The following collects the information of four talks in one-day meeting on persistent homology, Time: Aug. 2, 24’, TencentMeeting: 707-514-100; co-organizer: Xubing Zheng.
Some slides files are missed.
Xiang Liu (BIMSA):
This talk gives a brief introduction to history and development trend of persistent homology.
Ran Liu (BIMSA):
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 pathspectral 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.
Shuang Wu (BIMSA):
Human diseases involve metabolic alterations. Metabolomic profiles have served asa vital biomarker for the early identification of high-risk individuals and disease prevention. We have leveraged a statistical physics model to combine all metabolites into bidirectional, signed, and weighted interaction networks and trace how the flow of information from one metabolite to the next causes changes in health state. We integrate concepts from ecosystem theory and evolutionary game theory to model how the health state-dependent alteration of a metabolite is shaped by its intrinsic properties and through extrinsic influences from its conspecifics. We code intrinsic contributions as nodes and extrinsic contributions as edges into quantitative networks and implement GLMY homology theory to analyze and interpret the topological change of health state from symbiosis to dysbiosis. The application of this model to real data allows us to identify several hub metabolites and their interaction webs, which play a part in the formation of inflammatory bowel diseases.
Bingxu Wang (Peking University Shenzhen Graduate School)
高熵合金因其多样可调的活性位点所构成的广阔化学空间而在催化领域引起越来越多的关注。为了加速这一化学空间的探索，高效的深度学习模型变得至关重要。然而，在通过深度学习设计晶体材料的任务中，出现了三个主要障碍：缺乏晶体结构固有数学特征的表示，模型训练数据不足，以及由于可解释性问题导致对机器学习结果信心不足。
鉴于这些挑战，本文提出了一种基于持续GLMY同调的半监督预测和生成框架，由我们的PathVAEs提供支持。该框架旨在预测高熵合金催化剂的吸附能和设计潜力。它通过以下方式解决这些障碍：(1) 引入一种有效的拓扑方法以提取固有特征——配体和配位特征；(2) 利用半监督学习增强模型训练；(3) 将机器学习操作与催化含义对齐。结果令人鼓舞：预测组件达到了高准确性，生成方面产生了八种高性能催化剂设计。该工作不仅提供了一种优秀的基于拓扑的特征提取方法，还为晶体材料的设计引入了一种新的研究范式。Li Cai (Xi’an Jiaotong-Liverpool University (XJTLU)), Jan. 23-24, 24’, 10:30-11:45; TencentMeeting: 97551609660; totally on blackboard.
Consider a finite simple graph with set of vertices $V$ and set edges $E$, such that we assign a group $G_v$ to each vertex $v$ of $V$. The graph product of groups $G_{v}$ with respect to the graph is the free product of $G_v$ subject to the relations that $G_v$ and $G_{v’}$ commutes if and only if $v$ and $v’$ is connected by an edge from $E$. Our recent results imply that the graph pproduct of simplicial groups is a model of the loop space of a polyhedral products, whose homology (endowed with the Pontryagin product) is isomorphic to a graph product of Hopf algebras.
In the first talk we mainly focus on the homotopy properties of the graph product of simplicial groups and show its relation with loop spaces.
In the second one we illustrate the homology of such a graph products of simplicial groups and show that it is isomorphic to the graph product of the homology of each $G_v$.
Feifei Fan (South China Normal University), May 5, 23’, Friday, 15:00-16:00; TencentMeeting: 105432114.
This talk is toward the applications of toric topology in algebraic combinatorics, especially the face enumeration problems for triangulated spheres and further for triangulated manifolds. A motivational problem is the celebrated g-conjecture in the theory of face enumeration. First, we survey how the g-conjecture came about and how it relates to the topology of a class of toric spaces, which can be viewed as a generalization of toric varieties. Using the theory of toric topology, we give a topological interpretation for several fundamental results in algebraic combinatorics.
Hao Zhao (South China Normal University), May 11, 23’, Thursday, 15:00-16:00; TencentMeeting: 423681872.
In order to study the double suspension $E^2:S^{2n-1}\to \Omega^2 S^{2n+1}$ when localized at a prime $p$, Selick filtered $\Omega^2 S^{2n+1}$ by some H-spaces which geometrically realize a natural Hopf algebra filtration of the cohomology $H^\ast (\Omega^2 S^{2n+1};\mathbb{Z}/p)$. Later, Gray showed that the fiber $W_n$ of $E^2$ has an integral classifying space $BW_n$.
In this talk I will introduce how to correspondingly filter $BW_n$ in a manner which is compatible with Selick’s filtration and the homotopy fibration $S^{2n-1}\to \Omega^2 S^{2n+1}\to BW_n$. Furthermore, I will talk about our results on the multiplicative properties and homotopy exponents of the spaces in the filtrations.Qi Guo (Remin University of China), Dec. 7, 23’, Thursday, 15:00-16:00; TencentMeeting: 597930348.
In this talk, I will discuss some recent works on spectral gap of Dirac operator and its applications. First, I will go over the background of Dirac operator on Spin manifold. Then I will show three main theorems on spectrum of Dirac operator. At last, I will explain how these works can be applied to nonlinear problems. The key ingredient is the Clifford multiplication which comes from the representation of Clifford algebra.
Yang Ming (Hubei Polytechnic University), Dec. 7, 23’, Thursday, 16:10-17:20; TencentMeeting: 597930348.
Computational lithography is the set of mathematical and algorithmic approaches designed to improve the resolution attainable through photolithography. Computational lithography has become instrumental in further shrinking the design nodes and topology of semiconductor transistor manufacturing. In this talk, I will introduce the basic concepts and ideas in the patterning flow and explain how to use kernel methods to accelerate the numerical calculation.
Yang Shen (Fudan University), Dec. 8, 23’, Friday, 15:00-16:00; TencentMeeting: 276191606.
Let $M_{g,n(g)}$ be the moduli space of hyperbolic surfaces of genus $g$ with $n(g)$ punctures endowed with the Weil-Petersson metric. In this talk, we discuss the asymptotic behavior of the Cheeger constants and spectral gaps of random hyperbolic surfaces in $M_{g,n(g)}$ when $n(g)$ grows slower than $g$ as $g\to\infty$. This is a joint work with Yunhui Wu.
Xiang Liu (Nankai University), Dec. 8, 23’, Friday, 16:10-17:20; TencentMeeting: 276191606.
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. In this talk, I 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 simplicial complex and hypergraph. Then mathematical invariants can be calculated to give quantitative featurization of the molecules. 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.