Geometry and arithmetic of locally symmetric spaces

半单李群中的算术格是基本的数学对象，在几何、拓扑、群论、数论等领域具有重要意义。该项目考虑了这些场相互作用的方式，特别是对局部对称空间几何的影响。更具体地说，PI将应用数论、群论、代数几何和低维拓扑的见解来研究算术局部对称空间的几何和拓扑性质。特别的重点将放在格作用于实数和复双曲空间，其中更好地理解算术格可以阐明关于有限体积实数和复双曲流形的几何和拓扑的几个开放问题。二维和三维双曲流形的许多众所周知的性质导致了高维空间中类似的问题，研究这些性质中哪些是低维特有的，哪些在更神秘的高维空间中仍然成立是很有意义的。几何和数论是数学研究的两个最经典的领域，两者之间的相互作用自古以来就吸引着数学家。从欧几里得的《几何要素》到高斯的《费马大定理》，再到怀尔斯对费马大定理的证明，无数伟大的数学进步背后都蕴含着这种关系。对这种相互作用的研究继续取得丰硕成果。数域是与整数系数多项式方程的解相关的对象（例如，有理数），数域产生了无限的几何对象家族，称为算术局部对称空间，这是朗兰兹程序的核心，经常参数化数学和物理中的基本对象。这些数域的不变量，如多项式的判别式，以及相关对称群的性质，使我们对这些几何空间的本质有了深刻的认识。群论是现代几何和数论研究中最重要的资产之一，它可以追溯到拉格朗日和伽罗瓦对多项式的根及其对称性的研究，而现代观点起源于19世纪70年代克莱因的埃尔兰根计划。近年来，算术局部对称空间的几何在计算机科学、信息论、图论和密码学中的重要应用，以及与数论和群论的联系是许多这些应用的核心。

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.