Algebraic and Arithmetic Geometry via Stacks

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A long-term goal of my research career is to develop novel stack-theoretic techniques and apply these techniques to questions of fundamental importance in algebraic and arithmetic geometry. Over the next 5 years, I aim to accomplish 3 major objectives: 1. Greatly advance the program I initiated with Ellenberg and Zureick-Brown to obtain asymptotic growth rates for rational points on stacks, unifying the Manin and Malle Conjectures. 2. Prove a new smoothness criterion for quotients by semi-simple Lie groups, thereby making major headway on a longstanding open question of Popov; applying my solution, I plan to generalize a cornerstone result in stack theory due to Vistoli. 3. Prove a novel motivic change of variables formula for stacks, introducing powerful new techniques to study motivic integrals on singular varieties. Measuring the asymptotic growth rate of rational points on algebraic varieties is central to arithmetic geometry. Through recent joint work, Ellenberg, Zureick-Brown, and I have unified two major conjectures in the field by developing a theory of heights on stacks. Moreover, we initiated a program to provide great insight into asymptotics for rational points on stacks. I aim to further our program by proving our main conjecture for the wide class of horospherical stacks, as well as proving stack-theoretic analogues of recent deep results of Lehmann, Sengupta, and Tanimoto. Group quotients are ubiquitous in algebraic geometry, particularly because they often serve as local models for moduli spaces. In his 1986 ICM address, Popov asked whether one could give a criterion for when Lie group quotients are smooth. I aim to generalize recent work with Edidin and Whitehead to answer Popov's question for semi-simple Lie groups. As an application, I will generalize Vistoli's canonical stack construction to include a much larger class of singularities. Ever since its introduction, Vistoli's construction has had a profound impact on the field; I anticipate that my generalized canonical stacks will provide a new industry of techniques for studying group quotient singularities. Motivic integration has had a revolutionary impact on algebraic geometry and mathematical physics alike. Central to the theory is a change of variables formula. Based on joint work with Usatine, I aim to greatly generalize the existing change of variables formulas to include smooth Artin stacks. By viewing such stacks as resolutions of singular varieties, this will give a new industry of techniques for computing motivic integrals on varieties. My research program will build connections between communities of mathematicians studying seemingly different objects, and will make significant progress on numerous open problems. My program will train HQP at all levels to become world-class researchers in arithmetic and algebraic geometry, poised to have highly impactful careers in natural sciences and engineering.