Methods of algebraic geometry in algebraic topology

这个项目的目的是进一步研究稳定的同伦理论的方法和机械发展的代数几何。 有四个主要目标。 第一个目标是研究拓扑自守形式的光谱的具体例子，重点是计算判别式15和皮卡德模块化表面在色级3的志村曲线。 第二，Michael Hill和PI的共同工作旨在将拓扑模形式理论扩展到椭圆曲线模的对数位置，特别是泛函地产生具有层次结构的拓扑模形式。第三，PI希望进一步开发采用Zink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，与大卫Gepner的合作工作将研究皮卡德和Brauer群在派生设置使用最近发展的机器在高级范畴理论。这一研究领域的主要重点是系统地研究定性性质的形状从代数的方法。 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正信息，这是很难获得的。 这个研究项目的目标是采取一些这些最新的进展，发生在科目，如高等范畴理论和代数几何，并将它们发展成具体的工具，可以进一步推进我们的知识。

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.