的映射度和手性而引发的拓扑和数论之间的一些问题和相互作用。
Title: Equivariant formality of corank-1 homogeneous spaces and rational products of spheres
Abstract: For a flag manifold G/T, or more generally G/H with rank(G) = rank(H), it is well known that the rational cohomology concentrates in even degrees, so the isotropy action of H on G/H is always equivariantly formal. In this talk, we will completely characterize whether a G/H with rank(G) - rank(H)=1 is equivariantly formal. The analysis requires us to correct and extend an existing partial classification of homogeneous spaces G/K with the rational homotopy type of a product of an odd- and an even-dimensional sphere. Joint work with J. Carlson, arXiv:2204.00135.
Title: Cubic forms, anomaly cancellation and modularity
Abstract: Recently Freed and Hopkins developed an algebraic theory of cubic forms, which is an analogy to the theory of quadratic forms in topology. They are motivated by the Witten-Freed-Hopkins anomaly cancellation formula in M-theory, which equals a cubic form arising from an E8 bundle over a 12 dimensional spin manifold to the indices of twisted Dirac operators on the manifold. In this talk, we will first review the Witten-Freed-Hopkins anomaly cancellation formula and the algebraic theory of cubic forms, and then show that the cubic forms as well as the anomaly cancellation formula can be naturally derived from modular forms that we construct inspired by the Witten genus and the basic representation of affine E8. Following this approach, we obtain new cubic forms and anomaly cancellation formulas on non-spin manifolds and thus provide a unified way to obtain anomaly cancellation formulas of this type. This is based on our joint work with Prof. Ruizhi Huang, Prof. Kefeng Liu and Prof. Weiping Zhang.
Title: The Hirzebruch genera, symmetric functions and multiple zeta values
Abstract: In this talk we first review several classical oriented and complex genera, and then explain how to derive the coefficients in front of the characteristic numbers, where the symmetric function theory and mutiple-(star) zeta values are involved.
Title: Heegaard Floer homology and the fundamental group
Abstract: Heegaard Floer homology was introduced in 2000 by Peter Ozsvath and Zoltan Szabo. It has many applications in the low dimensional topology. In this talk, we will introduce the Heegaard Floer theory and its applications, and the relation between Heegaard Floer homology and the fundamental group. In particular, we show that the hat version Heegaard Floer homology of a (1,1) knot is determined by a particular presentation of its fundamental group. This is joint work with Xiliu Yang.
Title: On embedding periodic maps of surfaces into those of $S^m$
Abstract: It is known that in the smooth orientable category any periodic map of order $n$ on a closed surface of genus $g$ can extend periodically over some $m$-dimensional sphere with respect to an equivariant embedding. We will determine the smallest possible $m$ when $n\geq 3g$. We will also show that for each integer $k>1$ there exist infinitely many periodic maps such that the smallest possible $m$ is equal to $k$. This is a joint work with Chao Wang.
Title: Mathematical AI for molecular data analysis
Abstract: Artificial intelligence (AI) based molecular data analysis has begun to gain momentum due to the great advancement in experimental data, computational power and learning models. However, a major issue that remains for all AI-based learning models is the efficient molecular representations and featurization. Here we propose advanced mathematics-based molecular representations and featurization (or feature engineering). Molecular structures and their interactions are represented as various simplicial complexes (Rips complex, Neighborhood complex, Dowker complex, and Hom-complex), hypergraphs, and Tor-algebra-based models. Molecular descriptors are systematically generated from various persistent invariants, including persistent homology, persistent Ricci curvature, persistent spectral, and persistent Tor-algebra. These features are combined with machine learning and deep learning models, including random forest, CNN, RNN, GNN, Transformer, BERT, and others. They have demonstrated great advantage over traditional models in drug design and material informatics.
Title: Legendrian knots and Lagrangian fillings
Abstract: Legendrian knots and their exact Lagrangian fillings are center of the low dimensional contact and symplectic topology. They can be used to classify 3 dimensional contact structures and 4 dimensional Weinstein manifolds. In this talk, I will explain the geometric construction of exact Lagrangian fillings and the algebraic machinery to distinguish them. This is a joint work with Roger Casals.
Title: Equivariant computation of homotopy groups of topological modular forms
Abstract: The computation of homotopy groups of topological modular forms usually needs nontrivial topology information. In this talk, we present a new equivariant approach of the 2-primary computation. This new approach uses more algebraic input and provides richer information. The complete computation is determined by equivariant structure maps and the computation of subgroup cases. In particular, the computation avoids the use of Toda brackets. This is joint work with Zhipeng Duan, Hana Jia Kong, Yunze Lu, and Guozhen Wang.
Title: On moduli spaces of smooth 4-manifolds
Abstract: The moduli space of a smooth manifold is defined as the classifying space of its diffeomorphism group. Understanding the homotopy type of this space helps us to classify families of manifolds. In this talk, I will discuss some new properties of the moduli spaces of 4-manifolds. Some of them are special in dimension 4 (e.g. the homological instability phenomena), while some of them also appear in higher dimensions (e.g. a discrepancy between the smooth moduli space and the topological moduli space). The talk is based on a joint work with Hokuto Konno and a joint work with Yi Xie.
Title: Gromov norm on nonpositively curved manifolds
Abstract: Given a topological space, the Gromov norm of a singular real homology class measures certain topological complexity. In the case of connected closed oriented manifolds, the simplicial volume is defined as the Gromov norm of the fundamental class. For hyperbolic manifolds, the simplicial volume is proportional to its hyperbolic volume and it is in particular positive. In this talk, I will discuss some positivity results on the Gromov norm for certain nonpositively curved manifolds.
Title: 等变形式的$\mathbb{Z}_2$环面流形
Abstract: 空间中等变形式的(equivariantly formal)群作用是拓扑变换群理论中重要的研究对象，在几何学、拓扑学和代数几何学中都有广泛的应用。$\mathbb{Z}_2$环面流形($2$-torus manifold)是具有光滑的(非自由的)有效$\mathbb{Z}_2^n$群作用的连通闭$n$维流形。本报告将首先介绍拓扑变换群理论中一些基本的概念和构造，然后介绍如何用$\mathbb{Z}_2^n$群作用的轨道空间的拓扑性质来判断一个$\mathbb{Z}_2$环面流形什么时候是等变形式的，其证明涉及到等变上同调的局部化和$\mathbb{Z}_2$环面群作用的GKM理论等内容。该研究成果平行于Masuda-Panov对具有平凡奇数维上同调的环面流形(torus manifold)的研究。另外，利用该结果我们可以决定什么样的$\mathbb{Z}_2$环面流形上存在正规的极大对合(即不动点集是离散集合且不动点个数达到极大的对合)。
Title: New approaches to discovering symplectic non-convexity
Abstract: In this talk, we will provide new examples of star-shaped (toric) domains in \C^2 that are dynamically convex but not symplectically convex. Our examples are based on two approaches: one is from Chaidez-Edtmair’s criterion via Ruelle invariant and systolic ratio; the other is from the ECH capacities and an analog non-linear version of Banach-Mazur distance in symplectic geometry. In particular, from the second approach, we derive the first family of examples that can be numerically verified (instead of taking a certain limit from the first approach). We will also illustrate that the information given by these two approaches is in general independent of each other. This talk is based on joint work with Dardennes, Gutt, and Ramos.
Title: Involution on pseudoisotopy spaces
Abstract: The involution on pseudoisotopy spaces is closely related to the homotopy type of the diffeomorphism group of a smooth compact manifold. In this talk, we will introduce some background, a result on computing the involution on pseudoisotopy spaces and its application to the space of nonnegatively curved metrics on open manifolds. This is joint work with Mauricio Bustamante and Francis Thomas Farrell.
Title: Topology of Complex Network
Abstract: High-order interaction is the most challenging in complex network. Algebraic topology might represent the next revolution in high-order complex network science. In this talk, we will give an introduction to the topological approaches to (di-)graphs and hypergraphs beyond topological data analysis (TDA), including GLMY homology introduced by S. T. Yau et al, and it generalizations. After reviewing some topological approaches on the subject, we will report some of our recent progress on the applications of GLMY theory in biology and materials. Moreover, we will discuss the homotopy theory of digraphs and its connections to the fundamental structures and dynamics of complex network.
Title: The Homotopy Group and Puppe Sequence of Digraphs
Abstract: High-order structure in (di-)graphs plays an important role in many kinds of applied fields. The homology and homotopy theory of (di-)graphs can help us to capture the high-dimensional structure in (di-)graphs. Prof. Yau et al introduced the GLMY homology theory of digraphs in 2013, which has already got a lot of effective results in many different applied fields. Moreover, they also established homotopy theory of digraphs. In this talk, based on the homotopy theory of digraphs established by them, we aim to give an intuitive descpition for the homotopy group of digraphs, i.e. the network description for the homotopy group of digraphs. Furthermore, we continue exploring the Puppe sequence of digraphs for any based digragh map.