NSF Award
2429055
This award will fund US-based participants in the conference "Hyperbolic manifolds, their submanifolds fundamental groups", which will consist of a conference January 6-10, 2025 and a follow-up workshop aimed at early-career mathematicians January 13-17, 2025, both held at Instituto de Matematica Pura e Aplicada (IMPA) in Rio de Janeiro, Brazil. The conference will bring together around 100 participants from around the world to learn about the latest exciting developments in the fast-moving and very active area encompassed by the title of the conference. This will include 23 lectures by world-renowned experts of various career stages all at the forefront of the field, along with an extended lightning talk session to provide young participants an opportunity to communicate their work. The follow-up workshop is aimed at early-career participants and will be dedicated to four detailed mini-courses given by rising experts. The primary use of this grant will be to fund applicants that are early-career (graduate student or junior faculty), with a particular emphasis on those from underrepresented groups in the mathematical sciences and those from institutions with extremely limited access to resources and research opportunities.<br/><br/>The primary purpose of the conference is the disseminate the most current results in and around the study of hyperbolic manifolds, which are fundamental objects in differential geometry and topology. Hyperbolic manifolds, i.e., manifolds of constant curvature -1, are among the most important and basic examples in differential geometry. Yet, they remain profoundly mysterious in comparison with their flat and positively-curved analogues. Forty years ago, Thurston laid out a grand vision for geometrization of 3-manifolds which, if validated, identified hyperbolic 3-manifolds as the essential case to unlock. Twenty years ago, Perelman substantiated this by proving Thurston's geometrization conjecture, and ten years ago Ian Agol made another profound breakthrough by proving Thurston's conjecture that all compact hyperbolic 3-manifolds virtually fiber over the circle. Despite this immense progress, much remains to be done before we have a satisfactory understanding of hyperbolic 3-manifolds, and the field continues to flourish and move forward at a rapid pace and find new deep connections with other areas of mathematics like profinite groups, geometric analysis, and number theory. In comparison, our understanding of higher-dimensional hyperbolic manifolds remains far more unsatisfactory. However - often guided by the significant progress in dimension three - the last decade has seen an explosion of fundamental new results leading to significant optimism that a much deeper understanding in higher dimensions is within reach. These new advances are coming from researchers with a wide range of perspectives and toolkits, but also from mathematicians from Europe and throughout the Americas, for example in the chosen location of Brazil. A primary aim of this conference and workshop is to provide a significant opportunity for both experts and young researchers to learn about the newest developments in an environment fostering international collaborations that will drive the next advances. The follow-up workshop will boost the intellectual development of early-career participants through opportunities to make new professional connections, collaborate with their peers, and learn the details of four distinct topics of contemporary interest from a rising expert.<br/>https://impa.br/en_US/eventos-do-impa/2025-2/hyperbolic-manifolds-their-submanifolds-and-fundamental-groups/<br/><br/>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.
