## At the Interface of Mathematics and Physics

In the higher reaches of physics, says Federico Camia, the border with mathematics has become remarkably porous. And he is living proof of that: holder of a Ph.D. in physics from NYU New York, he is today a visiting associate professor of Mathematics at NYU Abu Dhabi.

Camia's work in theoretical statistical mechanics focuses on systems such as the Ising model, a way of thinking about ferromagnetism. Named after 1920s German physics student Ernst Ising, this is perhaps the most-studied model of statistical mechanics, said Camia. And now the Ising model is yielding important new insights, thanks to the use of recently developed mathematical tools.

"I study stochastic models with a spatial structure," Camia said. That is, he is interested in "both randomness and geometry."

If those sound mutually exclusive, consider that probability theory began as the study of single random variables. But "things have evolved. One of the most interesting fields of modern probability theory is the study of not just random numbers but random geometric structures: systems made up of many components that interact randomly with each other, in a way that depends on how they are arranged in space. That requires an interplay between geometry and randomness," Camia said.

To explain, he cites percolation, referring to the way fluids filter through a porous material, like water moving through coffee grounds. The process is not regular, so developing a model to describe it is a challenge. One way is by starting with a regular structure of channels, organized as in a lattice. Randomness is then introduced by deeming each channel to have a given probability of being closed. The process of percolation reveals that at a certain critical value of that probability, a startling phase transition occurs. Camia offered an analogy with water: at critical points of temperature, water changes from solid to liquid to gas. Similarly, as the density of closed channels increases, at a certain point, the probability that water will permeate a porous material completely will suddenly jump from almost one to almost zero.

Camia's research seeks to explain the behavior of the whole system at one of these critical points. "Systems at or very near the critical point have special properties," he said, "properties they share with certain systems from particle physics and quantum field theory. There's not an obvious physical connection, but the mathematics you need to model these two subjects is the same."

It's quite amazing that physics turns out to be described by mathematical language that many of us find quite beautiful.

Federico Camia, NYUAD visiting associate professor of Mathematics.

This has been known for a long time, but today "new mathematical tools have proved very useful," he said, producing new results so that "we start to understand a lot more." This realization has galvanized scholars around the world, he adds. In 2006 and again in 2010, the International Medal for Outstanding Discoveries in Mathematics, known as the Fields Medal, was awarded in this subject area. One such new tool is an equation that describes random fractal curves with very special properties that seem to characterize two-dimensional systems at the critical point.

At the critical point, Camia goes on, a system acquires a new symmetry known as scale invariance: if one takes a portion of the system and blows it up to the same size as the whole, it looks just like the whole system. Exploiting this fact (in the opposite direction), one can study scaled-down versions of a system, and as the system becomes denser — its points closer together — "you get a continuum, and it has fractal properties, so that brings in the mathematics of fractals…it's quite amazing that physics turns out to be described by mathematical language that many of us find quite beautiful."

Working at the interface between mathematics and physics brought Camia to a fruitful partnership in 2013, when for a semester at NYUAD he was able to work with two visiting scholars, Matthew Kleban, NYUNY associate professor of Physics, and Alberto Gandolfi, a professor of Probability and Mathematical Statistics at the University of Firenze, Italy. The three joined forces to work on a problem in cosmology and mathematical physics, Camia recalls. He has been busy editing the paper they wrote.

He believes that in many fields there is much to be accomplished and discovered at the places where one field of knowledge meets another. "It is unfortunately true that we can tend to work in silos," he acknowledges. "As we get more specialized, problems become more complicated and so do the tools we use; it takes time to learn new tools.

"So cross-pollination is not easy. Once you become an expert in something, it can be a temptation to just keep using the tools you've learned to make incremental progress. But as you find out more, there are new questions that arise, and you realize there are connections with other fields."