Conference: Low-Dimensional Manifolds, their Geometry and Topology, Representations and Actions of their Fundamental Groups and Connections with Physics

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This award provides support for U.S. based participants to attend a sequence of workshops, to be held at ICMAT Madrid and El Barco de Ávila, Spain, in April-July 2023. As evidenced by the title, “Low-dimensional manifolds, their geometry and topology, representations and actions of their fundamental groups and connections with physics”, the themes are broadly classical, while relating to central research areas in modern mathematics. The workshops will include introductory lectures and research talks by leading experts as well as offer the early career researchers the professional development opportunity to present their own work and form new international collaborations.The multiple connections between the seemingly disparate themes of these workshops can be articulated by the central role played by spaces of representations of a discrete group into a larger group: for example a Lie group, groups of diffeomorphisms or groups of homeomorphisms. These are powerful tools in unpacking the structure of the original group, particularly when it is the fundamental group of a compact manifold. The classical case of surface group representations is particularly noteworthy, providing connections with Teichmuller Theory, three-manifold topology and more recently Higher Teichmuller theory. In the setting of Higher Teichmuller theory there is a very different perspective given through the lens of Higgs bundles over a compact surface (equipped with a complex structure). Since Hitchin's original insight around thirty years ago, this viewpoint has had a tremendous impact in geometry, topology and theoretical physics. These and many other research areas where spaces of representations of a discrete group are ubiquitous will be at the core of the thematic program. More information about the workshops is available at https://sites.google.com/view/javier-aramayona/agol-lab/activity-period-2023.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.