New Momentum and Optimism at the Convergence of Geometry and Analysis
SLMath - April 2025
When contemplating degeneration and mess—such as wrinkles at points or cusps, or even wrinkling behavior that propagates to infinity on otherwise smooth surfaces known as manifolds—the Swiss-French mathematician Tristan Rivière, a professor at ETH Zurich, is upbeat and hopeful. “Understanding these degenerations is inspiring,” Dr. Rivière said.
Dr. Rivière’s optimism is rooted in the momentum generated by some 100 mathematicians—among them geometers, studying spaces, and analysts, studying continuous change via calculus—who converged at the Simons Laufer Mathematical Sciences Institute in Berkeley for the Fall 2024 program on “Special Geometric Structures and Analysis.”
The idea for a program exploring the convergence around geometry and analysis coalesced during a January 2020 conference—a “Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics.” “We understood that there is a huge opportunity for development,” said Dr. Rivière. “Because in a sense, geometry without the input of sophisticated analytical ideas and techniques is a bit stuck.” Antecedents of this convergence date to further back, to the work of Fields medalists Sir Michael Atiyah and Sir Simon Donaldson, and Abel Prize laureates Louis Nirenberg and Karen Uhlenbeck, who are counted among the most prominent founders of modern geometric analysis.
A special geometric structure possesses a unique property, a special characteristic that holds true only under certain circumstances. For instance, while our everyday reality transpires in three dimensions, a special structure on a manifold might only reside in seven dimensions, or eight.
Manifolds with special structures in greater than four dimensions often generate wild aspects, thus posing problems that pure geometry alone cannot solve.
Indeed, as strange as it may seem to non-geometers, understanding such special geometric structures happens to be possible through understanding special equations posed by these manifolds—and more precisely through understanding the space of solutions to these equations. Originating in theoretical physics, these equations are called instanton equations; their solutions are called simply instantons. It is something of a circular scenario: The manifold generates the space of solutions, the space of instantons, and the space of instantons in turn generates retrospective information about the manifold. “The equations produce a new space, which, as it turns out, is the object of interest,” said Dr. Rivière, an analyst by training who now considers himself a geometric analyst. The boundary of the space is especially compelling—like an ocean’s variegated coastline terrain. Said Dr. Rivière: “While approaching this boundary, how do the instantons behave? Are they blowing up? And if so, can one describe the process?” This is where pure geometry leverages the support of deep analysis.
Analysts, using tools such as partial differential equations (PDEs) and geometric measure theory (GMT), “put some order to the mess and show that these singularities are not as wild and terrible as the preliminary analysis suggests,” said Dr. Rivière.
The Spanish geometer Izar Alonso attended the program as an SLMath postdoc and is now at Rutgers. Dr. Alonso received her Ph.D. in special geometric structures from Oxford in 2023, with an emphasis on the seventh dimension. Manifolds in dimension seven are “exceptional,” Dr. Alonso said, and as a result they have connections to other disciplines, such as theoretical physics: because adding the exceptional seven dimensions to the everyday three dimensions produces 10 dimensions. “Our universe can be thought of as a 10-dimensional manifold,” said Dr. Alonso. In order to make investigations easier, physicists decompose such a 10-dimensional structure into a three-dimensional part, and another part, “the seven extra dimensions that are known as the internal space—they’re compactified and we interpret them as being very small,” she said. “It’s interesting to be able to give mathematical sense to these structures that physicists have been finding in their calculations.”
During the SLMath program, Dr. Alonso got to know the analysts’ way of thinking. She met Dr. Rivière and his former Ph.D. student, the Italian analyst Riccardo Caniato, an SLMath member during the term, and a postdoctoral scholar and teaching fellow at Caltech.
“We soon understood that there was real potential for working together,” said Dr. Caniato. “I love PDEs, but I also love finding out what you can prove with them, especially solutions to geometric problems. And Izar is the perfect fit for this. She is very sharp geometrically, and I come from a solid analytical background. So, we started discussing Yang–Mills fields, which is a hard topic for both of those two fields.”
Some program participants benefited from concrete collaborations during the term that moved projects decisively toward publication. Dr. Caniato posted to the arXiv a paper that was “born and raised at SLMath”—in collaboration with Davide Parise, an SLMath postdoc for the term, now at the University of California San Diego. He posted a paper with Dr. Rivière, advancing research in supercritical gauge theory, a first step towards a 30-year-old proposal put forward by Simon Donaldson and Richard Thomas about the holomorphic invariants of Calabi–Yau three-folds and four-folds—which, Dr. Caniato said, “might be the dream for this community. Tristan and I have some ideas, but we definitely need people to help. It’s a long way to go.”
“I see a germ of something new emerging out of this program,” said Dr. Rivière. “I’m very optimistic about the future.”
“I feel especially optimistic about the people,” said Dr. Alonso. Nevertheless, she also has a sense of uncertainty. “There are some things that I feel are out of one’s control,” she said. “In mathematics, sometimes you don't know how difficult things are until they’re done.”
