Conference: Conference and School on Trisections and Related Topics

This grant will support workshops, a research school and a conference on Trisections and their Generalizations, to be held at the Centre International de Recontres Mathematiques (CIRM) in Marseille, France, during the weeks July 7-11, October 6-10, Oct 13-17 and November 17-21, 2025. CIRM is a well-known international mathematics conference center, and the Principal Investigator of this grant, Professor David T. Gay of the University of Georgia, has been selected as CIRM's "Jean Morlet Chair" for the period July-December 2025 for the purpose of organizing a semester's worth of international activities at CIRM related to his research program in topology. This grant will support the travel and accommodation expenses of US-based mathematicians to participate in some of these events. The research topic, in topology, concerns new methods to study the intrinsic large-scale shapes of a wide range of spaces called manifolds: Manifolds are spaces which at the small scale, look just like the ordinary space around us, perhaps with more dimensions, but which can be tangled up with themselves in strange ways on the large scale. Interesting manifolds arise, for example, when thinking about all the possible configurations of a mechanical system, such as a robot or a human limb, when thinking about all the possible states of a large language model under training in an artificial intelligence system, or when understanding how a protein folds. Trisections, and generalizations of these, give universal ways to decompose manifolds into a small number of simple pieces (often that number is 3) that are easy to understand, so that understanding the large-scale topology reduces to understanding the rules for how these small number of simple pieces can fit together to make a single closed manifold. Besides increasing our understanding of fundamental mathematical objects underlying many scientific applications, a key broader impact of this grant will be to provide opportunities for early career US-based mathematicians from a broad range of institutions to interact with the international scientific community, establish new collaborations and engage in a rapidly developing research field. Compared to many fields within mathematics, the subject matter of this semester of activities presents a relatively low bar for entry and a fairly quick access for the beginner to rich, deep and challenging mathematics, and also presents many opportunities for engagement with the broader public through interesting visualization and illustration activities.


Since Gay’s initial work on trisections of smooth 4–manifolds in 2013, there has been an explosion of activity in the field leading to at least 92 published papers on trisections and related topics listed on MathSciNet. Fundamental questions have been answered and extensive generalizations have been developed, connecting the field to many other important areas in topology. Two distinct relative versions of trisections have been developed, trisections of 4–manifolds with boundary, relative to open book decompositions, and trisections of 4–manifolds with embedded surfaces. Connections with algebraic geometry and symplectic geometry have been developed as well as connections with piecewise linear topology and higher-dimensional topology. In work with Abrams and Kirby, Gay used trisections of 4–manifolds to define group trisections and thus show that the study of smooth 4–manifolds could in principal be completely reduced to group theoretic questions. This is just the tip of the iceberg, and the time is right for an international meeting to survey progress in the field, hear about the latest results, and develop a collective plan for future steps. This is the purpose of the main conference in October, which will be preceded by a week-long Research School to help early career researchers get up to speed. The two workshops, one in July and one in November, will bring together smaller groups of researchers for more focused exploration and collaborative research work on some specific topics related to higher-dimensional generalizations and symplectic geometry. The webpage for the semester of events is https://www.chairejeanmorlet.com/2025-gay-moussard-2nd-semester.html

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

Award Number: 2525283
Principal Investigator: David Gay
Funds Obligated: $47,500
State: GA