“Original and profound” —that’s how the Society for Industrial and Applied Mathematics (SIAM) characterized the work of Karl Liechty when awarding him the 2015 Gábor Szegő prize, which recognizes outstanding research in orthogonal polynomials and special functions. Liechty’s research focuses on the six-vertex model, also called the "square ice" model (seen on the board).
The assistant professor in mathematics explains:
“There are six types of H2O molecules. When water forms ice, they crystalize into a matrix pattern. How many different ways can these H2O molecules align with each other? That’s a mathematical counting question. But physicists are interested in a slightly different issue, and here’s where my work comes in: If each molecule type has a different kind of energy, what is the total energy of the sheet of ice? The lowest energy state is called the ground state, but since systems don’t stay steady, there are fluctuations around the ground state. And that raises a second question: How big are the fluctuations?
“With my research partner, Pavel Bleher of Indiana University-Purdue University Indianapolis, I’ve used discrete orthogonal polynomials, which are complex and difficult to calculate, to identify classes of energy and their corresponding thermodynamic behaviors. Simply said, I’m using math to understand and describe ‘universal’ phenomena in the physical world. Certain objects appear in disparate places both in theory and in nature, and the question is why? This is an exciting field of research.”
Three of the mathematicians’ published works, which were cited by the prize committee, cover multiple phases of the “square ice” problem. Each phase posed specific and different mathematical difficulties that had to be resolved. A fourth work centered on Liechty’s original analysis of non-intersecting random paths, which is also connected to random matrix theory.
At the heart of Liechty’s research is the fact that physicists and mathematicians solve problems in ways that are different but complementary.
“Science is all about making and testing hypotheses, but math is all about rigorous proofs," he explains. "So, physicists come up with good ideas, brilliant ideas, but they aren’t trying to prove anything by mathematical standards. I like to think that they’re giving mathematicians a gift. Working on a problem like the ‘square ice’ challenge is satisfying because it’s not easy to find a model that’s robust enough to represent something physical, yet simple enough to be expressed and explored mathematically. I find the concrete nature of the inquiry very appealing.”
Before becoming a mathematical analyst Liechty was a jazz saxophonist.
“To me, music was always about creatively forming interesting patterns with some kind of a constraint in the background,” he says, explaining the leap from one discipline to the other. “If a jazz musician is playing Bye-Bye Blackbird, the chord changes in the background are the form; but the ‘something interesting’ that happens also has to have some structure. The same kind of interplay happens when math is used to solve problem in physics and probability. I know quite a few musicians, jazz and classical, who are mathematicians and vice versa. There’s a definite connection.”