CAREER: From Equivariant Chromatic Homotopy Theory to Phases of Matter: Voyage to the Edge

Homotopy theory studies properties of geometric objects that are unchanged under continuous deformations by associating quantities called invariants to these objects. Some invariants are stable in the sense that they are independent of certain spatial dimension shifts. They are easier to compute because they are in some sense more algebraic.

This project has two main themes. The first line of investigation is in chromatic homotopy theory, a field of mathematics that studies structural properties of stable invariants. This part of the project seeks to answer questions such as: Are there families of stable invariants that share common properties?

What kind of symmetries exist for these families? How can these symmetries be used to do explicit computations and learn new things about fundamental geometric objects such as higher dimensional spheres? The second line of investigation is part of a multi-disciplinary collaboration with mathematicians and physicists, which uses stable invariants to study the phase of matter.

A quantum system is a collection of interacting particles and, roughly, a phase is a family of quantum systems that may be different microscopically, but share certain macroscopic properties. For certain types of quantum systems, the phase type can be detected by stable invariants. This project aims to construct new stable invariants of phases and to study stable invariants of quantum systems equipped with certain symmetries.

The broader goal is to make progress on the classification of phases of matter to better understand the fundamental properties of materials. The project has an integrated educational component, one goal of which is to make the two areas of research accessible to graduate students and advanced undergraduates through a series of graduate workshops. The educational plan also includes undergraduate and graduate research.

In particular, the project will conduct research in collaboration with existing initiatives at the University of Colorado, Boulder that work to promote diversity, equity and inclusion in STEM. An important goal is to increase the accessibility of research for underrepresented minorities in mathematics.

Stable invariants are studied using generalized cohomology theories or, more specifically, mathematical objects called spectra. Chromatic homotopy theory aims to classify families of spectra according to different periodic behaviors exhibited by the stable invariants they compute. There is a close relationship between periodic behaviors and the symmetries of the spectrum.

This project uses equivariant techniques to better understand this relationship. It has applications to the study of stable homotopy groups of spheres. Specifically, the project explores theoretical and computational properties of equivariant generalizations of Lubin-Tate theories that are built from the Real bordism spectrum, an equivariant generalization of complex bordism.

The project develops techniques to compute the equivariant stable invariants arising from these theories. In a different direction, the project examines a conjecture that parametrized gapped invertible phases of matter form a generalized cohomology theory. The project proposes the construction of a ground-state bundle for parametrized quantum systems and explores how equivariant homotopy theory can inform the study of systems with symmetries.

The project aims to increase accessibility of these topics to students in two ways. First, the project includes two five days mathematical events that combine a graduate workshop with research talks with a focus on the main areas of research of the project. These are to be part of an ongoing series that will continue beyond the five year duration of the project.

These events will create an interactive and collaborative environment between participants and experts by incorporating active-learning in the program structure. Secondly, the project will engage in academic term and summer undergraduate research opportunities in topology and phases of matter with a goal to increase access to undergraduate research for students from historically excluded groups in mathematics.

The project will also support graduate students working on diversity, equity and inclusion initiatives.

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.