Euler characteristics, length spectra, deformations and lifting problems

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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).