Algebraic Topology and Algebraic K-theory

这个项目使用同伦理论来研究数论、代数和微分拓扑学中的问题。同伦理论研究那些在小变形下不变的数学对象的性质。这些数学对象通常是几何性质的，但同伦理论的方法也越来越多地应用于代数性质的对象。同伦的理论性质往往可以通过利用小变化下的不变性来进行计算。由于它们通常也保留了关于原始数学对象的重要信息，同伦理论是解决广泛数学问题的有效工具。该项目有三个主要焦点。第一个焦点涉及球谱的代数K-理论，它与微分拓扑学和数论相联系。该研究执行了一个程序来识别球谱的K-理论和整数的K-理论之间的线性化映射的纤维的同伦类型；一个程序来识别割圆迹的同伦纤维；以及通过构造适当的割圆谱的谱范畴来接近Hesselholt猜想。第二个重点是把复余边谱研究为E_无穷或更一般的E_n环谱。该项目调查了关于协边乘法的一些具体问题以及解决这些问题的方法。第三个焦点是E_n可微分次代数与单连通空间的(不稳定)同伦理论之间的关系。这项工作包括一个探索“形式”E_n代数概念的项目，特别是一个空间的余链复形何时可以是形式的问题，以及它何时等价于交换微分分次代数的问题。

This project uses homotopy theory to study questions in number theory and in algebraic and differential topology. Homotopy theory studies those properties of mathematical objects that do not change under small deformations. These mathematical objects are often of a geometric nature, but the methods of homotopy theory have been increasingly applied to objects of an algebraic nature as well. Homotopy theoretic properties tend to be accessible to computation by taking advantage of the invariance under small changes. Since they also generally retain important information about the original mathematical objects, homotopy theory is an effective tool for a wide range of mathematical problems.The project has three main foci. The first focus concerns the algebraic K-theory of the sphere spectrum, which connects to both differential topology and number theory. The research pursues a program to identify the homotopy type of the fiber of the linearization map between the K-theory of the sphere spectrum and the K-theory of the integers; a program to identify the homotopy fiber of the cyclotomic trace; and an approach to a conjecture of Hesselholt by constructing an appropriate spectral category of cyclotomic spectra. The second focus is the study of the complex cobordism spectrum as an E_infinity or more generally E_n ring spectrum. The project investigates some concrete questions on the multiplication on cobordism and approaches to solving them. The third focus concerns the relationship between the E_n differential graded algebras and the (unstable) homotopy theory of simply connected spaces. The work includes a project for exploring the notion of "formal" E_n algebra, particularly the question of when the cochain complex of a space can be formal and the related question of when it is equivalent to a commutative differential graded algebra.