Geometry, Arithmetic, and Groups.

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A conference on Geometry, Arithmetic, and Groups organized by researchers from across the United States will be held at the University of Texas, Austin, June 20-24, 2022. This conference will feature a diverse mix of speakers whose work connects the topics listed in the title in crucial and important ways. The goal of the conference is to expose the inter-connectivity of these topics to a wide range of researchers whose work may not necessarily touch each of the areas, including researchers from underrepresented groups, early career researchers, and researchers from liberal arts and R2 universities. The format for the conference includes one hour plenary research talks, multiple lightning-talk sessions, and dedicated time for informal mathematical discussions. This event will provide the impetus for new collaborations between researchers from diverse mathematical fields and institutions, and from a wide variety of career stages.The conference will focus on the interactions between geometric topology, geometric group theory, number theory, representation theory, and spectral geometry. At the heart of this interplay is the theory of Lie groups and their discrete subgroups, where one can produce manifolds via the actions of discrete groups on associated homogeneous spaces. Of particular interest are discrete groups acting isometrically on hyperbolic space. These actions play a fundamental role in low dimensional geometry and topology via geometrization, and have provided the impetus for much current research in geometric group theory and the deformation theory of geometric structures. Among the discrete subgroups of simple Lie groups more generally, arithmetic lattices play a central role, utilizing technology from algebraic/analytic number theory and the theory of algebraic groups. These interactions provide deep and important connections between the geometry of the associated spaces and the algebra of the arithmetic lattices. All these interactions have driven mathematical research for more than one hundred years, and this conference will bring together researchers from these areas and closely related fields to search for new and exciting connections for the future. The website for the conference is at https://sites.google.com/view/awr-conference/homeThis 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.