Chris Rogers, an associate professor in the Department of Mathematics and Statistics, was recently awarded a grant by the Division of Mathematical Sciences in the National Science Foundation. The work Rogers and his doctoral students are doing is very abstract but has real implications.
“To me, this kind of math is a fundamental part of our reality,” Rogers said. “We may invent the tools and give names to the theories, but the patterns and relationships that we are uncovering, they feel like real things that have been waiting to be discovered.”
For Rogers, the best math is applied math, but “applied” doesn’t have to mean used in physics, chemistry or other sciences. In Rogers’ mind, applied math also describes “pure math” that can be used to solve problems in other areas of math. Rogers’ own research studies how groups, a technical term for sets of numbers or other mathematical objects which describe symmetries, are related to one another.
According to Rogers, the power of group theory is that it can reveal highly non-obvious relationships between very different kinds of mathematics. Two seemingly different objects, say, an equilateral triangle and a polynomial equation, can sometimes have symmetry groups that are closely related or even equal to each other. When this happens, it suggests that there is a hidden connection between the two objects.
Mathematical groups can represent real-world mysteries
“The whole project is about studying the relationship between the symmetry groups of two kinds of mathematical objects that have, at first glance, really no business being related to each other,” Rogers said.
"I have some ideas about how to make these groups talk to each other in ways that no one has ever done before."
The two groups Rogers is speaking about are the KRV group, named for mathematicians Masaki Kashiwara and Michèle Vergne, and the GT group, named for mathematicians Alexander Grothendieck and Oswald Teichmüller and discovered by mathematical physicist Vladimir Drinfeld. The GT group contains the symmetries of objects called “braids” (think woven-together pieces of rope), which, along with knots, are important objects of study in a branch of math called topology.
Over the past three decades, mathematicians have been able to manipulate the symmetries buried inside the KRV and GT groups and show that they are related, not just to one another, but to symmetries of other mathematical objects. These results are considered “deep,” which, according to Rogers, is a “revered, almost sacred word” in mathematics that means the results have foundational implications.
The first of these deep results was discovered by Drinfeld in 1990, who showed that the GT group contains what is known as the “absolute Galois group,” a massive, mysterious group which is related to number theory. Rogers described this group as the ultimate generalization of the quadratic formula learned in high school.
“The absolute Galois group contains, in a very compressed and highly organized way, the information needed to solve every polynomial equation in existence,” Rogers said.
Even more remarkable than Drinfeld’s result is, according to Rogers, how Drinfeld discovered it.
“Drinfeld realized that the symmetries of certain equations used in quantum physics have the same properties as both the symmetries of braids and the symmetries found in the absolute Galois group,” said Rogers.
Results like this are the reason why Rogers believes that this kind of mathematics is really “built into” our universe.
The KRV group, on the other hand, is more explicit and concrete, involving symmetries of equations whose solutions aren’t numbers but rather objects called matrices. The important thing about matrices is that they have strange multiplication laws. Unlike numbers, “A times B” is not necessarily equal to “B times A,” when A and B are matrices. This, Rogers said, is what makes matrices the right kind of math to use in quantum physics.
“These are equations that involve trig and log functions whose inputs and outputs are matrices, just like the matrices we teach STEM undergrads about in linear algebra,” said Rogers.
A second deep result, by Anton Alekseev and Charles Torossian, proved in 2012 that all of the symmetries contained in the GT group can also be found inside the KRV group.
Now, Rogers is working to prove that the groups aren’t just related to one other.
“The conjecture is that KRV and GT are actually the same group,” Rogers said. “I have some ideas about how to make these groups talk to each other in ways that no one has ever done before. That’s why I wrote the grant proposal.”
Rogers and collaborators have already made significant progress in showing that the KRV group and the GT group act in very similar ways. These results use technical ideas from areas called “homotopy theory” and “algebraic operads,” which Rogers likens to specialized tools that can be used to deform and transform symmetries, allowing them to pass from one group to the other.
Homotopy theory is a hot topic in mathematics right now, Rogers said, because it can be used to make different areas of mathematics interact with each other in highly non-trivial ways. He said that is the big draw for math students who want to work on deep problems. Despite being so abstract, Rogers emphasized that many of the basic ideas behind his research are accessible even to undergraduate students.
“Introductory group theory is the topic of our Math 331 course, which many math majors and minors take in their junior year,” Rogers said. “And a student first meets homotopy theory typically in their senior year in our Math 441 course on algebraic topology.”
Math requires creativity
Mathematicians like Rogers spend most of their working time thinking about unsolved, or “open,” mathematical problems. Some open problems have been around only a few months, others have remained unsolved for hundreds of years. In either case, the solution is always expressed using a mathematical proof: a rigorous, airtight logical argument that justifies why the solution is indeed the correct one and thereby rules out a potentially infinite number of other possibilities.
“A proof leading to the solution of an open problem, especially a longstanding one, usually requires the invention of brand-new mathematical ideas and original insights,” Rogers said. “It is here where mathematics becomes a highly creative enterprise.”
Because computers can’t comprehend infinity, they can only provide examples and not a complete proof, so their usefulness in this kind of math is limited. Quantum computers, when they eventually become practical, might be able to provide examples more quickly and closer to infinity but they still won’t be able to come up with new relationships. The same goes for artificial intelligence. While AI is good at recognizing patterns in mathematics that already exist, it can’t build new mathematics and construct new proofs like the human brain can.
“I’m skeptical that meaningful progress in my field will come from a computer any time soon,” Rogers said. “If I told an AI to prove Fermat's Last Theorem, and it gave me a proof using the classical mathematics from the 1800s that we somehow missed, then that would not be as exciting as Andrew Wiles’s proof from 1990 because he, and the others before him, had to really invent a whole new way of thinking about the math involved with that problem.”
What makes the deep results in mathematics most interesting, Rogers said, is the new ways of thinking that they generate, not just the solution.
“Day to day, it's a lot of writing out special cases, it's a lot of playing around with examples but only to the point where you gain intuition and insight about them so that you can actually build up to the general statement and then prove it,” Rogers said. “There's no way computers can really do that last step for us.”
Encouraging diversity in mathematics through interdisciplinary collaborations
The NSF grant will support graduate students working with Rogers, and he feels lucky to have worked with a more diverse group of graduate students than math has historically welcomed.
“I want to challenge the traditional narratives of what math is for, who does it and what doing math looks like.”
Rogers hopes that students who have not historically been welcomed into mathematics will see the University’s graduate mathematics program and realize there is a space for them at the University, and in mathematics.
“This money is supporting not just graduate math research, it's supporting the people that should have been supported a long time ago,” Rogers said.
In addition, many of the collaborators who Rogers has been working with to understand these symmetry groups happen to be female mathematicians. Rogers feels these mathematicians, who have made substantial contributions to the field, have been underappreciated. Rogers plans to organize with one of those colleagues a day-long symposia during which ΒιΆΉΣ³» math students can discuss whatever research they’re working on in a supportive, informal setting.
Another major part of the grant will involve coordinating research and career development events with the University’s chapter of the Association for Women in Mathematics. Rogers hopes to show students that there are a wide variety of job types for mathematicians. He also wants to help faculty understand that many students don’t have the luxury of applying for positions in certain regions of the country or world because of safety issues pertaining to their identity.
Rogers wants to build interdisciplinary connections and work with advocacy groups on campus to discuss stereotypes of mathematicians and how those stereotypes perpetuate the lack of diversity within mathematics.
“We think of mathematicians as looking a certain way, behaving a certain way. It’s easy to have a trope replace the presentation of a real person,” Rogers said. “I want to challenge the traditional narratives of what math is for, who does it, and what doing math looks like.”
Rogers hopes that the work and the programming he plans to do with the grant support will help to combat stereotypes and limitations in mathematics, and of course, to advance group theory.
