Noncongruence Modular Farms and Supercongruences

模形式是高度对称的壮观函数。它们在数学和物理中大量出现。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，正如费马大定理的证明所证明的那样。模形式的对称性是由整数环上特殊的2次线性群的元素所捕获的。在所有模形式中，大多数模形式是不同余的，即它们的对称性不能用同余来描述。由于缺乏令人满意的赫克理论，非同余模形式的研究一直落后于同余模形式的研究。然而，近年来非同余模形式的研究进展揭示了它们与p进模形式、自同构形式和伽罗瓦表示等几个富有成果的研究前沿之间的丰富联系。非同余模形式理论对共形场论和组合学等领域产生了深远的影响。本课题的目的是在理论上进一步发展非同余模形式，并将其应用于超同余问题，如许多有趣的组合或等差数列所满足的特殊同余。科学目标是：研究附加在非同余模形式上的伽罗瓦表示如何通过朗兰兹对应与自同构形式相关联；理解真正的非同余模形式的一个基本猜想；利用起源于非同余模形式研究的Atkin和Swinnerton-Dyer同余的观点探索超同余。目前，超同余的证明往往需要寻找辅助恒等式，这一过程往往涉及有趣的猜测。对超同余的更概念化的理解可能会揭示如何系统地寻找这些身份。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，继续扩大她的研究项目的影响。近年来，PI一直认真参与促进妇女在数学方面的进步的活动。她是班夫国际研究站2008年和2011年妇女人数研讨会（WIN）和WIN2的小组共同负责人，也是2014年即将举行的win3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.