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

世纪以来，数学家们已经知道，由整数算术产生的某些量（如将一个自然数表示为四个平方和的方法的数量）与由表示论和调和分析产生的量（傅立叶系数）有着微妙的关系，调和分析是理解函数的图形如何被实现为简单波形的叠加的过程。这个项目的首要目标是通过所谓的theta对应来研究这些关系的几何学，从而在解决算术几何学中长期存在的问题方面取得进展。该项目为研究生提供培训机会。这些广泛的解释揭示了爱森斯坦级数和某些志村簇的内部几何之间的深刻关系，这些簇已经成为现代算术几何的基本对象。 首席研究员打算使用方法从推导出的代数几何在全球设置和最近的事态发展，在当地设置，以获得一个概念性的理解几何巩固的真理，这些programmings.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.