# Math and Biology Seminar

**Organizers:** Jie Wu, Jingyan Li, Xiang Liu

**Date:** 2022/9/23- 2022/12/30

**Time: **10:30-11:30 am

**Frequency:** It will be held on Friday, biweekly

**Zoom:** 293 812 9202 **Passcode:** BIMSA

**Introduction: **

人工智能（AI）和现代生物学是当今世界最重要的科学领域之一，它们可以带来技术革命，并从根本上改变社会格局。二十一世纪生物科学的主要趋势是从现象学、描述性科学向定量和预测性科学的转变。构建生物学的“fundamental laws”已经成为本世纪数学发展的中心问题之一。借助人工智能和数学工具，生物科学已经获得了巨大进步和发展。但更多具有挑战性的问题还亟需更好的数学方法和人工智能工具。近年来，来自于代数拓扑，计算拓扑，微分几何等数学领域的方法正逐渐被应用于数据分析，包括备受关注的拓扑数据分析（TDA），在人工智能和分子生物学的各个方面已经展示出巨大的应用潜力。

本讨论班将会以“生物为主线”，分享交流数学方法的用武之地，以报告的形式进行，每次会有一位报告人介绍数学理论或者数学理论的应用。

**Reference: **

[1] Grigor'yan, A., Lin, Y., Muranov, Y., & Yau, S. T. (2012). Homologies of path complexes and digraphs. arXiv preprint arXiv:1207.2834.

[2] Chowdhury, S., & Mémoli, F. (2018). Persistent path homology of directed networks. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 1152-1169). Society for Industrial and Applied Mathematics.

[3] Sun, Z., Pei, S., He, R. L., & Yau, S. S. T. (2020). A novel numerical representation for proteins: three-dimensional chaos game representation and its extended natural vector. Computational and structural biotechnology journal, 18, 1904-1913.

[4] Dong, R., Pei, S., Yin, C., He, R. L., & Yau, S. S. T. (2020). Analysis of the hosts and transmission paths of SARS-CoV-2 in the COVID-19 outbreak. Genes, 11(6), 637.

[5] Bressan, S., Li, J., Ren, S., & Wu, J. (2019). The embedded homology of hypergraphs and applications. Asian Journal of Mathematics, 23(3), 479-500.

[6] Liu, X., Wang, X., Wu, J., & Xia, K. (2021). Hypergraph-based persistent cohomology (HPC) for molecular representations in drug design. Briefings in Bioinformatics, 22(5), bbaa411.

[7] Kishimoto, D., & Takeda, M. (2021). Spaces of commuting elements in the classical groups. Advances in Mathematics, 386, 107809.

[8] Farber, M., Kishimoto, D., & Stanley, D. (2020). Generating functions and topological complexity. Topology and its Applications, 278, 107235.

[9] Luo, X., & Shvydkoy, R. (2017). Addendum: 2D homogeneous solutions to the Euler equation. Communications in Partial Differential Equations, 42(3), 491-493.

[10] Luo, X., & Yau, S. S. T. (2018). The suboptimal method via probabilists’ Hermite polynomials to solve nonlinear filtering problems. Automatica, 94, 9-17.

[11] Gao, Y., Li, F., Liang, L., & Lei, F. (2021). Weakly reducible H-splittings of 3-manifolds. Journal of Knot Theory and Its Ramifications, 30(10), 2140004.

[12] Yue, Y., Wu, J., & Lei, F. (2019). The evolution of non-degenerate and degenerate rendezvous tasks. Topology and its Applications, 264, 187-200.

** **

**Schedule: **

2022/12/30

**Speaker:** Dr. Pengcheng Li

**Title: **TBA

**Abstract:** TBA

2022/12/16

**Speaker: **Dr. Fedor Pavutnitskiy

**Title: **Quadric hypersurface intersection for manifold learning in feature space

**Abstract:** The knowledge that data lies close to a particular submanifold of the ambient Euclidean space may be useful in a number of ways. For instance, one may want to automatically mark any point far away from the submanifold as an outlier, or to use its geodesic distance to measure similarity between points. Classical problems for manifold learning are often posed in a very high dimension, e.g. for spaces of images or spaces of representations of words. Today, with deep representation learning on the rise in areas such as computer vision and natural language processing, many problems of this kind may be transformed into problems of moderately high dimension, typically of the order of hundreds. Motivated by this, we propose a manifold learning technique suitable for moderately high dimension and large datasets. The manifold is learned from the training data in the form of an intersection of quadric hypersurfaces — simple but expressive objects. At test time, this manifold can be used to introduce an outlier score for arbitrary new points and to improve a given similarity metric by incorporating learned geometric structure into it.

2022/12/2

**Speaker:** Dr. Pengcheng Li

**Title: **Suspension Homotopy of 4-manifolds And the Second 2-local Cohomotopy Sets

**Abstract:** In this talk we study the homotopy type of the (double) suspension of an orientable, closed, connected $4$-manifold $M$, whose integral homology can have $2$-torsion. This gives a complete solution to a prior research problem of So and Theriault. Moreover, the decomposition results are applied to give a characterization of the second $2$-local cohomotopy set $\pi^2(M;\mathbb{Z}_{(2)})$, which is the set of homotopy classes of based maps from $M$ into the $2$-local sphere $S^2_{(2)}$.

2022/11/18

**Speaker:** Prof. Xiaoxian Tang (Beihang Uinversity)

**Title: **Multistability of Small Reaction Networks

**Abstract: **The multistability problem of biochemical reaction systems is crucial for understanding basic phenomena such as decision-making process in cellular signaling. Mathematically, it is a challenging real quantifier elimination problem. We present some recent progress on multistability of small reaction networks. 1) For reaction networks with two reactions (possibly reversible), we find the multistable networks those have the minimum numbers of reactants and species. 2) For reaction networks with one-dimensional stoichiometric subspaces, we give the relation between the maximum numbers of stable steady states and steady states. 3）For bi-reaction networks, we completely characterize the bi-reaction networks that admit at least three positive steady states. 4) For zero-one networks, we prove that if a network admits multistationrity, then its rank is at least three.

2022/11/4

**Speaker: **Dr. Mengmeng Zhang (Hebie Normal University & BIMSA)

**Title:** The $Delta$-twisted homology and fiber bundle structure of twisted simplicial sets

**Abstract:** Different from classical homology theory, Alexander Grigor'yan, Yuri Muranov and Shing-Tung Yau recently introduced $delta$-(co)homology, taking the (co)boundary homomorphisms as $\delta$-weighted alternating sum of (co)faces. For understanding the ideas of $delta$-homology, Li, Vershinin and Wu introduced $delta$-twisted homology and homotopy in 2017. On the other hand, the twisted Cartesian product of simplicial sets was introduced by Barratt, Gugenheim and Moore in 1959, playing a key role for establishing the simplicial theory of fibre bundles and fibrations. The corresponding chain version is twisted tensor product introduced by Brown in 1959. In this talk, I will report our recent progress for unifying $delta$-homology and twisted Cartesian product. We introduce $\Delta$-twisted Carlsson construction of $\Delta$-groups and simplicial groups, whose abelianization gives a twisted chain complex generalizeing the $delta$-homology, called $\Delta$-twisted homology. We show that Mayer-Vietoris sequence theorem holds for $\Delta$-twisted homology. Moreover, we introduce the concept of $\Delta$-twisted Cartesian product as a generalization of the twisted Cartesian product, and explore the fiber bundle structure. The notion of $\Delta$-twisted smash product, which is a canonical quotient of $\Delta$-twisted Cartesian product, is used for determining the homotopy type of $\Delta$-twisted Carlsson construction of simplicial groups.

2022/10/21

**Speaker: **Dr. Yichen Tong (Kyoto University)

**Title:** Rational self-closeness numbers of mapping spaces

**Abstract: **For spaces X and Y , let Map(X, Y ) denotes the free mapping space from X to Y . Classifying components up to homotopy type for a given mapping space is aclassical problem in algebraic topology and has been studied since at least 1940s. For Map(M, S2n) where M a closed simply-connected 2n-dimensional manifold, it was proved that its components have exactly two rational homotopy types. However, since this result is proved by algebraic models of components, we do not know whether a rational homotopy invariant distinguishes these two types or not. In this talk, we completely determine the rational self-closeness numbers of components ofMap(M, S2n) and prove that they do distinguish different rational homotopy types.

2022/10/7

**Speaker: **Prof. Xue Luo (Beihang University)

**Title:** Proper orthogonal decomposition method for forward Kolmogorov equation and its application to nonlinear filtering problems

**Abstract: **In this talk, we discuss the proper orthogonal decomposition (POD) method to numerically solve the forward Kolmogorov equation (FKE). This method aims to explore the low-dimensional structures in the solution space of the FKE and to develop efficient numerical methods. As our primary motivation to study the POD method to FKE, we use it to solve the nonlinear filtering (NLF) problems combined with the real-time algorithm. This algorithm consists of off-line and on-line stages. In the off-line stage, we construct a small number of POD basis functions that capture the dynamics of the system and compute propagation of the POD basis functions under the FKE operator. In the on-line stage, we synchronize the coming observations in a real-time manner. Its convergence analysis has also been discussed. Some numerical experiments of the NLF problems are performed to illustrate the feasibility of our algorithm and to verify the convergence rate.

This is joint work with Z. Wang, S. S.-T. Yau and Z. Zhang.

2022/9/23

**Speaker: **Prof. Daisuke Kishimoto (Kyoto University)

**Title: **Tverberg’s theorem for cell complexes

**Abstract: **Tverberg’s theorem states that any affine map from (d+1)(r-1)-simplex into the Euclidean d-space, there are pairwise disjoint faces of the simplex whose image in the Euclidean space have a point in common. A topological generalization which replace an affine map with a continuous map is known to hold as long as r is a prime power. We further generalize the topological version to a continuous map out of a certain CW complex including a simplicial sphere. A key is a homotopy decomposition of a discretized configuration space.

This is joint work with S. Hasui, M. Takeda, and M. Tsutaya.