Categorical Methods for Classical, Equivariant, and Motivic Homotopy Theory

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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.