Algebraic Topology and Algebraic K-theory

这个项目使用同伦理论来研究数论、代数和微分拓扑中的问题。同伦理论研究数学对象在小变形下不改变的性质。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。利用小变化下的不变性，同伦理论性质易于计算。由于它们通常也保留了原始数学对象的重要信息，因此同伦理论是解决各种数学问题的有效工具。该项目有三个主要重点。第一个重点关注球谱的代数k理论，它与微分拓扑学和数论相联系。研究了球谱的k理论与整数的k理论之间线性化映射的光纤同伦类型的确定方案；一种识别切眼轨迹同伦纤维的程序以及通过构造一个合适的分环光谱的光谱类别来探讨Hesselholt猜想。第二个重点是作为e_∞或更一般的E_n环谱的复共协谱的研究。本课题研究了乘上协数的几个具体问题及其求解方法。第三个重点关注E_n微分梯度代数与单连通空间的（不稳定）同伦理论之间的关系。他的工作包括一个探索“形式化”E_n代数概念的项目，特别是当一个空间的协链复形可以是形式化的问题，以及当它等价于一个交换微分渐变代数的相关问题。

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.