Geometrization of the Local and Global Theta Correspondence with Applications to the Kudla Program

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For well over a century now, mathematicians have known that certain quantities arising from the arithmetic of the integers (such as the number of ways of representing a natural number as a sum of four squares) have subtle relationships with quantities (Fourier coefficients) arising from representation theory and harmonic analysis, which is the process of understanding how graphs of functions can be realized as superpositions of simpler waveforms. The overarching goal of this project is to study the geometry of these relationships, via the so-called theta correspondence, thereby making progress towards resolving longstanding conjectures in arithmetic geometry. The project provides training opportunities for graduate students. These wide-ranging conjectures posit a deep relationship between Eisenstein series and the internal geometry of certain Shimura varieties, which have turned out to be fundamental objects in modern arithmetic geometry. The principal investigator intends to use methods from derived algebraic geometry in the global setting and recent developments in the local setting to gain a conceptual understanding of the geometry undergirding the truth of these conjectures.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.