活動時間：9:30

活動日期：2018-12-2

地點：長安校區(qū)數(shù)學(xué)與信息科學(xué)學(xué)院學(xué)術(shù)交流廳

主辦單位：數(shù)學(xué)與信息科學(xué)學(xué)院

講座題目1：Four dcpos, a theorem, and an open problem

講座時間：9:30-10:30

報告人：Achim Jung

講座內(nèi)容簡介：

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.

講座人簡介：

Achim Jung是英國伯明翰大學(xué)計算機科學(xué)學(xué)院理論計算機科學(xué)教授，曾任計算機科學(xué)學(xué)院院長。系雜志《Theoretical Computer Science》《Categories and General Algebraic Structures with Applications》和《Electronic Notes in Theoretical Computer Science》編輯。主要從事Domain理論、拓撲學(xué)、程序語言語義學(xué)、概率論及Lambda計算方面的研究。Achim Jung教授在domain范疇的分類問題上做出了杰出的貢獻。他提出了FS-domain范疇與L-domain范疇的概念并證明它們在domain范疇中的極大性，成功解決了domain范疇的分類問題。在概率性程序語言的計算模型方向，Achim Jung教授證明了穩(wěn)定緊空間范疇、Lawson緊domain范疇、QFS-domain范疇的概率冪domain構(gòu)造的封閉性。此外，在domain的邏輯表示方面，他將G. Plotkin教授對代數(shù)domain范疇的邏輯表示的工作推廣到了domain范疇，通過提出Proximity lattice的概念，給出了穩(wěn)定緊空間，F(xiàn)S-domain的有限結(jié)構(gòu)表示，建立了domain與邏輯的對偶理論。Achim Jung教授與牛津大學(xué)Samson Abramsky教授合著《Domain Theory》一書，成為domain理論研究方向的經(jīng)典書籍之一。

講座題目2：Equality of the Isbell and Scott topologies on function spaces

講座時間：10:30-11:30

報告人：李慶國教授

講座內(nèi)容簡介：

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 spaceXto 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]. IfXis locally compact, then the compact-open topology is equal to the Isbell topology. In 1990, Lawson and Mislove posed the following problem:

Problem.LetXbe a topological space andLa dcpo equipped with the Scott topology. Under what conditions onLdo 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 spaceXto a dcpoLwith the pointwise order. The main results are:

(1) IfLis a sober dcpo which is bi-complete, then (i) that the Isbell and Scott topologies coincide on [X®L] for all c-spacesXimplies thatLis a pointed L-dcpo; (ii) that the Isbell and Scott topologies coincide on [X®L] for all irreducible c-spacesXimplies thatLis an L-dcpo.

(2) LetLbe a quasicontinuous UBC-domain andXa c-space. IfLhas a least element orXis connected, then the Isbell and Scott topologies coincide on [X®L].

(3) LetLbe a quasicontinuous UFL-domain and the topological spaceX=Xi, where everyXiis an irreducible Scott c-space andIis a nonempty _nite set. IfLhas a least element orIis a singleton, then the Isbell and Scott topologies coincide on [X®L].

講座人簡介：

李慶國，男，漢族。生于1963年6月。博士，數(shù)學(xué)學(xué)院二級教授，博士生導(dǎo)師，校學(xué)術(shù)委員會委員。曾任湖南大學(xué)研究生院院長，湖南大學(xué)科技處處長，湖南省人民政府學(xué)位委員會委員。1999年7月至2000年6月及2008年11月至2009年11月分別在美國科羅拉多大學(xué)數(shù)學(xué)系和康涅底克大學(xué)數(shù)學(xué)系作訪問教授。2000年12月起擔任湖南大學(xué)應(yīng)用數(shù)學(xué)專業(yè)博士生導(dǎo)師。現(xiàn)為中國系統(tǒng)工程學(xué)會模糊數(shù)學(xué)與模糊系統(tǒng)委員會副理事長，湖南省數(shù)學(xué)學(xué)會副理事長，《模糊系統(tǒng)與數(shù)學(xué)》雜志編委，美國“數(shù)學(xué)評論”特約評論員。入選湖南省121人才第一層次，國務(wù)院政府特殊津貼獲得者。曾獲2013年湖南省自然科學(xué)一等獎，排名第一。2009年湖南省自然科學(xué)二等獎，排名第二。已完成國家自然科學(xué)基金課題《廣義連續(xù)格上拓撲及應(yīng)用研究》、《模糊概念格理論及在信息科學(xué)中的應(yīng)用》、《量子邏輯和模糊邏輯的相關(guān)問題研究》、《Domain結(jié)構(gòu)與信息系統(tǒng)的表示理論研究》四項，及教育部博士點基金課題，湖南省自然科學(xué)基金重點課題二項。現(xiàn)在正承擔國家自然科學(xué)基金課題《連續(xù)偏序集的拓撲性質(zhì)、笛卡爾閉性及函數(shù)空間的研究》，湖南省自然科學(xué)基金重點項目《量子邏輯的基礎(chǔ)結(jié)構(gòu)研究》等。

目前主要研究領(lǐng)域為格上拓撲、模糊數(shù)學(xué)理論與應(yīng)用。重在研究計算機與信息科學(xué)中所涉及的數(shù)學(xué)問題，主要從以下三個方面著手進行研究：計算機程序語言的指稱語義—Domain理論，計算機與信息科學(xué)的邏輯基礎(chǔ)，形式概念分析及粗糙集理論等新的數(shù)學(xué)理論在信息科學(xué)中的應(yīng)用。至今為止，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》等國際期刊上發(fā)表論文近100篇。累計培養(yǎng)博士畢業(yè)生41名，全部進入高校工作。兩次獲得湖南省優(yōu)秀博士學(xué)位論文指導(dǎo)獎。