Noncongruence Modular Farms and Supercongruences

模数形式是高度对称的壮观函数。它们在整个数学和物理学中大量出现。一个多世纪以来，模形式理论一直在数论中发挥着核心作用，费马大定理的证明证明了这一点。模形式的对称性被整数环上的2次特殊线性群的元素捕获。在所有的模形式中，它们中的大多数是非同余的，因为它们的对称性不能用同余来描述。由于缺乏令人满意的Hecke理论，对非同余模形式的研究一直落后于同余模形式的研究。然而，非同余模形式的最新进展揭示了它们与几个卓有成效的研究前沿的丰富联系：P-进模形式、自同构形式和伽罗瓦表示。非同余模形式理论对保形场理论和组合学等其他领域有着深远的影响。该项目旨在进一步发展非同余模形式的理论，并将其应用于超同余等问题，超同余是许多有趣的组合或算术序列所满足的特殊同余。其科学目标是：研究附加到非同余模形式上的Galois表示何时通过朗兰兹对应与自同构形式相关；理解表征真非同余模形式的一个基本猜想；使用起源于非同余模形式研究的Atkin和Swinnerton-Dyer同余的观点来探索超同余。目前，超同余关系的证明通常需要寻找辅助恒等式，这一过程往往涉及有趣的猜测。对超同余的更概念性的理解可能有助于系统地寻找这些恒等式。研究成果将在高质量的期刊上发表，并在各种会议和研讨会上传播。PI将通过指导本科生、研究生和博士后，以及组织科学会议，继续扩大HIS研究计划的影响。近年来，国际数学联合会认真参与了促进提高妇女数学水平的活动。她是2008年班夫国际研究站数字女性研讨会(WIN)和2011年WIN2研讨会的小组联合负责人，并是即将于2014年举行的WIN 3会议的组织者之一。

Modular forms are spectacular functions that are highly symmetric. They occur in abundance throughout mathematics and physics. For more than one century, the theory of modular forms has being playing a central role in number theory, as witnessed in the proof of Fermat's Last Theorem. The symmetries of modular forms are captured by the elements of the special linear group of degree 2 over the ring of integers. Among all modular forms, majority of them are noncongruence in the sense that their symmetries cannot be described by congruences. The study of noncongruence modular forms has been fallen behind its congruence counterpart due to the lack of a satisfactory Hecke theory. However, recent progresses on noncongruence modular forms have revealed their rich connections with several fruitful research frontiers: p-adic modular forms, automorphic forms, and Galois representations. The theory of noncongruence modular forms has far reaching impacts on other areas like the conformal field theory and combinatorics. The project aims at a further theoretic development of noncongruence modular forms with applications to problems like supercongruences, which are special congruences satisfied by many interesting combinatorial or arithmetic sequences. The scientific objectives are: study when Galois representations attached to noncongruence modular forms are related to automorphic forms via Langlands correspondence; understand a fundamental conjecture that characterizes genuine noncongruence modular forms; explore supercongruences using the perspective of Atkin and Swinnerton-Dyer congruences that were originated in the study of noncongruence modular forms. Currently, the proofs of supercongruences often require the finding of auxiliary identities, which procedure often involves intriguing guesses. A more conceptual understanding of supercongruences may shed lights on how to search for these identities systematically.The research outcomes will be published by high quality journals and disseminated at various conferences and seminars. The PI will continue to widen the impacts of the her research program by mentoring undergraduate students, graduate students, and postdocs, as well as organizing scientific conferences. In recent years, the PI has been seriously involved in activities to promote the advancement of women in mathematics. She was a group co-leader for Banff International Research Station workshops on Women in Numbers (WIN) in 2008 and WIN2 in 2011 and is one of the organizers for a coming WIN 3 conference in 2014.