Arithmetic lattices in semisimple Lie groups are fundamental mathematical objects that have deep significance in geometry, topology, group theory, and number theory. This project considers the way in which these fields interact, particularly the effect on the geometry of locally symmetric spaces. More specifically, the PI will apply insights from number theory, group theory, algebraic geometry, and low dimensional topology to study the geometric and topological properties of arithmetic locally symmetric spaces. Special focus will be placed on lattices acting on real and complex hyperbolic space, where a better understanding of arithmetic lattices can shed light on several open questions about the geometry and topology of finite volume real and complex hyperbolic manifolds. Many well-known properties of 2 and 3-dimensional hyperbolic manifolds lead to analogous questions in higher dimensions, and it is of significant interest to study which of these properties are peculiar to low dimensions and which remain true for the more enigmatic higher-dimensional spaces.
Geometry and number theory are two of the most classical areas of mathematical study, and the interaction between the two has fascinated mathematicians since ancient times. This relationship lies behind countless great mathematical advances from Euclid's Elements to Gauss to Wiles's proof of Fermat's Last Theorem. Studying this interplay continues to bear deep fruit. Number fields are objects related to solutions to polynomial equations with integer coefficients (e.g., the rational numbers), and number fields give rise to an infinite family of geometric objects called arithmetic locally symmetric spaces, which lie at the heart of the Langlands program and often parameterize fundamental objects in mathematics and physics. Invariants of these number fields, like the discriminant of a polynomial, and properties of associated symmetry groups lead to deep insight about nature of these geometric spaces. Group theory is one of the most significant assets in the modern study of geometry and number theory, which goes back to Lagrange and Galois's study of the roots of polynomials and their symmetry, and the modern viewpoint originates in Klein's Erlangen program from the 1870s. Recent years have also seen significant applications of the geometry of arithmetic locally symmetric spaces to computer science, information theory, graph theory, and cryptography, and the connections with number theory and group theory are central to many of these applications.
