“拓扑与应用”数学学科学术交流研讨会

　　为推进新疆地区数学学科的建设与发展，推动当地拓扑应用研究与国内外前沿接轨，促进相关领域的交流与合作研究，为“拓扑与应用”交叉研究的青年学者和研究生搭建一个学习和交流的平台，兹定于2023年5月20-22日在新疆永利集团3044官网欢迎您举办“拓扑与应用”数学学科学术交流活动，活动内容包括“拓扑与应用研讨会”、面向本科生和研究生的通俗数学学术讲座及数学学科发展建设与人才培养研讨。会议报到时间为5月19日，离会时间为5月23日。

　　本次学术交流活动由永利集团3044官网欢迎您理学院主办。

“拓扑与应用研讨会”日程安排学术讲座

第一会场

“拓扑与应用研讨会” 报告与通俗讲座的题目和摘要

　　报 告 人：吕志教授（复旦大学）

　　报告题目：Equivariant bordism of G-actions fixing isolated points

　　报告摘要：In this talk, we first recall the history and development of (G-equivariant) unoriented bordism, especially for G to be a mod 2-torus.Then　we introduce the recent some progresses for the equivariant unoriented bordism classification of mod 2-torus-actions fixing isolated points, which are joint works with Bo Chen and Qiangbo Tan.

　　报 告 人：赵学志教授（首都师范大学）

　　报告题目：曲面上曲线相交与填充

　　报告摘要：在闭曲面上, 给定两个闭曲线同伦类, 它们各取代表元, 处于一般位置时最少的交点个数被称为这两个闭曲线类的几何相交数, 这是一个由来已久的数学问题, 其更多的信息至今依然尚待挖掘. 我们将介绍几何相交数的一种新的符号计算方法, 进一步解释如何计算和说明曲线族如何分割、填充所在曲面.

　　报 告 人：刘祥（南开大学与BIMSA）

　　报告题目：Persistent function based machine learning for drug design

　　报告摘要：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 successes 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, persistent spectral and persistent Tor-algebra. These persistent functions are used as molecular descriptors for the machine learning models. The state-of-the-art results can be obtained by these persistent functions based machine learning models.

　　报 告 人：张旭东（大连理工大学）

　　报告题目：Identification of autism spectrum disorder using topological data analysis and the ABIDE database

　　报告摘要：We use TDA to extract topological features from the data of the well-known Autism Brain Imaging Data Exchange (ABIDE) database. By using machine learning classifiers, we identify subjects with Autism Spectrum Disorder (ASD) to these features. Our results show that the average accuracy of 10-folds cross validation is 90.9% that is higher than any other existing methods. Two local regions of the frontal lobe and parietal lobe are also used as ROIs to extract topological features. The resulting of these regions (average accuracy of 89.4% and 91.5%, respectively) is also higher than that of any other existing methods. The average accuracy of the local region in the parietal lobe (88.4%) is quite close to the highest average accuracy of existing methods. Moreover, we need less time to run the whole process.

　　报 告 人：岳云光 （永利集团3044官网欢迎您，BIMSA）

　　报告题目：Asynchronous computability theorem in arbitrary solo models 

　　报告摘要：In this talk, I will give an asynchronous computability theorem in d-solo system by borrowing concepts from combinatorial topology, in which it states a necessary and sufficient conditions for a task to be wait-free computable in that system. Intuitively, a d-solo system allows as many d processes to access it as if each were running solo, namely, without detecting communication from any peer. As an application, we completely characterize the solvability of the input-less tasks in such systems. This characterization also leads to a hardness classification of these tasks according to whether their output complexes hold a d-nest structure. As a byproduct, we find an alternative way to distinguish the computational power of d-solo objects for different d. 

　　报 告 人：刘西民教授（大连理工大学）

　　报告题目：Cyclic group actions on elliptic surfaces

　　报告摘要：In this talk, we first give a survey of some topics around finite group actions on 4-manifolds then we consider cyclic group actions on elliptic surface, we prove the non-existence of cyclic group actions of odd prime order that act trivially on the cohomology of elliptic surfaces.

　　报 告 人：刘兴武教授（大连理工大学）

　　报告题目：不动点定理—计算的视角

　　报告摘要：本文将从计算的角度介绍不动点理论，并给出一些应用。

　　报 告 人：付鑫（北京应用数学研究院）

　　报告题目：The integral cohomology rings of four-dimensional toric orbifolds

　　报告摘要：Toric orbifolds introduced by Davis and Januszkiewicz are topological analogs of projective toric varieties. When a toric orbifold is smooth, its integral cohomology ring is isomorphic to a quotient ring of the Stanley-Reisner ring. Such a formula holds for the singular case over rational coefficients, but integrally it becomes more complicated. For instance, the cohomology of a weighted projective space is additively isomorphic to the cohomology of a complex projective space, but the ring structure differs. In this talk, we focus on toric orbifolds X in four dimensions. If X has a smooth fixed point, we construct a basis for its integral cohomology and present their cup products in a matrix whose entries are explicitly determined by the characteristic function. This is joint work with Tseleung So and Jongbaek Song.

　　报 告 人：高亚茹（大连理工大学）

　　报告题目：Face-to-face interaction analysis from persistent hypergraph model

　　报告摘要：Close proximity interactions between individuals are usually measured and analyzed using the model of connectivity graphs. Recent researches show that the hypergraph model reveals more global and geometric features in high dimensions. We generalizes classical persistent homology on simplicial complexes to hypergraphs. Our theory is demonstrated by analyzing face-to-face interactions of different populations. We select data sets of baboons in primate center and people from rural Malawi, scientific conference, workplace and high school.

　　报 告 人：李京艳 副研究员（北京应用数学研究院）

　　报告题目：GLMY同调的概述

　　报告摘要：GLMY理论是丘成桐先生及其合作者发展的一套新理论，试图在有向图上建立象拓扑空间上定义的同伦与同调。这个报告中我主要介绍一下丘先生他们已经建立的有向图的同调理论和同伦理论以及我们在这方面取得的一些成绩。

　　报 告 人：张蒙蒙（河北师范大学）

　　报告题目：The $Delta$-twisted homology and fiber bundle structure of twisted simplicial sets

　　报告摘要：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.

　　报 告 人：吴杰研究员（北京应用数学研究院BIMSA）

　　报告题目：Algebraic topology and its applications in big data and complex network

报告摘要：This is an introductory talk on algebraic topology and its applications. The talk will consist of two parts. In the first part, we will give a brief introduction to algebraic topology. In the second part, we will give a brief review for the applications of algebraic topology in data analytics and complex network.

　　报 告 人：雷逢春教授（大连理工大学）

　　报告题目：低维拓扑概览

报告摘要：拓扑学是当代核心数学的一个重要前沿领域，其渊源可追溯到欧拉早期的工作（1836年，哥尼斯堡七桥问题的解答）。拓扑学奠基于19世纪末, 在20世纪取得了梦幻般的发展, 进入21世纪更加兴旺发达。本报告将对拓扑学的发展作一个跨越世纪的通俗概要浏览, 重点介绍低维拓扑学的发展，即二、三和四维流形拓扑学的主要成就。