Homotopy Theory and its Applications

9977089高斯在数学的各个领域中，重要的是要了解不同的对象是如何获得某种乘法的。例如，在同伦理论中，它是研究在连续变形下保持不变的数学现象，人们想要确定一个对象是否有一个乘法对所有“高同伦”都是可交换的——这是一种在变形下不变的乘法。讨论这个问题的一种方法是用歌剧的方法，它使“高同伦”这个词变得严格。Goerss打算在同伦理论的背景下发展和探索这些思想；例如，他想讨论谱何时具有操作数乘法，并讨论拓扑空间上的协链具有这种乘法的事实的含义。该项目的总体目标是进一步理解同伦理论，这是拓扑的一个分支，涉及在连续变形下保持不变的高维表面和其他几何物体的性质。同伦理论与数论和代数几何等看似无关的数学领域之间存在着一些令人惊讶且相当神秘的联系，人们希望该项目最终能阐明这些联系

9977089Goerss Throughout different fields of mathematics, it is important tounderstand how various objects acquire some sort of multiplication.For example, in homotopy theory, which is the study of mathematicalphenomena that remain invariant under continuous deformation, onewants to decide if an object has a multiplication that is commutativeup to all ``higher homotopies'' -- this is the type of multiplicationthat is invariant under deformation. One way to discuss this issue iswith the device of opera, which make rigorous the phrase ``higherhomotopies.'' Goerss intends to develop and explore these ideas inthe context of homotopy theory; for example, he would like to discusswhen a spectrum has an operad multiplication, and to discuss theimplications of the fact that the cochains on a topological space havesuch multiplications. The broad goal of this project is to further understanding ofhomotopy theory, the branch of topology concerned with properties ofhigher-dimensional surfaces and other geometric objects that remaininvariant under continuous deformation. There are some surprisingand rather mysterious connections between homotopy theory and seeminglyunrelated fields of mathematics such as number theory and algebraicgeometry, and one hopes the project will ultimately illuminate these.***