Categorical Methods for Classical, Equivariant, and Motivic Homotopy Theory

经典同伦理论在非常粗糙、灵活和近似的意义上研究空间的形状。严格的几何概念，如距离和曲率，并不起作用；相反，本质上是这样的概念，即一个空间可以连续变形成另一个空间。在过去的50年里，随着数学家们认识到在许多情况下可以找到或定义类似的“连续变形”概念，同伦理论的方法和范围得到了显著的推广和扩展。事实上，我们现在知道，在数学中有许多不同的同伦理论，它们出现在广泛的数学学科中。在这个项目中，PI的目标是将张量三角几何的方法应用于经典、等变和动机同伦理论中四个截然不同但主题相关的问题。该提案旨在解决的问题在各自的学科中都是重要的问题，但提出的解决这些问题的技术并不是从代数拓扑学的核心出发的主流方法。通常使用的方法是非常跨学科的，使用张量三角分类的平台在不同的数学学科之间传递思想和技术。事实上，拟议的工作将通过在代数拓扑学、代数几何和表示论之间传递思想和技术来加强现代数学的互联性。此外，拟议的研究计划将扩展和发展张量三角几何技术，这是一个广泛的研究计划，旨在开发代数几何方法来推理数学中出现的各种同伦理论。在第一个项目中，PI旨在继续最近在理解等变稳定同伦理论谱方面的势头，目标是完成所有有限群的谱的计算，并追求对紧李群情况的推广。第二个项目提出了一种方法来证明Palmieri关于Steenrod代数上的模分类的一个20年前的猜想，Steenrod代数是经典代数拓扑学的一个重要对象，已经被深入研究了60多年。在第三个项目中，PI将使用一种策略来计算动机稳定同伦范畴的一个重要部分的谱。最后，PI还将使用一种范畴方法来处理Motivic稳定同伦范畴的Picard群，这有可能澄清对这些群知之甚少的东西。所提出的方法的一个共同主题是通过穿越张量三角范畴的统一世界来推广和应用在模表示理论中取得成功的方法。第二个共同主题是在张量三角几何中使用etale拓扑学的类比。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Classical homotopy theory studies the shape of space in a very coarse, flexible and approximate sense. Rigid geometric notions such as distance and curvature play no role; instead, at heart is the notion that one space can be continuously deformed into another. Over the last 50 years the methods and scope of homotopy theory have been significantly generalized and expanded as mathematicians have realized that there are many situations where analogous notions of "continuous deformation" can be found or defined. Indeed we now know that there are many different homotopy theories to be found in mathematics and that they arise in a wide range of mathematical disciplines. In this project, the PI aims to apply the methods of tensor triangular geometry to four distinct but thematically-related problems in classical, equivariant, and motivic homotopy theory. The problems the proposal aims to solve are significant questions in their respective subjects and yet the techniques proposed to tackle these problems are not mainstream methods from the heart of algebraic topology. The general approach used is extremely interdisciplinary, using the platform of tensor triangulated categories to pass ideas and techniques between diverse mathematical subjects. Indeed, the proposed work will strengthen the interconnectedness of modern mathematics by transferring ideas and techniques between algebraic topology, algebraic geometry and representation theory. Moreover, the proposed research program will extend and develop the techniques of tensor triangular geometry, a broad research program which aims to develop algebro-geometric methods for reasoning about a wide variety of homotopy theories arising throughout mathematics.In the first project, the PI aims to continue recent momentum in understanding the spectrum of equivariant stable homotopy theory with the goal of completing the computation of the spectrum for all finite groups and for pursuing generalizations to the case of compact Lie groups. The second project proposes a method to prove a twenty-year-old conjecture of Palmieri concerning the classification of modules over the Steenrod algebra, an important object of classical algebraic topology that has been intensely studied for over sixty years. In the third project, the PI shall use a strategy to compute the spectrum of an important piece of the motivic stable homotopy category. Finally, the PI will also use a categorical approach to the Picard group of motivic stable homotopy categories, which has the potential to clarify what little is known about these groups. A common theme in the proposed methods is the generalization and application of methods which have been successful in modular representation theory by passing through the unifying world of tensor triangulated categories. A second common theme is the use of an analogue of the etale topology in tensor triangular geometry.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.