Emmy Noether Lectures

2015 Lecturer: Wen-Ching Winnie Li

Modular Forms for Congruence and Noncongruence Subgroups

The arithmetic of modular forms for congruence subgroups of SL(2, Z) has been a central theme in number theory for over one century. It has close connections with many branches of mathematics. Wiles’s proof of Fermat’s Last Theorem has brought the field to a new climax. The arithmetic of modular forms for noncongruence subgroups, on the other hand, has not attracted much attention in the past. However, the research in this area has been reinvigorated in the past decade. This talk is an overview of the progress on modular forms for both congruence and noncongruence subgroups as well as the connections between these two kinds of forms.
Brief Biography:
Wen-Ching (Winnie) Li is a Distinguished Professor of Mathematics at Penn State University, where she joined the faculty in 1979 As well her position at Penn State, Li serves as Director of Taiwan’s National Center of Theoretical Sciences. Previous honors and awards include a Sloan Foundation Fellowship in 1981 and the 2010 Chern Prize in Mathematics, awarded by the International Congress of Chinese Mathematicians for her outstanding contributions to mathematics.  She was named a Fellow of the American Mathematical Society in 2012.
Li’s research focuses on number theory, in particular modular forms and automorphic forms, as well as broad applications to coding theory and spectral graph theory. She has more than 90 publications. Li’s thesis work on the “new” space of modular forms based on the renowned work of Atkin-Lehner was cited in Andrew Wiles’ proof of Fermat’s Last Theorem.  Li studies the rich interplay between combinatorics, group theory, and number theory through associated zeta functions. In particular, she has applied her research results in automorphic forms and number theory to construct efficient communication networks called Ramanujan graphs and Ramanujan complexes. In recent years, she has done important work on the arithmetic of modular forms for noncongruence subgroups, which revitalized the field.
Li has always been seriously involved in activities to promote the advancement of women in mathematics. In particular, she was a mentor for the Women Mentoring Program at IAS in 1999 and the BIRS workshop on Women in Numbers in 2008, and was the Distinguished Women in Mathematics Lecturer at UT Austin in 2011.
Prof. Li’s work is impressive for its depth, the connections it makes between different areas of mathematics, and its continuing influence.  With a career that has already spanned over 35 years of excellent research, Prof. Li stands out as someone who truly deserves the honor of being chosen as the Noether Lecturer.