Dima Arinkin didn’t fully realize it at the time, but even as an undergraduate student, the work he was doing was part of something much, much bigger.
Arinkin, a professor in the Department of Mathematics, has spent his career studying algebraic geometry, a discipline that uses algebraic equations to describe and reveal truths about geometric shapes. Put another way, Arinkin uses one branch of mathematics to shed light on another, creating a bridge between the two.
That approach is also the essence of something called the Langlands Program, a multi-pronged conjecture first proposed by the Canadian mathematician Robert J. Langlands in the 1960s that relates to the mathematical description of complex waves and frequencies. Exploring its impact has been the work of multiple decades for armies of modern mathematicians, united in their pursuit of what Edward Frenkel, a mathematician known for his efforts to popularize difficult concepts, called "the grand unified theory of mathematics."
“Some of the first things that I worked on were already motivated by the Langlands conjecture,” says Arinkin, who came to UW–Madison 11 years ago following stints at the University of North Carolina and the California Institute of Technology. “I didn’t understand the topic or the big picture then, but now I do.”
A paper Arinkin co-authored in 2012 on ind-coherent sheaves — an algebraic tool for studying geometric spaces — became one of the main building blocks that allowed a different group of nine mathematicians to prove one of the trickiest parts of the Langlands Program, the Geometric Langlands Conjecture, earlier this year. Creating connections between multiple fields of mathematics was central to the proof.
“That’s what makes this area so exciting for me, because it’s like you start speaking one language, and then you realize what you are saying actually makes sense in a different language,” Arinkin explains. “You can say something algebraic and then jump into a geometric explanation or geometric argument. That’s one of the big reasons why it’s so hard to talk about, because if we’re going to use several languages, we need to understand them all.”
He’s not wrong about the Geometric Langlands Conjecture (and its significance to the field of mathematics) being difficult to describe — especially to a non-mathematician audience. Its particulars involve complex terminology like automorphic functions and eigensheaves.
“What made people excited about this conjecture is that it said that something you can compute through manipulating with numbers, should match some other computation that you can do by studying symmetries of different kinds of objects,” says Arinkin.
The Geometric Langlands Conjecture may have been proven, but the research surrounding the other branches of the Langlands Program is far from complete. Arinkin believes there’s now work to be done looping geometric methods back to their origins in number theory, leveraging connections to different areas of mathematics and beyond, including fields such as quantum physics. The Geometric Langlands Program represents just one of the ways in which mathematicians are collaborating across disciplines.
“Now that there is all this development, now that there are all these exciting things we can prove in the geometric field, we can ask whether we can bring all this understanding back,” Arinkin says. “We are trying to understand how to use something developed in one area in another area of mathematics. But as we try to answer it, we have to develop new tools.”
In doing so, there’s the possibility of creating even more bridges, this time between fundamental and applied mathematics. Arinkin points to number theory, a discipline that was developed multiple centuries ago with no practical application in mind. Today, the principles of number theory have become the central basis for internet encryption, an essential piece of the modern technological landscape. Arinkin’s own research is often more fundamental than applied, but the tools he ends up creating through that research could have critical practical applications in the future.
“Mathematics is so abstract, it also means that it’s universal,” says Arinkin, pointing out that something as straightforward as practically solving equations can become useful to something like predicting epidemic distributions during an outbreak. “It’s like you invent a tool that can magically drill through a thick wall of rock. I might not immediately know how to use it, but it’s amazing that we have figured out how to do it.”
Arinkin makes a point of coaching the PhD students he teaches to take an approach that ignores boundaries and incorporates perspectives from multiple areas of mathematics, just as he has done with his own research.
“Sometimes it’s very easy to get overly focused on your own area. As students, it’s the best time to expand your horizon a little bit so that you at least know what’s happening with the big picture of mathematics,” he says. “The boundaries between different areas of mathematics are not rigid in the abstract world, and they are not rigid in our department.”