CAREER: Periodicities and symmetries in stable homotopy theory

Symmetry is a fundamental organizing principle in mathematics and science. This project will use symmetry to study some of the central objects in topology: spheres and their higher-dimensional patterns. Although spheres are among the simplest geometric objects, the algebraic structures that describe their stable behavior are extraordinarily rich and difficult to compute. Understanding these structures has led to major advances in topology, geometry, and mathematical physics. The PI will develop new methods that combine symmetry, algebraic topology, and computer programming to make these computations more systematic and accessible. The project will also support education and outreach through the Husky Math Academy Summer Program, which will provide high-quality mathematical enrichment for middle and high school students in the greater Seattle area, and through WCATSS, which will provide training and mentoring for graduate students and early-career researchers in algebraic topology. These activities will advance fundamental science, strengthen the STEM workforce, support young researchers, and broaden access to advanced mathematical education. The Husky Math Academy Summer Program will be held at the University of Washington in Seattle, Washington, July 12–23, 2027, and the West Coast Algebraic Topology Summer School (WCATSS) will be held at the University of Washington in Seattle, Washington, July 10–14, 2028. Specific research projects include the study of higher real K-theories and Real bordism theories using equivariant homotopy theory and the equivariant slice spectral sequence. The PI will investigate structural patterns in these theories, including transchromatic phenomena, duality, periodicity, and vanishing lines. The PI will also adapt synthetic techniques to the equivariant slice filtration, develop improved methods using restriction, transfer, and norm maps, and implement these methods in computer programs to automate spectral sequence computations. These tools will be applied to compute fixed point spectra associated with Lubin—Tate theories and Real bordism theories across different groups and chromatic heights. Finally, the PI will study equivariant Hurewicz images and detection theorems, with applications to families in the stable homotopy groups of spheres and questions related to the Kervaire invariant problem. The expected outcomes include new computations in stable homotopy theory, new structural results in equivariant and chromatic homotopy theory, and computational tools that can be used by researchers in future work. Related applications include the study of invertible framed topology quantum field theories (TQFTs), where physical anomalies correspond to non-trivial stable homotopy classes by the cobordism hypothesis. Webpage listing: PI webpage: https://sites.google.com/view/dannyshi Husky Math Academy: https://www.huskymath.org/ 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. NSF Award ID: 2541934 | Program: 01002930DB NSF RESEARCH & RELATED ACTIVIT,01003031DB NSF RESEARCH & RELATED ACTIVIT,01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: XiaoLin Danny Shi | Institution: University of Washington, SEATTLE, WA | Award Amount: $245,971 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2541934 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2541934.html