# Caltech’s Tom Hutchroft Awarded Prestigious Packard Fellowship for Math Research

Since 1988, the Packard Fellowships have “encouraged visionary work by providing maximum flexibility through unrestricted funds that can be used in any way the Fellows choose, including paying for necessities like child care,” according to the David and Lucile Packard Foundation, which bestows the awards. Hutchcroft is among 20 early-career scientists who are receiving the fellowships this year.

Hutchcroft studies probability theory—more specifically percolation theory. This is a study of the complex math that occurs when percolating systems, such as water traveling through ground espresso beans or diseases spreading through populations, reach phase transitions. During these critical phases, fractal-like mathematical objects emerge.

“One of the things that makes this area so interesting, in my view, is that the same mathematics often describes different physical systems that don’t have anything to do with each other,” says Hutchcroft.

Hutchcroft’s work has made significant progress in understanding phase transitions in curved geometries, or what mathematicians call non-Euclidean geometries. In 2017, he proved that percolation in negatively curved spaces, such as those resembling a Pringle potato chip, always undergoes two phase transitions, in which there first emerges infinitely many infinite clusters that only later merge into a single infinite cluster. In a preprint posted in 2023, with Caltech graduate student Philip Easo, Hutchcroft also solved Schramm’s locality conjecture, a celebrated problem about percolation in non-Euclidean geometries.

More recently, Hutchcroft has been focusing on one of the toughest problems in percolation theory—namely to find the mathematics that describe what happens during phase transitions in three dimensions. While the problem has been solved for two dimensions and even higher dimensions beyond 11, the cases for three, four, five, and six dimensions have proved particularly hard to crack.

“The most basic question is to figure out if the phase transition occurs with a jump, what we call discontinuous, or more smoothy, what we call continuous,” Hutchcroft said in a Caltech Q&A about his work. “This problem has been open for a really long time and needs to be solved before we can move on to understanding all the cool fractal stuff that should be happening at the phase transition.”

One avenue for approaching the problem involves what is called long-range percolation. Percolation problems are often described with grids, in which the edges are given different probabilities of being open or closed. If, for instance the percolation model is describing liquid traveling through a porous medium, then the critical phase would occur when the edges are open just the right amount to let liquid meander across the grid.

With long-range percolation, edges are not restricted to neighboring points in the grid but can appear between any two nodes with a probability that decays as the distance between nodes gets larger. Changing this decay parameter, it turns out, has similar effects to changing dimensions, which allows mathematicians to, in essence, take a back road into possibly solving the percolation problem in three dimensions.

“The Packard Fellowship will give me a huge amount of flexibility to pursue high-risk speculative projects and engage with the most fundamental problems in the area,” Hutchcroft says. He notes that one of the best things about math is getting totally lost in a problem for hours and hours. “You get these little incremental bits of knowledge and then eventually you build up and actually can solve it,” he says. “It’s a really thrilling experience to solve a problem.”

Hutchcroft earned his bachelor’s degree in mathematics from Cambridge University in 2013 and his PhD in mathematics from the University of British Columbia, Canada, in 2017. He held internships at Microsoft Research Theory Group during his graduate studies and later completed postdoctoral fellowships at the University of Cambridge from 2017 to 2021. He joined the Caltech faculty in 2021.

Recent Packard Fellows on the Caltech faculty include Zachary Ross, François Tissot, Kimberly See, Matt Thomson, Mansi Kasliwal (PhD ’11), Konstantin Batygin (PhD ’12), Hosea Nelson (PhD ’13), Mikhail Shapiro, and David Hsieh.