This CAREER award falls under the umbrella of the NSF-wide Mathematical Sciences Priority Area and supports integrated research and education in theoretical condensed matter physics. The PI plans to address localization in superconducting wires with broken spin-rotation symmetry, network models for quantum Hall effects, rare fluctuations in disordered superconductors, and stochastic Loewner evolution and its generalizations. Superconducting wires with broken spin-rotation symmetry exhibit delocalized quasiparticle states leading to unusual thermal transport properties. This happens at a critical point. The PI plans to study the nature of this critical point using scattering theory and supersymmetry methods. Quantum Hall transitions are challenging examples of critical points. Network models for quantum Hall transitions and their generalizations will be studied with an aim to understanding the nature of these transitions. The supersymmetry method adapted for network models provides the formal approach for the research. Density and spatial structure of quasiparticle states in disordered superconductors are essential parts of our understanding of these fascinating systems. The optimal fluctuation method and self-consistent treatment of the suppression of superconductivity by impurities will be combined to study rare fluctuations in disorder configurations and ensuing low-energy states in disordered superconductors. The PI plans to study stochastic Loewner evolution and its generalizations. The stochastic Loewner evolution is a recent development in the study of criticality in two dimensions and fractal structures emerging as results of stochastic growth phenomena. Possible applications include such long-standing open problems as diffusion-limited aggregation and other growth phenomena. The education component involves developing graduate level courses in modern condensed matter physics, supervision of graduate and undergraduate research students, and making novel contributions to the Research Experiences for Undergraduates, Young Scholars, and Seminars for Elementary Specialists And Mathematics Educators programs as well as working with professionals at the Museum of Science and Industry (MSI). These activities will be closely coupled with feedback from education professionals. Courses will focus on conceptual understanding rather than theoretical techniques, and will be accessible to experimentally inclined students. The REU and Young Scholars programs reach out to underrepresented groups, while the MSI each year serves about one half million school students. %%% This CAREER award falls under the umbrella of the NSF-wide Mathematical Sciences Priority Area. It supports integrated research and education in theoretical condensed matter physics. The PI will use advanced theoretical methods to study two problems at the heart of condensed matter physics connected by a common theme of randomness. The first involves electrons in a random environment, for example, in materials in which the atoms are randomly displaced from a regular periodic lattice. The PI focuses on interesting physical cases when the motion of the electrons is further restricted to a plane or a wire, and when novel states of matter and transformations among them are involved, like superconducting or quantum Hall states. The second area involves materials growth processes, like those at work in the formation of snowflakes. The PI seeks intellectual advances by pursuing a recent breakthrough in the mathematical description of these processes. The research brings together ideas from physics and mathematics including localization, superconductivity, statistical mechanics of random systems, critical phenomena, complex analysis, probability theory, fractals, and random matrices. The education and outreach component of the proposal will integrate research into teaching of physics from high school to graduate level, provide new course material for high school students, and help keep professionals at a major science museum scientifically up to date. ***
