Emmy Noether Lectures

2024 Lecturer: Anne Schilling

The Ubiquity of Crystal Bases

Crystal bases are combinatorial skeletons of Lie algebra representations. They appeared in the work of Kashiwara, Lusztig and Littelmann on quantum groups and the geometry of flag varieties. Crystal bases arise in many unexpected places, from mathematical physics to probability and number theory. In this talk, I will showcase ten reasons and applications of how crystal theory can be used to solve problems in representation theory, geometry and beyond.

Professor Anne Schilling is recognized for sustained leadership and impact in algebraic combinatorics and related fields, including globally recognized research and inspiring contributions to the broader mathematics community.

Schilling is Full Professor and Chair of the Mathematics Department at UC Davis, which she joined in 2000. Previously she held a Moore Instructorship at MIT and was a postdoc at the Institute for Theoretical Physics at the University of Amsterdam. Schilling has been awarded a Humboldt Research Fellowship (2002), a Simons fellowship (2012-2013), and AMS Fellowship (2019).

Schilling’s work in algebraic combinatorics is highly regarded, combining deep results with elegant solutions and appearing in the leading journals of mathematics. Continuously supported by the NSF through both collaborative and individual grants, the work spans problems that many would agree are the most fundamental and difficult in the area, such as the restriction problem, the plethysm problem and the combinatorial analysis of other structure coefficients. Schilling is the co-author of nearly 100 papers, two research texts, and a linear algebra textbook. Her recent book with Dan Bump on crystal bases has been highly praised by others using it in the community.

In addition to her research accomplishments, Prof. Schilling is an inspiring mentor and an extremely active leader in the research community. She has organized over 27 conferences and workshops in her field, most recently as the main organizer for the Formal Power Series and Algebraic Combinatorics conference, the largest annual conference in Algebraic Combinatorics known for highlighting contributions of junior mathematicians. In addition, she is one of the main contributors to SageMath, a free and open source computer algebra system, which is an invaluable tool for experimentation in mathematics. She is an editor on the board of two “flipped” journals, Combinatorial Theory and Algebraic Combinatorics, and she helped to write the constitution for Algebraic Combinatorics as it was getting established. Schilling’s energetic mentorship has been visible in several venues, complementary to her supervision of eleven graduate students and several postdocs. She has been active in two Algebraic Combinatorixx workshops at Banff and in the Research Community in Algebraic Combinatorics at ICERM, mentoring untenured women faculty and leading research groups that support junior women researchers in their careers.