Homotopy Theory and It's Applications

米切尔的项目有两个主要组成部分。第一部分研究K（1）-局部同伦理论及其应用，包括在代数K-理论和一般线性群同调中的应用。该方法结合了Iwasawa理论的K（1）-局部谱。第二部分是对环群和对称空间的拓扑和几何的研究，包括Pontrjagin环的同调Schubert演算和有关稳定同伦类型的问题。代数拓扑是几何学的分支，研究几何对象的定性性质，特别是在连续变形下保持不变的性质。拓扑学与数学的其他分支有许多联系，近年来，拓扑学以令人惊讶的频率出现在科学领域，包括物理学甚至分子生物学。Mitchell的项目是研究代数拓扑、数论和李群几何（流形和其他几何对象的变换群）之间有趣的新联系。

Mitchell's project has two principal components. The first is a study ofK(1)-local homotopy theory and its applications, including applicationsto algebraic K-theory and general linear group homology. The methodsincorporate an Iwasawa theory for K(1)-local spectra. The secondcomponent is an investigation of the topology and geometry of loopgroups and symmetric spaces, including a homology Schubert calculus forthe Pontrjagin ring and questions about stable homotopy type.Algebraic topology is the branch of geometry that studies qualitativeproperties of geometric objects, especially properties that remaininvariant under continuous deformation. Topology has many connections toother branches of mathematics, and in recent years has appeared withsurprising frequency in the sciences, including physics and evenmolecular biology. Mitchell's project is an investigation of intriguingnew connections between algebraic topology, number theory, and thegeometry of Lie groups (groups of transformations of manifolds and othergeometric objects).