报告一

报告题目：Aronszajn-type Topological Regularity for Nonlinear Delay Evolutions in Fr\'{e}chet Spaces

报告人：王荣年 上海师范大学教授

报告时间：5月12日9：00

报告地点：腾讯ID：705-143-702

摘要：The Aronszajn-type regularity, also called $R_\delta$\text{-}structure, was proposed for characterizing the solution set of differential equation having no uniqueness. The solution set with such regularity is homeomorphic to the intersection of a decreasing sequence of compact absolute retracts. In the scale of infinite-dimensionality, we consider, in the present paper, a nonlinear delay evolution equation involving multivalued nonlinearity and possibly unbounded operator in the principal part. Both a compact interval and a noncompact one are considered.  The nonlinearity, having nonempty, convex and closed values, is upper hemicontinuous with respect to the solution variable and the operator is nonautonomous linear, multivalued and/or nonlinear or autonomous semilinear. One considering the Cauchy problem, a basic question about whether there exists solution set carrying Aronszajn-type regularity remains unsolved when the evolution family (or nonlinear semigroup) generated by the operator lacks the compactness. One of our main attention is paid to settling this question in the affirmative and providing a feasible roadmap for how such a result could be obtained. Moreover, we prove that  the solution map, having nonempty and compact values, is an $R_{\delta}$-map, which sends any connected set into a connected set. These geometric features of the solution map are then used to deal with the existence in the large for the corresponding nonlocal Cauchy problem. Finally, several examples are worked out in detail, illustrating the applicability of our abstract results.

报告二

报告题目：Stochastic dynamics of an SIS epidemic on networks

报告人：刘桂荣 山西大学教授

报告时间：5月12日14：30

报告地点：腾讯会议：459-999-086

摘要：We derive a stochastic SIS pairwise model by considering the change of the variables of this system caused by an event. Based on approximations, we construct a low-dimensional deterministic system that can be used to describe the epidemic spread on a regular network. The mathematical treatment of the model yields explicit expressions for the variances of each variable at equilibrium. Then a comparison between the stochastic pairwise model and the stochastic mean-field SIS model is performed to indicate the effect of network structure. We find that the variances of the prevalence of infection for these two models are almost equal when the number of neighbors of every individual is large. Furthermore, approximations for the quasi-stationary distribution of the number of infected individuals and the expected time to extinction starting in quasi-stationary are derived. We analyze the approximations for the critical number of neighbors and the persistence threshold based on the stochastic model. The approximate performance is then examined by numerical and stochastic simulations. Moreover, during the early development phase, the temporal variance of the infection is also obtained. The simulations show that our analytical results are asymptotically accurate and reasonable.

报告三

报告题目：自 动 机 序 列 短 课 程（一）

A short introduction to the theory of automatic sequences （1）

主 讲 人：姚家燕  清华大学教授

时报告间：5月12日15:00-18:00

报告地点：腾讯 ID：582-143-511

摘要：In this minicourse, we shall present the basic properties of automatic sequences and their different generalizations, then apply them in the study of transcendence theory.

报告四

报告题目：Exponential convergence of hp-discontinuous Galerkin method for nonlinear Caputo fractional differential equations

主 讲 人：陈艳萍  华南师范大学教授

报告时间：5月12日16:00

报告地点：腾 讯 ID：191-883-959

摘要：We present an hp-discontinuous Galerkin method for solving nonlinear fractional differential equations involving Caputo-type fractional derivative. The main idea behind our approach is to fifirst transform the fractional differential equations into nonlinear Volterra or Fredholm integral equations, and then the hp-discontinuous Galerkin method is used to solve the equivalent integral equations. We derive a-priori error bounds in the L2 -norm that are totally explicit with respect to the local mesh sizes, the local polynomial degrees, and the local regularities of the exact solutions. In particular, we prove that exponential convergence can be achieved for solutions with endpoint singularities by using geometrically refined meshes and linearly increasing approximation orders. The theoretical results are confirmed by a series of numerical experiments.

专家简介：

  王荣年, 博士, 上海师范大学教授, 博士生导师（应用数学）。目前主要从事非线性发展方程适定性、多值扰动及解集的拓扑正则性、不变流形理论等问题的研究，完成的研究结果已被"Mathematische Annalen"、“Int. Math. Res. Notices 、"Journal of Functional Analysis"、"Journal of Differential Equations""J. Phys. A: Math. Theo."等学术期刊发表。主持承担了2项国家自然科学基金面上项目、国家自然科学基金青年项目、4项省自然科学基金项目和2项省教育厅基金项目。曾获聘广东省高等学校“千百十人才工程”省级培养对象、江西省高校中青年骨干教师等。

刘桂荣，山西大学教授，博士生导师。2007年毕业于山西大学，获博士学位。主要从事微分方程与生物数学方向的研究工作，主持国家自然科学基金4项、人社部留学回国人员科技活动择优资助项目(优秀类) 1项、省级项目5项。2010年，以第一完成人获得山西省教学成果一等奖；2011年，入选山西大学青年英才计划；2012年，入选山西省高等学校优秀青年学术带头人支持计划；2015年，以第一完成人获山西省科学技术奖 (自然科学类) 二等奖。

  姚家燕，清华大学数学系教授，博士生导师。主要从事数论及其相关领域的研究，包括代换序列、Drinfeld 模上的超越数论、p-adic 动力系统、有限自动机的理论及其在数论中的应用等。师从法国著名数学家Michel Mendès France 教授， 2003 年在法国南巴黎大学获“指导研究资格”(计算机科学-数学)，曾多次应邀赴法国进行学术访问和合作研究。主持完成多项国家科研项目，在Adv. Math.与J. Reine Angew. Math.等著名刊物上发表论文二十多篇，相关工作被Goss，Waldschmidt，Thakur，Allouche 等国际上的著名数学家多次引用 ，其部分研究结果被完整录入自动机序列专著。

  陈艳萍，华南师范大学博士生导师。2008年被聘为广东省高等学校珠江学者特聘教授，2004年享受国务院颁发的政府特殊津贴、入选教育部首批新世纪优秀人才支持计划，2002年评为教育部首批全国高等学校优秀骨干教师。2017年获教育部自然科学二等奖、2012年获广东省科学技术二等奖、2011年获湖南省自然科学一等奖、2008年获教育部自然科学一等奖、2004年获湖南省科学技术进步二等奖。出版专著《最优控制问题高效算法理论》（英文版，科学出版社，2015年11月，由国家出版基金资助、入选信息与计算科学丛书）和教材《最优控制问 题高效算法理论》（英文版）、《偏微分方程数值解》、《数值计算方法》和《数学建模教学与评估指南》，在SIAM Numer. Anal.和Math. Comp.等国际一流学术期刊上发表了学术论文285篇，是ESI 高被引论文作者，2014年至2020年连续7年入选Elsevier中国高被引学者榜单。正在主持1项国家自然科学基金重点项目、1项国家自然科学基金面上项目和1项国家自然科学基金访问学者项目，主持并完成6项国家自然科学基金面上项目和1项国家自然科学基金重大研究计划培育项目等。