Around Non-Positive Curvature

几何学关注空间的定量特征，而拓扑学研究空间的定性方面。例如，茶杯和甜甜圈在拓扑上是相同的（它们都有一个“孔”），但在几何上是不同的。非正曲率是一种几何性质，大致意味着在小尺度上空间不会弯曲回自身。这些空间有一个有趣的相关动力学系统：人们可以试着想象台球在这样的空间中如何随时间移动。该提案的重点是从所有三个角度（几何，拓扑和动力学）研究这些空间，为不同的方法和技术提供了一个丰富的熔炉。虽然非正曲率的概念可能看起来很奇怪，但这些空间实际上在数学和自然界中都很普遍。因此，它们是数学家研究和理解的基本空间类。这个项目将解决一系列的项目是松散的非正曲率的概念为中心。该项目将为研究生的研究和培训提供机会。此外，PI还将在2022年帮助组织EDGE（增强研究生教育的多样性）计划，这是一项旨在加强女性在数学研究和教育领域追求职业生涯的能力，并让更多女性在数学界担任可见的领导角色的招聘计划。首席研究员（PI）将从事非正曲率的各种项目，分为四大类。(1)局部对称空间上的项目：PI将考虑高阶半单李群中格的同构问题，将在PO（n，1）中构造新的非算术格，将产生可分的积分同调类，并将研究浸入子流形上的分支覆盖。(2)粗糙几何体中的项目：PI将证明通过Charney-Davis双曲化得到的群是线性的，将开发空间对的新的QI-不变量，并研究具有边界的流形的QI刚性。(3)动力学性质的项目：PI将研究具有约束的连分数的Khinchine定理的一个版本，并将研究紧致局部CAT（-1）空间上测地流相关性的衰减。(4)关于K理论的文章：PI将用同构积分K-理论产生不同的代数数域，并将表明所有由PI产生的Q-线性群都具有有限的VC几何维数。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Geometry is concerned with quantitative features of a space, while topology studies qualitative aspects of a space. As an example, a teacup and a donut are topologically the same (they both have a single “hole”), but geometrically different. Non-positive curvature is a geometric property, and roughly means that on small scales the space does not curve back on itself. These spaces have an interesting associated dynamical system: one can try to imagine how a billiard ball in such a space would move over time. This proposal is focused on studying these spaces from all three points of view (geometry, topology, and dynamics), providing a rich melting pot for different approaches and techniques. While the notion of non-positive curvature might seem strange, these spaces are in fact pervasive, both in mathematics and in nature. For this reason, they are a fundamental class of spaces for mathematicians to study and understand. This project will address a series of projects that are loosely centered around the notion of non-positive curvature. This project will provide opportunities for graduate student research and training. In addition the PI will help organize the EDGE (Enhancing Diversity in Graduate Education) program in 2022, a recruitment program with the goals of strengthening the ability of women to pursue careers in mathematical research and education and placing more women in visible leadership roles in the mathematics community.The Principal Investigator (PI) will work on various projects in non-positive curvature that fall into four broad categories. (1) Projects on locally symmetric spaces: the PI will consider the isomorphism problem for lattices in higher rank semisimple Lie groups, will construct new non-arithmetic lattices in PO(n,1), will produce divisible integral homology classes, and will study branched covers over immersed submanifolds. (2) Projects in coarse geometry: the PI will show that groups obtained via the Charney-Davis hyperbolizations are linear, will develop new QI-invariants of pairs of spaces, and study QI rigidity for manifolds with boundary. (3) Projects of a dynamical nature: the PI will work on a version of Khinchine’s theorem for continued fractions with constraints, and will study the decay of correlations for geodesic flows on compact locally CAT(-1) spaces. (4) Projects on K-theory: the PI will produce distinct algebraic number fields with isomorphic integral K-theory, and will show that all finitely generated Q-linear groups have finite VC geometric dimension.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.