University of Michigan student Sophie Kriz was awarded an Honorable Mention for the 2023 Morgan Prize for outstanding research in mathematics by an undergraduate student.

Kriz has written nearly a dozen research papers in representation theory and category theory and has co-authored a textbook, Introduction to Algebraic Geometry. She was awarded an honorable mention for this prize last year as well.

“Mathematics has been my main interest since an early age,” she said. “It is a field that combines creativity and imagination with practicality and real-world applications.”

The daughter of mathematicians, Kriz, who was born and raised in Ann Arbor, grew up around abstract mathematics and quickly developed a fascination for it.

“It’s really quite interesting,” she said. “It balances things that exist within your mind and things which you can work with, in a more material sense. It is essential for students to learn more about it and to develop their skills. That skill set is important in work, in the progress of science and knowledge as we know it.”

The Noether problem

Kriz said the first dilemma that drew her into this field was the Noether Problem. Simply put, it asks if rational functions—ratios of polynomials—that do not change under some specifically allowed swaps of variables can always be expressed as rational functions of some fixed parameters. It is known not to work in general, but the question for what systems of variable swaps the parametrization exists is still open. The systems of swaps are called permutation groups.

“Emmy Noether is one of the most important female mathematicians in the world,” Kriz said. “She created the basic framework of algebra, which we use today, and she still worked on other completely different topics. Her mathematical theorem on conservation laws in physics changed the way physicists study the universe.”

Inspired by Noether, Kriz found a positive solution to Noether’s rationality problem for an infinite set of particular groups, which resulted in her first paper, published by Communications in Algebra.

“I was so happy to have done some work that impacted the broader scientific community, research that added to some collective knowledge we have about mathematics,” she said. “It meant so much to me.”

Now, in her senior year as an Honors Mathematics major, Kriz has published five papers and has another already accepted for publication. She also has several other articles in the works.

“Sophie is one of the most talented, hard working and ambitious students I have ever met,” said her mentor, Andrew Snowden, U-M professor of mathematics. “She’s already written more research papers than most postdocs. And her papers cover a lot of different subjects: topology, algebraic geometry, representation theory, and more. It is an extremely impressive body of work for an undergraduate and her honorable mentions for the Morgan Prize are very well-deserved.”

The prize is made jointly by the American Mathematical Society, the Mathematical Association of America and the Society for Industrial and Applied Mathematics. Kriz will receive the honor during the Joint Mathematics Meetings in Boston in January 2023.