Beijing Institute of Mathematical Sciences and Applications

Given two complex topological bundles over $\mathbb CP^n$, one can ask whether the bundles are topologically equivalent. The first test is to compare their Chern classes, since equivalent bundles must have the same Chern data. The converse fails in general, which leads to the following question: given $n$ and $k$ positive integers, what invariants beyond Chern classes are needed to distinguish complex rank $k$ topological bundles on $\mathbb CP^n$, up to topological equivalence? In this talk, I will discuss the subtleties of using methods from stable homotopy theory to answer this question. I'll start by explaining how Atiyah--Rees classified all complex rank 2 topological vector bundles on $\mathbb CP^3$ via an invariant valued in the generalized cohomology theory of real K theory. I will then discuss my work classifying complex rank 3 topological vector bundles on $\mathbb CP^5$ using a generalized cohomology theory called topological modular forms. As time allows, I will discuss work in progress (joint with Hood Chatham and Yang Hu) to address other ranks and dimensions.

In this talk, we introduce a wide framework for studying magnitude of metric spaces. That is, a theory of categories enriched over filtered sets, which contains all generalized metric spaces and all small categories. We discuss the homology and the Euler characteristic (magnitude) of filtered set enriched categories. Such a homology theory contains category homology, magnitude homology and the path homology. We also give some applications to the digraph theory from this view point.

We develop a theory of covering digraphs, similar to the theory of covering spaces. By applying this theory to Cayley digraphs, we build a ``bridge'' between GLMY-theory and group homology theory, which helps to reduce path homology calculations to group homology with coefficients in certain modules. We show some cases where this approach allows us to fully express path homology in terms of group homology. To illustrate this method, we provide a path homology computation for the Cayley digraph of the additive group of rational numbers.

We define the notion of a Hodge Laplacian acting on the spaces of $\partial$-invariant paths. We also state some results of about the spectrum of the Hodge Laplacian.

The Gromov-Hausdorff distance between two metric spaces is an important tool in geometry, but it is difficult to compute. I will show how to provide new lower bounds on the Gromov-Hausdorff distance between unit spheres of different dimensions by combining Vietoris-Rips complexes with Borsuk-Ulam theorems. This is joint work with Johnathan Bush, Nate Clause, Florian Frick, Mario Gómez, Michael Harrison, R. Amzi Jeffs, Evgeniya Lagoda, Sunhyuk Lim, Facundo Mémoli, Michael Moy, Nikola Sadovek, Matt Superdock, Daniel Vargas, Qingsong Wang, Ling Zhou, available at https://arxiv.org/abs/2301.00246. Many questions remain open!

Molecular descriptors are essential to quantitative structure activity/property relationship (QSAR/QSPR) models and machine learning models. In this talk we will introduce our recently proposed persistent path-spectral (PPS), PPS-based molecular descriptors, and PPS-based machine learning model for the prediction of the protein-ligand binding affinity. For the graph, simplicial complex, and hypergraph representation of molecular structures and interactions, the path-Laplacian can be constructed and the derived path-spectral naturally gives a quantitative description of molecules. Further, by introducing the filtration process of the representation, the persistent path-spectral can be derived, which gives a multiscale characterization of molecules. Molecular descriptors from the persistent path-spectral attributes then are combined with the machine learning model, in particular, the gradient boosting tree, to form our PPS-ML model. We test our model on three most commonly used data sets, i.e., PDBbind-v2007, PDBbind-v2013, and PDBbind-v2016, and our model can achieve competitive results.

Artificial intelligence (AI) based drug design has demonstrated great potential to fundamentally change the pharmaceutical industries. However, a key issue in all AI-based drug design models is efficient molecular representation and featurization. Recently,topological data analysis (TDA) has been used for molecular representations and its combination with machine learning models have achieved great success in drug design. In this talk, we will introduce our recently proposed persistent models for molecular representation and featurization. In our persistent models, molecular interactions and structures are characterized by various topological objects, including hypergraph, Dowker complex, Neighborhood complex, Hom-complex. Then mathematical invariants can be calculated to give quantitative featurization of the molecules. By considering a filtration process of the representations, various persistent functions can be constructed from the mathematical invariants of the representations through the filtration process, like the persistent homology and persistent spectral. These persistent functions are used as molecular descriptors for the machine learning models. The state-of-art results can be obtained by these persistent function based machine learning models.

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.

We consider the terminal monad among those preserving the objects of a subcategory $D$ of $C$, and in particular preserving the image of a monad over the category C. Relations to the double duals are explored. Several common monads $M : C \to C$, such as pro-finite completions, are shown to be uniquely characterized by the property of being terminal objects in the category of co-augmented endo-functors. Once extended to infinity categories, this gives, for example, a complete characterization of the well-known Bousfield-Kan R-homology completion $R_\infty$. This allows one to generalize the nice properties of this tower to a general infinity monad. Joint work with Sergei Ivanov.

During last decades were introduced and intensively studied groups elements of which can be present by braids with special type of crossings. These groups are some analogous of the Artin braid group $B_n$ and are called by braid-like groups. Examples of braid-like groups: universal braid group $UB_n$, virtual braid group $VB_n$, singular braid group $SB_n$, welded braid group $WB_n$ and so on. Fred Cohen and Jie Wu defined simplicial structure on the set of pure braid groups $P_* = \{ P_n \}_{n = 1}^{\infty}$, $P_n \leq B_n$, and proved that the Milnor's free construction $F[S^1]$ of simplicial circle $S^1$ can be embedded into $P_*$. This gives a possibility to present generators of homotopy groups $\pi_k(S^2)$ by braids. As co-product was defined a new system of generators (cabling generators) of $P_n$, have found some properties of Lie algebra $L_n$ that corresponds to $P_n$. Similar constructions for virtual pure braid groups $VP_n$ are studied in series of papers by V. Bardakov, R. Mikhailov, V. Vershinin and Jie Wu. On my talk we will discuss simplicial structures, cabling generators and Lie algebras for other braid-like groups.

The monopole h-invariants are numerical invariants of rational homology 3-spheres (i.e. closed, oriented 3-manifolds with vanishing first Betti number) that arise in the context of Seiberg-Witten theory, more specifically monopole Floer homology as defined by Kronheimer and Mrowka. It is an open question whether or not the h-invariants depend on the choice of coefficient ring used to define monopole Floer homology. We use Manolescu's homotopy theoretic approach to Seiberg-Witten theory on 3-manifolds to provide some evidence that the h-invariants depend on the choice of coefficients. We also discuss additivity, duality, and monotonicity properties of the h-invariants for different coefficients.

In this talk I will discuss a recognitio nprinciple for iterated suspensions as coalgebras over the little cubes operad. This is joint work with Oisín Flynn-Connolly and José Moreno-Fernández.

Homotopy theory, higher structures and related facets have reshaped many modern areas of mathematics and have provided new techniques for studying old problems. One such area that is beginning to benefit from these homotopical methods is the geometric theory of non-linear partial differential equations. Much like Algebraic analysis and the corresponding D-module theory for linear systems fits naturally into the framework of homological algebra, derived categories etc. it is becoming evident that the analogous non-linear theory fits naturally into the framework of higher algebra, simplicially enriched/infinity-categories etc. In my talk I will describe a functor-of-points approach to non-linear PDEs in the presence of symmetries as they are naturally described by (higher) stacks using the language of derived analytic geometry.

Localization is a foundational method in homotopy theory. It is easy to believe that life is much easier after localization. In particular, integral/unstable homotopy theory should be much more complicated than rational homotopy theory. Indeed, by the famous work of Quillen and Sullivan, rational homotopy theory is purely algebraic. Therefore, one can expect that it is much easier to apply Quillen or Sullivan’s theory to study rational homotopy than to apply the wilder unstable homotopy theory. However, in several recent joint works with Stephen Theriault, we are able to apply unstable homotopy theory techniques to study some rational homotopy properties of manifolds. In this talk, I will explain this with examples, ideas and its advantages.

We will discuss spatial embeddings of graphs into the 3-sphere. Even a graph can be simple combinatorically, its embedding can be very complicated since any cycle of a graph will be embedded as a knot in the 3-sphere. Two spatial graphs are said to be equivalent if there is an ambient isotopy of the 3-sphere which transforms one spatial graph to another. As well as knots and links, spatial graphs can be studied from their diagrams. The Yamada and Jagger polynomials are most useful invariants of spatial graphs. Let $K_4$ be the complete graph on 4 vertices. We will present a relation between normalized Jagger polynomials of spatial $K_4$-graph and its spatial subgraphs with Jones polynomial of the associated link. The obtained results are joint with O. Oshmarina.

Stable homotopy is intimately related to the geometry of formal groups through the Adams-Novikov spectral sequence. Franke took a step towards making this precise by proposing a category of certain sheaves on the moduli stack of formal groups as an algebraic analog of the spectra localized at chromatic height $\leq h$. He conjectured that for sufficiently large primes the homotopy truncation of the two categories coincide. Franke's conjecture, and its recent resolution by Patchkoria-Pstragowski, neglects the natural tensor product structure on both sides. I will discuss recent work in which I formulate and prove a monoidal version of Franke's conjecture.

In this talk we recall some recent results of Borsuk-Ulam type theorems, where these theorems consist in generalizations of the classical Borsuk-Ulam theorem. For the generalizations in consideration, we describe the concept of the Borsuk-Ulam property with respect to the homotopy classes of maps and present a few recent results which consist of such study for maps between two surfaces of Euler characteristic $0$. Then we move to the situation where we have a bundle over $S^1$ where the fibre is the Torus. We restrict ourself to the cases where the bundle $T\to M_A\to S^1$ has as monodromy a matrix $A$ which is not Anosov. We discuss the machinery necessary to study the Borsuk-Ulam property of maps over $S^1$. We discuss the free involutions over $S^1$ for such torus bundles $M_A$ and for all involutions which are the homotopy classes of maps which satisfy the Borsuk-Ulam property. This is joint work with Vinicius Casteluber Laass (Universidade Federal da Bahia) and Weslem Liberato Silva (Universidade Estadual de Santa Cruz).

Work of Quillen, Mandell, Behrens-Rezk, and Heuts shows that unstable homotopy theory is strongly tied to the homotopy theory of commutative and Lie algebras. This dual description is a reflection of the Koszul duality between the operads com and lie. Recently, Ching-Salvatore showed the Koszul duality of com and lie can be upgraded to the statement that the sequence of operads $\mathrm{lie}[n] \rightarrow E_n \rightarrow \mathrm{com}$ is Koszul self dual. We use this to show manifold calculus is Koszul self dual and admits both a covariant and a contravariant comparison to the Goodwillie calculus of spaces.

An $(n-1)$-connected finite CW-complex with dimension $\leq n+k$ is called an $A_n^k$-complex. The homotopy types of indecomposable (up to wedge product ) $A_n^2$-complexes were classified by Prof. Chang Sucheng in 1950, containing spheres, elementary Moore spaces and elementary Chang complexes. These are elementary homotopy types in homotopy theory. In this talk, I introduce some homotopy of $A_n^2$-complexes, especially the homotopy decomposition problems of them. These are joint works with Prof. Pan Jianzhong.

To understand the homotopy type of a space, it is standard to study the homotopy classes of its self-maps. In 2015, Choi and Lee introduced the self-closeness number of a connected CW complex, which is the least integer $k$ such that any of its self-map inducing an isomorphism in $\pi_*$ for $*\le k$ is a homotopy equivalence. This homotopy invariant is relatively new, and there is few results on explicit determination for a given space. In this talk, we introduce some interesting features of the self-closeness numbers, and determine them for a class of non-simply-connected finite complexes with finite fundamental groups.

In the 1980-s Michael Atiyah rigorously defined topological field theory (TFT) with values in $\mathcal{C}$ as symmetric monoidal functors from a bordism category to $\mathcal{C}$. A natural generalization comes from looking at \emph{embedded bordism categories} where all the manifolds are embedded in $\mathbb{R}^m$ and bordisms are embedded in $\mathbb{R}^m\times I$ for some fixed $m$. Monoidal functors from an embedded bordism category are called enhanced TFTs. An example for an embedded bordism category is the category $\mathrm{Tang}_{0,2}$. This category has oriented $0$-dimensional manifolds in $\mathbb{R}^2$ as objects and oriented $1$-manifolds with boundary in $\mathbb{R}^2\times I$ as morphisms. The category $\mathrm{Tang}_{0,2}$ is also interesting for knot theorists. The main reason is that the space $\mathrm{Map}_{\mathrm{Tang}_{0,2}}(\emptyset,\emptyset)$ is by definition the space of links up to isotopy. The following two questions naturally arise. Do knot theoretic gadgets allow us to classify enhanced TFTs with values in some category $\mathcal{C}$? Can we use TFTs in order to define or give a more pleasing definition of knot invariants? We will answer this questions when $\mathcal{C}=\mathrm{Span}(\mathrm{Set})$ and the associated knots invariants are biquandle colorings.

I will discuss the proof of a conjecture by Bhattacharya and Ricka on the Stable Picard Group of $A(n)$. This is a joint work with my student Rujia Yan.

We compare three different definitions of the equivariant cohomological dimension of a group with operators, coming from Takasu, Adamson and Bredon relative cohomologies of a subgroup pair, giving examples of strict inequalities in all cases that can occur. We also show that Farber’s topological complexity of a group $G$ is not given any of the relative cohomological dimensions of $G\times G$ relative to the diagonal subgroup. This is joint work with Kevin Li, Ehud Meir and Irakli Patchkoria.

The topological modular forms spectrum tmf serves as an approximation to the sphere spectrum. Because it is computationally simpler than the sphere, tmf serves as a test case for new computational approaches. There are two ways to compute the homotopy groups of tmf: the Adams spectral sequence and the Adams-Novikov spectral sequence. Both approaches have separately been fully analyzed in previous work. We return to the analysis of these spectral sequences, but we add a new perspective. By studying both spectral sequences simultaneously, we are able to simplify the analysis to purely algebraic techniques, with a few exceptions. Along the way, we settle a previously unresolved detail about the multiplicative structure of the homotopy groups of tmf. This is joint work with Hana Jia Kong, Guchuan Li, Yangyang Ruan, and Heyi Zhu.