Number Theory and Combinatorics

这个项目的首要主题是划分理论的新发现。这些发现既涉及新的物体(例如Durfee符号和k标记的Durfee符号)，也涉及古老问题的新方面(例如没有序列的麦克马洪划分，阿尔德猜想等)。首先，作者考虑了k-标记Durfee符号的相关母函数的对称性研究。下一个主题是与具有短序列的划分相关的渐近性。然后，提出者研究了部分分数方法，其起源在于Ramanujan的Lost Notebook(资深PI广泛研究的手稿)。此外，提出者还考察了最近有关讲堂隔断的发现所产生的问题，并对长期存在的阿尔德猜想进行了进一步的研究，该猜想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对第一段中提到的对称性研究采取看似合理的组合方法。虽然这些主题是基于对划分理论(加法数论的一个分支)的研究，但值得注意的是，具有短序列的划分在概率论中具有有趣的含义。此外，最近也有大量工作揭示了k-标记Durfee符号与模形式理论的最新发展之间的有趣关系。

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.