Number Theory and Combinatorics

数论和组合学

• 批准号: 0801184
• 负责人: George Andrews
• 金额: $ 16万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0801184&HistoricalAwards=false
• 关键词: Number; Theory; Combinatorics

The overarching theme of this project is new discoveries in the theory of partitions. These discoveries concern both new objects (e.g. Durfee symbols and k-marked Durfee symbols) and new aspects of venerable problems (e.g. MacMahon's partitions without sequences, Alder's Conjecture, etc.). First the proposers consider symmetry studies of the relevant generating functions for the k-marked Durfee symbols. The next topic is the asymptotics related to partitions with short sequences. Then the proposers study partial fraction methods whose genesis lies in Ramanujan's Lost Notebook (a manuscript studied extensively by the senior PI). In addition the proposers look at questions arising from recent discoveries concerning lecture hall partitions and conclude with further investigations of the long standing Alder Conjecture on which the co-PI has made a major breakthrough. The proposal continues the training of graduate students, one of whom has started a plausible combinatorial approach to the symmetry study mentioned in the first paragraph. While these topics are based on studies in the theory of partitions (a branch of additive number theory), it is notable that partitions with short sequences have interesting implications in probability theory. Also there has been a spate of recent work revealing fascinating relations between k-marked Durfee symbols and recent developments in the theory of modular forms.

这个项目的首要主题是分区理论的新发现。这些发现既涉及新对象（如Durfee符号和k标记的Durfee符号），也涉及老问题的新方面（如MacMahon的无序列分割，Alder猜想等）。首先，作者考虑了k标记Durfee符号的相关生成函数的对称性研究。下一个主题是与短序列分区相关的渐近性。然后，提议者研究了部分分式方法，该方法起源于拉马努金的《丢失的笔记本》（高级PI广泛研究的手稿）。此外，提议者还研究了最近发现的有关演讲厅隔墙的问题，并以进一步研究长期存在的奥尔德猜想（Alder Conjecture）作为结论，而共同指数在这一猜想上取得了重大突破。该提案继续对研究生进行培训，其中一名研究生已经开始了对第一段提到的对称性研究的似是而非的组合方法。虽然这些主题是基于分区理论（加性数论的一个分支）的研究，但值得注意的是，短序列的分区在概率论中具有有趣的含义。最近也有大量的工作揭示了k标记的Durfee符号与模形式理论的最新发展之间的迷人关系。