报告题目1：Four dcpos, a theorem, and an open problem

报告时间：9:30-10:30

报告人：Achim Jung

报告摘要：

From the very early days of continuous lattice theory, the question of the sobriety of the Scott topology has been of interest. In this talk, I will review the 1981 construction by Peter Johnstone of a dcpo which has a non-sober Scott topology, then a conceptually simpler construction proposed by Xiaodong Jia which has the added property that its Scott topology is well-filtered. Jia's construction is a useful link for understanding the example of a non-sober complete lattice given by John Isbell in 1982. Isbell's paper is often cited but rarely read, which is a shame because the construction is ingenious. From recent results of Lawson and Xi we know that Isbell's lattice is well-filtered in the Scott topology.

Weng Kin Ho and Dongsheng Zhao considered a variation of the sobriety question, asking whether it is possible to reconstruct a dcpo from its lattice of open sets by other means than taking the spectrum. In joint work, Weng Kin Ho, Jean Goubault-Larrecq, Xiaoyong Xi, and the speaker showed that in general this is not possible. The counterexample is the fourth dcpo mentioned in the title. Nevertheless, the reconstruction problem is solvable for a very large class of dcpos as our theorem shows. Whether it is the maximal class of such dcpos is not known.

报告人简介：

Achim Jung是英国伯明翰大学计算机科学学院理论计算机科学教授，曾任计算机科学学院院长。系杂志《Theoretical Computer Science》《Categories and General Algebraic Structures with Applications》和《Electronic Notes in Theoretical Computer Science》编辑。主要从事Domain理论、拓扑学、程序语言语义学、概率论及Lambda计算方面的研究。Achim Jung教授在domain范畴的分类问题上做出了杰出的贡献。他提出了FS-domain范畴与L-domain范畴的概念并证明它们在domain范畴中的极大性，成功解决了domain范畴的分类问题。在概率性程序语言的计算模型方向，Achim Jung教授证明了稳定紧空间范畴、Lawson紧domain范畴、QFS-domain范畴的概率幂domain构造的封闭性。此外，在domain的逻辑表示方面，他将G. Plotkin教授对代数domain范畴的逻辑表示的工作推广到了domain范畴，通过提出Proximity lattice的概念，给出了稳定紧空间，FS-domain的有限结构表示，建立了domain与逻辑的对偶理论。Achim Jung教授与牛津大学Samson Abramsky教授合著《Domain Theory》一书，成为domain理论研究方向的经典书籍之一。

报告题目2：Equality of the Isbell and Scott topologies on function spaces

报告时间：10:30-11:30

报告人：李庆国

报告摘要：

The function spaces have their background in theoretical computer science and have been studied by Jung, Lawson, Xu, Erker , Liu, Liang, Kou, Luo, Mislove and others. There exist four famous topologies which are the pointwise convergence, compact-open, Isbell and Scott topologies on the set [X→L] of the continuous functions from topological space X to a dcpo with the pointwise order. In general, the pointwise convergence topology is coarser than the compact-open topology, the compact-open topology is coarser than the Isbell topology, and the Isbell topology is coarser than the Scott topology on [X→L]. If X is locally compact, then the compact-open topology is equal to the Isbell topology. In 1990, Lawson and Mislove posed the following problem:
Problem. Let X be a topological space and L a dcpo equipped with the Scott topology. Under what conditions on L do the Isbell and Scott topologies on [X→L] agree?
 In this talk, we mainly consider the question of when the Isbell and Scott topologies coincide on the set [X→L] of all continuous mappings from a topological space X to a dcpo L with the pointwise order. The main results are:
(1) If L is a sober dcpo which is bi-complete, then (i) that the Isbell and Scott topologies coincide on [X→L] for all c-spaces X implies that L is a pointed L-dcpo; (ii) that the Isbell and Scott topologies coincide on [X→L] for all irreducible c-spaces X implies that L is an L-dcpo.
(2) Let L be a quasicontinuous UBC-domain and X a c-space. If L has a least element or X is connected, then the Isbell and Scott topologies coincide on [X→L].
(3) Let L be a quasicontinuous UFL-domain and the topological space X =∪Xi, where every Xi is an irreducible Scott c-space and I is a nonempty _nite set. If L has a least element or I is a singleton, then the Isbell and Scott topologies coincide on [X→L].

报告人简介：

李庆国，男，汉族。生于1963年6月。博士，数学学院二级教授，博士生导师，校学术委员会委员。曾任湖南大学研究生院院长，湖南大学科技处处长，湖南省人民政府学位委员会委员。1999年7月至2000年6月及2008年11月至2009年11月分别在美国科罗拉多大学数学系和康涅底克大学数学系作访问教授。2000年12月起担任湖南大学应用数学专业博士生导师。现为中国系统工程学会模糊数学与模糊系统委员会副理事长，湖南省数学学会副理事长，《模糊系统与数学》杂志编委，美国“数学评论”特约评论员。入选湖南省121人才第一层次，国务院政府特殊津贴获得者。曾获2013年湖南省自然科学一等奖，排名第一。2009年湖南省自然科学二等奖，排名第二。已完成国家自然科学基金课题《广义连续格上拓扑及应用研究》《模糊概念格理论及在信息科学中的应用》《量子逻辑和模糊逻辑的相关问题研究》《Domain结构与信息系统的表示理论研究》四项，及教育部博士点基金课题，湖南省自然科学基金重点课题二项。现在正承担国家自然科学基金课题《连续偏序集的拓扑性质、笛卡尔闭性及函数空间的研究》，湖南省自然科学基金重点项目《量子逻辑的基础结构研究》等。
目前主要研究领域为格上拓扑、模糊数学理论与应用。重在研究计算机与信息科学中所涉及的数学问题，主要从以下三个方面着手进行研究：计算机程序语言的指称语义—Domain理论，计算机与信息科学的逻辑基础，形式概念分析及粗糙集理论等新的数学理论在信息科学中的应用。至今为止，已在《Topology and its Applications》,《Applied Categorical Structures》，《Proceedings of the Edinburgh Mathematical Society》，《Order》，《Fuzzy Sets and Systems》，《International Journal of Approximate Reasoning》，《Rocky Mountain Journal of Mathematics》，《Discrete Mathematics》，《Information Sciences》，《Information and Computation》，《Theoretical Computer Science》，《Discrete Applied Mathematics》《Knowledge-Based Systems》《Houston Journal of Mathematics》《Semigroup Forum》等国际期刊上发表论文近100篇。累计培养博士毕业生41名，全部进入高校工作。两次获得湖南省优秀博士学位论文指导奖。