Chromatic Derived Algebraic Geometry and Equivariant Homotopy Theory

拓扑学是关于（通常）高维空间的定性研究。代数拓扑学使用代数不变量来达到这个目的。传统上，这些不变量通常是数字。但更现代的改进使用代数分析对象，包含更多关于我们研究的空间的信息。我所关心的精化使用模和自守形式。一个重要的早期动机使用模块化形式的拓扑是Withen的工作分析循环空间（即空间的封闭循环或字符串的空间）有关的量子场论或弦理论。我的项目研究拓扑模形式和拓扑自守形式中的对称性和对偶性。这对于理解它们与其他更容易计算的变体之间的关系特别有用。

Topology is about the qualitative study of (often) high-dimensional spaces. Algebraic topology uses algebraic invariants to this purposes. Classically these invariants were usually numbers. But more modern refinements use algebraic-analytic objects that contain more information about the spaces that we study. The refinements I am concerned with use modular and automorphic forms. An important early motivation for the use of modular forms in topology was Witen's work on analysis on loop spaces (i.e. spaces of closed loops or strings in spaces) as relevant in quantum field theory or string theory. My project investigates symmetries and dualities in topological modular forms and topological automorphic forms. This is, in particular, useful for understanding their relationship to other variants of them that are easier to compute.