CAREER: Sharp Concentration and Probabilistic Methods

职业：高度集中和概率方法

• 批准号: 0239316
• 负责人: Van Vu
• 金额: $ 40万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-09-01 至 2006-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0239316&HistoricalAwards=false
• 关键词: CAREER; Sharp; Concentration; Probabilistic; Methods

DMS-0239316Van V. VuIn this proposal, the PI addresses some of the deepest and mostimportant issues in probabilistic combinatorics, including sharpconcentration inequalities and the semi-random method. The PI and hiscollaborators plan to develop new sharp concentration inequalities andapply them, together with the powerful semi-random (Rodl nibble) methodto attack several well-known and long standing conjectures incombinatorics. Among others, the PI will study a series of minimaxproblems in finite geometry, raised in the fifties. Another topic is thestudy of geometrical properties of random objects. Here is a typicalquestion: how many points does one need to pierce n random d-dimensionalboxes, where a random box is obtained as the Cartesian product of drandom subintervals of the unit interval ? The PI also plans to usemethods from probabilistic combinatorics to attack problems arose fromvery practical fields such as coding theory and the study of"real-world" graphs. For educational purposes, he intends to write atextbook on sharp concentration inequalities and its applications incombinatorics and theoretical computer science. He also plans to designand teach a graduate course at UCSD on the probabilistic methods. In the last twenty years, probability has become one of the mostpowerful weapons in discrete mathematics and computer science. Thefirst main reason is that randomness helps us to overcome algorithmicand existence problems which cannot be solved deterministic means. Thesecond main reason is that many important real-life objects (such as theinternet) can be modeled by a random process. On the other hand, inmany situations, the problems are so complex that traditional resultsfrom probability theory cannot be applied. The PI plans to develop newprobabilistic tools which would have a break-thought impact on many suchproblems, some of them have been open for decades. Among others, thesenew tools would help us to design very efficient randomized algorithms,to prove the existence of interesting objects, and to analyzecomplicated processes. For educational purposes, the PI intends towrite a textbook on discrete probability and its applications indiscrete mathematics and theoretical computer science.

DMS-0239316VAN V. VUIN本提案，PI解决了概率组合学中最深切，最重要的问题，包括敏感的不平等和半随机方法。 PI和Hiscollaborators计划培养新的急剧浓度不平等，并将其应用于强大的半随机（Rodl Nibble）方法，以攻击几种知名和长期存在的猜想。除其他外，PI将研究五十年代提出的有限几何形状中的一系列最小问题。另一个主题是随机对象的几何特性的详细信息。这是一个典型的问题：需要多少点才能刺穿n随机d二维框，其中将随机盒作为单位间隔的Drandom亚间隔的笛卡尔产物获得了一个随机盒？ PI还计划从概率组合制剂中进行使用，以攻击问题，这是由于编码理论和对“现实世界”图的研究而出现的。出于教育目的，他打算写关于急剧集中不平等及其应用程序和理论计算机科学的Atextbook。他还计划设计并在UCSD教授概率方法的研究生课程。在过去的二十年中，概率已成为离散数学和计算机科学中最大的武器之一。 最初的主要原因是随机性有助于我们克服无法解决确定性手段的算法和存在问题。第三个主要原因是许多重要的现实生活对象（例如theternet）可以通过随机过程进行建模。 另一方面，问题是如此复杂，以至于无法应用概率理论的传统结果。 PI计划开发新的企业家工具，这些工具将对许多此类问题产生分裂的影响，其中一些已经开放了数十年。除其他外，这些工具将帮助我们设计非常有效的随机算法，以证明有趣的对象的存在并分析过程。 出于教育目的，PI打算拖着一本关于离散概率及其应用程序的教科书，及其应用程序不可差的数学和理论计算机科学。