Real Math is Never Artificial: Professor Christian Gaetz

(Photo by Johnny Gan Chong)

Pure mathematics and theoretical physics have long had a P.R. problem. People think that they are pursuits exclusively for long-haired wizards with thick accents who stay cooped up under the eaves covered in chalk dust. Math takes the worst of the stereotypes because physics at least has a wow factor—billions of lightyears or thousands of suns worth of heat can outshine a lot of eccentricities, but what do mathematicians have to reveal? And how can anyone besides other mathematicians appreciate their accomplishments?

Christian Gaetz rejects the notion that math is in a different category than other pursuits. He joined the Berkeley Mathematics Department faculty in 2024 and is quite vocal in his insistence that math should “not be daunting to anyone.” It isn’t just for “rarefied geniuses” but for anyone who puts in the time to build good foundations and exercise the right muscles in their head. His specialization is in algebraic combinatorics and its connections with geometry. His last institution, Cornell, tries to spook prospective majors by warning that “Combinatorics is, arguably, the most difficult subject in mathematics, which some attribute to the fact that it deals with discrete phenomena as opposed to continuous phenomena, the latter being usually more regular and well behaved.” Gaetz considers this more bad P.R.

(Photo by Johnny Gan Chong)

Combinatorics is “relatively hands-on” compared to what a lot of Berkeley faculty are working on, muses Gaetz. It focuses on the relationships between certain kinds of polynomials or graphs or matrices rather than on their contents. Imagine a map that shows several towns connected by roadways. Rather than worry about what is in each town, combinatorics looks at how the connections relate to each other. In higher-level algebra, these relationships and patterns are often studied using combinatorial methods, and if these methods give the same result regardless of how you label or organize parts, it’s called combinatorial invariance. A common way of explaining this to younger students is to consider two LEGO structures that have the same arrangement of connections: they are essentially the same, even if the individual blocks are different colors or sizes. The LEGO picture is also helpful because combinatorics users often talk about ‘edges’ and ‘vertices’ when describing how parts relate.

Logic, algebraic topology, and other areas of mathematics involve more layers of abstraction than algebraic combinatorics, so Professor Gaetz feels himself to be on somewhat firmer ground. There is an especially puzzling problem called the Combinatorial Invariance Conjecture that “too many mathematicians to name” have been trying to solve since the 1980s which was first proposed by Lusztig and by Dyer. This one involves eponymous and appropriately titled Kazhdan-Lusztig polynomials. A group of investigators abbreviated BBDVW (Blundell, Buesing, Davies, Veličković, and Williamson) have recently been reconsidering the Conjecture using machine language (ML) algorithms. ML is usually associated with practical applications like image recognition or language translation, but BBDVW feed tons of descriptions of shapes and equations into ML to see if it can find invariants that will resolve the Conjecture.

Gaetz believes that he has a contribution to make towards the solution that adds perspective from other angles of approach, ones that he happens to be fluent with. He explains that the ML approach has found combinatorial structures called "hypercube decompositions" that might assist in proving the Combinatorial Invariance Conjecture, but the ML itself can't prove anything. Gaetz and his colleague Grant Barkley have proven new pieces of the puzzle, and they are working on proving the full Conjecture. He sums it up this way: the Combinatorial Invariance Conjecture states that Kazhdan--Lusztig polynomials, which are important in several areas of mathematics like algebraic geometry and representation theory, can be determined using only information contained in certain graphs called Bruhat graphs. The challenge is to figure out how to extract this information from the very complicated graphs. BBDVW employed ML to write down a recipe for extracting this information that they believe gives the correct answer, but they weren't able to prove that their recipe works. “Grant Barkley and I have proven that the recipe does work in many cases, and we are working on proving the full conjecture.”

Christian Gaetz caught the math bug early, so he has a solid foundation. He insists that mathematicians, like most professionals, are built rather than born. Maybe basketball players have either got it or they don’t, but the rest of us acquire expertise the hard way – layer by layer, hour by hour, of devoted learning and practice. The lucky part is connecting with good mentors. Gaetz’s summa cum laude B.S. (2016) from the University of Minnesota was under Victor Reiner, a sage for the ages. At MIT for his Ph.D., Alex Postnikov (former Miller Fellow here at UC Berkeley) lit the flame of algebraic combinatorics that serves as Professor Gaetz’s guiding light. After a productive postdoc at Harvard where he worked with former UC Berkeley professor Lauren Williams, he won a fellowship at Cornell (’22-’24) where he connected with MIT alums from the decade previous. He gives Karola Mészáros and Allen Knutson (also a former UC Berkeley professor) due credit for mentorship in that period. This is what explains his expertise, Gaetz says. “When someone says, ‘I’m not a math person’ it’s because they didn’t have the opportunities or the right kind of exposure at the start,” he believes. A good math education makes it “natural” to be a math person.

Professor Gaetz comments that math sits at a special position among the arts and sciences. Like artists, mathematicians employ a lot of creativity along the way to a solution. But a piece of art is finished when the artist decides to stop; it depends on the creator’s subjective impulses. Mathematical solutions are clearer: everyone agrees when you’ve reached the ‘right’ conclusion. The solution seems like it was discovered rather than made.

That combination of creativity and confirmation is part of why Christian Gaetz is happy with his choice of career. He wasn’t aiming for it from the start but he’s glad he made it here. The tools and methods math uses aren’t objects of worship. Gaetz keeps a computer by his desk to run examples of what he’s thinking about only as a “sanity check” to help stay on track. He recalls an article by esteemed physicist Eugene Wigner. The article notes that mathematicians are often guided by an aesthetic sense, rather than potential applications. “Yet the ideas they produce very often eventually find application in science,” Gaetz marvels. “I think that mathematicians have developed a remarkable collective aesthetic by which ideas that mathematicians find interesting, and well-motivated, and beautiful very often get at something deep about the way that the universe works. And that such ideas often are found later to be important in natural science.” With the right foundation, these are accomplishments that everyone can appreciate.

(Photo by Johnny Gan Chong)