Euler characteristics, length spectra, deformations and lifting problems

本课题涉及算术几何、双曲几何、表示论和数论。第一个目标是证明有有限群作用的相干束和Weil-etale束的黎曼洛克公式。这些公式与l级数的特殊值的猜想有关。第二个目标是研究算术群的可通约性类及其与算术局部对称空间的长度谱的联系。第三个目标是研究无限群的模复合体的变形。重点将放在一个猜想上，即由算术几何产生的通用变形可以用通用变形环上的完美复合体来表示。本课题的最后一个目标是研究正特征曲线上的有限群作用可以提升到特征零点。这个项目的统一主题是对对称性的研究。黎曼洛克公式量化了方程组的对称性如何反映在它们的解中。我们可以利用这类结果来极大地约束解。根据物体反射声音或无线电波的方式来识别物体的形状，这是一个著名的问题。我们将研究这个问题的一个变体，它也涉及到使用物体上某些路径的长度来尝试识别它。考虑对称性的一个基本问题是量化描述对称性需要多少信息。这个问题将在描述具有一组规定对称性的物体变形的所有方法的背景下进行研究。最后，将研究障碍物如何将对称从小物体（正特征曲线）扩展到大物体（特征为零的曲线）。

This project concerns arithmetic geometry, hyperbolic geometry, representation theory and number theory. The first goal is to prove Riemann Roch formulas for coherent and Weil-etale sheaves on which a finite group acts. Such formulas are relevant to conjectures about special values of L-series. A second goal is to study commensurability classes of arithmetic groups and their connection to the length spectra of arithmetic locally symmetric spaces. A third goal is to study deformations of complexes of modules for a profinite group. The focus will be on a conjecture that versal deformations arising from arithmetic geometry are representable by perfect complexes over the versal deformation ring. The last goal of the project is to study which finite group actions on curves in positive characteristic can be lifted to characteristic zero.The unifying theme of this project is the study of symmetries. Riemann Roch formulas quantify how symmetries of systems of equations are reflected in their solutions. One can use results of this kind to greatly constrain the Solutions. Symmetries enter into the famous problem of recognizing the shape of an object from how it reflects sound or radio waves. A variant of this problem will be studied which involves also using the lengths of certain paths on the object to try to recognize it. A basic problem in considering symmetries is to quantify how much information is needed to describe them. This problem will be investigated in the context of describing all ways to deform an object having a set of prescribed symmetries. Finally, obstructions will be studied to extending symmetries from a small object (a curve in positive characteristic) to a larger one (a curve in characteristic zero).