Michael Groechenig to receive the 2024 Coxeter-James Prize

February 22, 2024 by Canadian Mathematical Society

The Canadian Mathematical Society (CMS) is pleased to announce that Dr. Michael Groechenig (University of Toronto) has been named the recipient of the 2024 Coxeter-James Prize for his outstanding contributions to arithmetic and algebraic geometry.

Dr. Groechenig obtained his BSc in Mathematics from ETH Zurich (2009) before completing his PhD in Mathematics at the University of Oxford in 2013. In 2018, he transitioned to the University of Toronto where he worked as an Assistant Professor until 2023, when he was promoted to Associate Professor (with tenure). Still in the early stages of his career, Dr. Groechenig has already successfully addressed numerous enduring open problems, a feat complemented by his simultaneous contributions to the advancement of foundational theory in technical domains—a remarkable accomplishment in itself.

Dr. Groechenig’s fields of interest include algebraic and arithmetic geometry, moduli spaces of Higgs bundles and flat connections, p-adic integration, and K-theory.

In his second paper titled “Hilbert schemes as moduli of Higgs bundles and local systems”, Dr. Groechenig employed advanced geometric methods, primarily drawing on derived categories of sheaves and occasionally delving into more intricate aspects of derived algebraic geometry. Through these sophisticated techniques, he successfully validated the elegant conjectures put forth by Boalch, which had been inspired, in part, by the prior insights of mathematical physicists Gorsky, Nekrasov, and Rubtsov.

In a collaborative publication (with Wyss and Ziegler) on p-adic integration, Dr. Groechenig substantiated a conjecture proposed by Hausel and Thaddeus. This conjecture posited an equivalence of stringy Hodge numbers between two separate moduli spaces.

The higher K-groups represent crucial invariants within the realm of cohomological assertions, yet they persist as enigmatic entities, proving challenging to grasp explicitly or generate specific elements. Nevertheless, Dr. Groechenig, together with Braunling, has recently accomplished a significant milestone by demonstrating the existence of non-torsion classes. This achievement marks a pivotal initial stride toward understanding these objects.

In his research paper titled “Moduli of flat connections in positive characteristic,” Dr. Groechenig significantly expanded upon the methodologies introduced by Bezrukavnikov and Braverman. He achieved a novel formulation of the Geometric Langlands conjecture in positive characteristic, concurrently establishing the properness of the Hitchin-Mochizuki map. This accomplishment demanded a profound technical comprehension of Azumaya-algebra splittings, a mastery of the arithmetic geometry inherent in the Hitchin fibration, and an adept handling of positive characteristic challenges outlined in the works of Arinkin and Haiman. Notably, leaders in the field, including Bezrukavnikov, Travkin, and Zhu, have cited and acknowledged the impact of Dr. Groechenig’s work.

Dr. Groechenig’s most notable achievement to date involves his collaborative work with Esnault (2020) on rigid local systems. In their groundbreaking contribution, they substantiate a consequence of Simpson’s conjecture, demonstrating that irreducible cohomologically rigid connections on a smooth projective variety are integral. Additionally, they extend Grothendieck’s p-curvature conjecture by proving new cases. What sets their approach apart is the striking methodology employed – despite the conjecture residing solely within the realm of complex geometry, they navigate through arithmetic to establish its validity. Impressively, their results have garnered recognition from experts and have already found other significant applications.

In addition to the aforementioned contributions, Dr. Groechenig has (co-) authored over 20 publications, establishing an extensive portfolio of high-impact achievements and fundamental outcomes. Moreover, he has garnered recognition through various grants and fellowships, including a Marie Sklodowska-Curie individual fellowship, an NSERC Discovery Grant, and an Alfred P. Sloan fellowship.

The CMS Research Committee states:

“Michael Groechenig is a mathematician of exceptional breadth and depth.”

“Groechenig’s results have revolutionized the study of the topology of the Hitchin system with p-adic integration techniques. His articles on mirror symmetry via p-adic integration (Invent. 2020) and on geometric stabilization via p-adic integration (JAMS 2020) are groundbreaking.”

In summary, Dr. Groechenig emerges as a dynamic and brilliant early-career mathematician, whose innovative ideas exert a profound influence across a broad spectrum of mathematical domains. The CMS is proud to award him the 2024 Coxeter-James Prize and eagerly anticipates witnessing the evolution of his career in the coming years.

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.