Grochow at the Santa Fe Institute circa 2014

Seven years ago, during his first year as an Omidyar Fellow at SFI, computer scientist Josh Grochow shared an office with physicist (and fellow Omidyar Fellow) Yoav Kallus. In conversations they revisited regularly during their tenure in Santa Fe, the two compared notes on the staggering diversity of situations, from quantum entanglement to the behavior of organisms to the group isomorphism problem, could be modeled and analyzed using the same concept: Higher-order interactions. These were natural connections to make in the academic, complexity-rich environment of SFI, says Grochow.

People have long used networks to model a variety of phenomena, but networks represent relationships between members of a population as simple connections — the line between two nodes. Higher-order interactions allow for richer relationships. Social media provides one example: A group of people may form an online network, but the dynamics of their relationships may transcend the sum of the individual links (if some are related, or neighbors, or belong to the same religion, for example).

Now an assistant professor at the University of Colorado Boulder, Grochow continues to pursue ideas that emerged during his time at SFI. And earlier this year, he received some hefty support to continue that work: In February, the National Science Foundation announced he would receive its most prestigious grant for junior faculty members, a Faculty Early Career Development, or CAREER, award, which will fund the next five years of his research.

With the award, he will be investigating ways to use and optimize mathematical tools that probe higher-order interactions. These include hypergraphs (which unlike ordinary networks have edges that might connect multiple nodes) and tensors (mathematical descriptions of the relationships between groups of objects). He’s particularly focused on the isomorphism problem, which essentially asks, mathematically, whether or not two objects (like tensors, or algebraic groups, or even problems) have the same basic structure.

He’ll stay busy as a CAREER recipient. Even though higher-order network approaches have been around for decades, mathematicians and computer scientists are only just beginning to understand what they do and don’t reveal about the internal structure of a dataset. In addition to diving into problems like the isomorphism problem, Grochow says he wants to investigate broader issues facing the field. “What’s the basic language for talking about these things?” he asks. “What’s the fundamental theory?”

Read the CU Boulder press release (May 17, 2021)

View the award on the NSF website (March 11, 2021)