Dualizing modules in algebra and geometry

PI提出研究同伦对偶模在数论和代数几何中的存在性及其意义。具体来说，将有两个这样的对偶模块的有价值的实例。其中一个将与托马·施兰克共同研究;它将给出泰特-普瓦图算术对偶定理的同伦推广，该定理指出某些绝对伽罗瓦群的上同调具有扭曲形式的自对偶。PI和Tomer Schlank还将研究这种对偶结果在代数簇上存在有理点的问题中的应用。一个不同的例子是关于有或没有层次结构的拓扑模形式的谱。PI以前的工作表明，广义椭圆曲线的谱导出模栈具有可交换环谱的简单可描述的对偶层;所提出的项目的一个目标是证明这一结果。拓扑模形式对于理解球谱中的v2周期同伦是至关重要的;虽然有点脱离对偶的主题，PI将与Mark Behrens，凯尔奥姆斯比和纳撒尼尔斯台普顿合作，计算基于连接拓扑模形式的同调中的合作。对偶性是数学中一个普遍的概念;拟议的项目将在一个统一的框架中研究不同类型的对偶性，从而获得新颖的和其他无法获得的信息。特别是，该项目的动机是这样一个想法，即引入同伦观点可以为我们对几何和算术的理解提供新的启发。有一个对象具有如此奇特的自对偶性质，即拓扑模形式，它有着广泛的应用，有些已经被探索过了，有些还没有，因为它在同伦、代数几何、数论甚至量子场论中都反映出自己是维滕属的容器。

The PI proposes to investigate the existence of homotopical dualizing modules in number theory and algebraic geometry and the implications thereof. Specifically, there will be two valuable instances of such dualizing modules. One will be studied in joint work with Tomer Schlank; it will give a homotopical extension of the arithmetic duality theorems of Tate-Poitou, which state that the cohomology of certain absolute Galois groups has a twisted form of self-duality. The PI and Tomer Schlank will also investigate applications of such duality results to problems of existence of rational points on algebraic varieties. A different example is related to the spectra of topological modular forms with or without level structures. Previous work of the PI suggests that the spectrally derived moduli stack of generalized elliptic curves has a simply describable dualizing sheaf of commutative ring spectra; an objective of the proposed project is to prove that result. Topological modular forms are crucial for understanding v2 periodic homotopy in the sphere spectrum; though somewhat removed from the theme of duality, the PI will collaborate with Mark Behrens, Kyle Ormsby, and Nathaniel Stapleton to compute the cooperations in the homology based on connective topological modular forms. Duality is a pervasive concept in mathematics; the proposed project will study different types of duality in a unified framework, thereby arriving at novel and otherwise inaccessible information. In particular, the project is motivated by the idea that introducing a homotopical viewpoint can shed new light on our understanding of geometry and arithmetic. One of the objects with such curious self-duality properties, namely topological modular forms, lends itself to vast applications, some already explored and others not, as it mirrors itself in homotopy, algebraic geometry, number theory, and even quantum field theory as the receptacle of the Witten genus.