Media Release –
March 31, 2023
Canadian Mathematical Society |

# DR. ROBERT HASLHOFER TO RECEIVE THE 2023 COXETER-JAMES PRIZE

OTTAWA, ON – The Canadian Mathematical Society (CMS) is pleased to announce that Dr. Robert Haslhofer (University of Toronto) has been named the recipient of the 2023 Coxeter-James Prize for his outstanding contributions to Riemannian geometry and geometric analysis, especially mean curvature and Ricci flows.

Dr. Haslhofer was awarded his Ph.D. in Mathematics from ETH Zurich in 2012. Since then Dr. Haslhofer has continued on an impressive trajectory. After three years as a Courant Instructor at New York University’s Courant Institute of Mathematical Sciences, he joined the Department of Mathematics at the University of Toronto in 2015. Recent recognitions of Dr. Haslhofer’s work include the Andre Aisenstadt Prize (2020), a Sloan Research Fellowship (2018-2022) and an NSERC Discovery Grant (2016-2023).

Lauded by his colleagues as “one of the most distinguished and most promising mathematicians worldwide in Riemannian geometry and geometric analysis,” Dr. Haslhofer’s scientific work with various collaborators include novel characterizations of Ricci flows, study of mean curvature flows through neck singularities, and impressive contributions to stochastic analysis on path spaces.

In a remarkable paper with Bruce Kleiner, Dr. Haslhofer’s work on mean curvature flow is largely set in a framework he developed that dramatically simplifies and unifies much of the classical theory on singularity formation. They also significantly strengthened earlier results, by establishing an interior gradient estimate that played a crucial role in their subsequent work (and independently that of Simon Brendle with Gerhard Huisken) constructing mean curvature flow with surgery for 2-convex hypersurfaces in arbitrary dimension. For surfaces in R3, this resolved a long-standing open problem.

Haslhofer’s work with Aaron Naber on Ricci flow solves a deep and long-standing question in this active area: An ingenious notion of weak solution of Ricci flow is introduced through stochastic analysis on the Ricci-flow spacetime. This allows for the definition of Ricci-flow on singular spaces, and in particular yields the first satisfactory notion of Ricci flow through singularities. This work uses ideas from stochastic analysis in a profound and original way. It is a major result, likely to facilitate many important further developments. (One such is the Bochner formula that Haslhofer and Naber subsequently obtained for martingales on path space PM , a vast generalization of the classical Bochner formula for the heat flow on a manifold M . Their new formula is related to two-sided bounds on Ricci curvature in much the same manner that the classical Bochner formula on M is related to lower bounds on Ricci curvature. This breakthrough provides a new fundamental tool for the study of spaces with two-sided Ricci bounds, including Einstein manifolds and the Ricci flow.)

With Dan Ketover, Haslhofer used min-max theory to establish that every generic metric on the 3-sphere admits at least two embedded minimal two-spheres, thus disproving a conjecture of Shing-Tung Yau concerning ellipsoids.

However, his most spectacular achievement to date is the resolution of the mean-convex neighbourhood conjecture for singularities of mean-curvature flow, a twenty-year old conjecture of his PhD advisor Tom Ilmanen. Together with his collaborator Kyeongsu Choi and his former PhD student Or Hershkovits, Haslhofer resolved this conjecture first for surfaces (Acta Mathematica 2022), and then with the addition of Brian White in higher dimensions (Inventiones 2022). Instead of assuming that the initial condition possesses some form of symmetry or convexity, Ilmanen’s conjecture states that it develops mean-convexity in a spacetime neighbourhood of any asymptotically cylindrical singularity (after which ex- isting theory can then be applied).

“Barely a decade after his PhD, Robert Haslhofer has established himself among the leading geometric analysts of his generation. A long list of accomplishments make him richly de-serving of the Coxeter-James award. Most spectacular among them is the resolution of the mean-convex neighbourhood conjecture for singularities of mean-curvature ﬂow, a twenty-year old conjecture of his PhD advisor Tom Ilmanen. Together with his postdoctoral fellow Kyeongsu Choi and his former PhD student Or Hershkovits, Haslhofer resolved this conjecture ﬁrst for surfaces (Acta Math 2022), and then with the addition of Brian White in higher dimensions (Inventiones 2022). Instead of assuming that the initial condition possesses some form of symmetry or convexity, Ilmanen’s conjecture states that it develops mean-convexity in a spacetime neighbourhood of any asymptotically cylindrical singularity (to which existing theory can then be applied).” – Dr. Robert J. McCann, FRSC

The CMS is delighted to present Dr. Haslhofer with the 2023 Coxeter-James Prize for his incredibly important addition to Riemannian geometry and geometric analysis, especially mean curvature and Ricci flows.

**About the Coxeter-James Prize**

The Coxeter-James Prize was inaugurated in 1978 to recognize young mathematicians who have made outstanding contributions to mathematical research. The award is named for two former CMS presidents, Donald Coxeter, who is recognized as one of the world’s best geometers, and Ralph Duncan James, who was a great contributor to mathematical development in Canada.

For more information, visit the Coxeter-James Prize page.

**About the Canadian Mathematical Society (CMS)**

The CMS is the main national mathematical organization whose goal is to promote and advance the discovery, learning and application of mathematics. The Society’s activities cover the whole spectrum of mathematics including: scientific meetings, research publications, and the promotion of excellence in mathematics competitions that recognize outstanding student achievements.

**For more information, please contact**:

Dr. Susan Cooper (uManitoba) Chair, CMS Research Committee Canadian Mathematical Society chair-resc@cms.math.ca |
or | Dr. Termeh Kousha Executive Director Canadian Mathematical Society tkousha@cms.math.ca |