Methods of algebraic geometry in algebraic topology

本项目旨在利用代数几何中发展起来的方法和机器来进一步研究稳定同伦理论。有四个主要目标。第一个目标是研究拓扑自同构型的谱的具体例子，重点是计算判别式15的Shimura曲线和色水平3的Picard模曲面。第二个目标是将拓扑模形式的理论推广到椭圆曲线模的对数位置，特别是功能上产生具有层次结构的拓扑模形式。第三，PI希望进一步发展应用Zink的显示理论的机器，以在色同伦理论中的各种重要光谱上产生高度结构化的乘法。最后，与David Gepner合作，将利用高等范畴理论中最新发展的机器来研究派生环境下的Picard群和Brauer群，这一研究领域的主要重点是用代数的方法系统地研究形状的定性性质。在过去，这些方法导致了许多不同的数学领域之间令人惊讶的联系，这些领域的发展已经转化为关于几何结构的真实信息，否则很难获得这些信息。这个研究项目的目标是将这些最新的进展中的一些，发生在更高范式论和代数几何等学科中，并将它们发展成可以进一步推进我们知识的具体工具。

This project aims to further the study of stable homotopy theory by methods and machinery developed in algebraic geometry. There are four main goals. The first goal is the study of specific examples of spectra of topological automorphic forms, focusing on computations for the Shimura curve of discriminant 15 and Picard modular surfaces at chromatic level 3. Second, joint work of Michael Hill and the PI aims to extend the theory of topological modular forms to the log-etale site of the moduli of elliptic curves, and in particular to functorially produce topological modular forms with level structure. Third, the PI hopes to further develop machinery employing Zink's theory of displays to produce highly structured multiplications on various spectra of importants in chromatic homotopy theory. Finally, joint work with David Gepner will study Picard and Brauer groups in the derived setting using recently-developed machinery in higher category theory.The main focus of this research field is to systematically study qualitative properties of shape by methods from algebra. In the past, these methods have led to surprising connections between many disparate fields of mathematics, and developments in these fields have translated into genuine information about geometric structures which is difficult to obtain otherwise. The goal of this research project is to take some of these most recent advances, taking place in subjects like higher category theory and algebraic geometry, and develop them into concrete tools that can advance our knowledge further.