CAREER: Unifying representation stability via Fl-categories

Stanford University, Stanford CA

Investigators

Abstract

Award: DMS 1350138, Principal Investigator: Thomas F. Church The physical distinction between bosons and fermions explains why photons can cohere into a laser, or helium-4 nuclei into a superfluid, whereas neutrons resist each other, as is necessary for the stability of neutron stars. Mathematically, this distinction is explained by the shape of their configuration space, which describes all possible ways that a collection of indistinguishable particles (such as photons or neutrons) can be configured in space. The investigator studies configuration spaces via their symmetries, proving that no matter what space the particles move within, the topology of configurations will stabilize, yielding identical behavior no matter how many particles are present. Configurations in restricted spaces have widespread applications: when electrons are confined to 2-dimensional surfaces as in field-effect transistors, new topology on the configuration space predicts new types of quasiparticles, which manifest in the fractional quantum Hall effect; in robotics, the topology of robots configured on a factory floor describes when one robot can be trapped by others. The proposed project would extend this stability to new types of configuration spaces possessing even more symmetry, whose stability cannot currently be understood or proved. The investigator will also develop and teach Math Discovery Lab, a new discovery-based course for undergraduates which develops critical skills in independent research. Students in MDL will investigate open-ended problems, collect data via computer experimentation and simulation, formulate conjectures, and prove theorems about their results, presenting their findings through written reports and lectures. The investigator's research focuses on representation stability, a technique applying representation theory to stability problems in topology and algebra. Representation stability has been very successful, leading to 20+ papers since its introduction by the investigator three years ago. The investigator proposes to extend and strengthen the power of representation stability by undertaking a long-term project to develop the combinatorial, categorical, and homological properties of FI-categories. These provide a coherent framework for studying a wide variety of symmetry groups, including many which could not be handled using previous methods. The proposed project will unify four distinct strands of stability as facets of the same theory: representation stability, homological stability, twisted stability, and central stability. A key advance is that the dependence on representation theory is removed, replaced instead by the combinatorial and homological properties of FI-categories, so that representation stability can be applied even to integral or modular representations.