Recent advances in condensed matter physics have ushered in a new era of cross-fertilization between mathematics and physics. The study of topological materials, a radically new class of quantum materials discovered in the mid-2000s and the subject of the 2016 Nobel Prize in Physics, has enriched the band theory of solids with concepts from algebraic and differential topology such as homotopy theory, characteristic classes, K-theory, and index theory. Advances in the difficult problem of the classification of symmetry-protected topological phases, a generalization of free-fermion topological materials to interacting systems of fermions or bosons, have involved aspects of group cohomology, topological quantum field theory, and cobordism theory. The description of topological phases with anyon excitations using modular tensor categories and rational conformal field theory, originally limited to 2+1 spacetime dimensions, has been extended to 3+1 dimensions in some cases. Beyond purely topological aspects, geometrical aspects of topological phases have also recently come to the fore. Proposed classifications of free-fermion or interacting topological phases protected by crystalline symmetries have involved sophisticated mathematical tools such as equivariant cohomology and the Atiyah-Hirzebruch spectral sequence. Studies of geometrical aspects of the fractional quantum Hall effect, such as its thermal or elastic responses and the physics of its neutral collective excitations, have led to a renewed interest in complex geometry and the theory of Riemann surfaces.
While this exciting interplay between condensed matter physics and pure mathematics has produced and undoubtedly will continue to produce transformative advances, it is often difficult for beginning researchers with a training focused on one or the other discipline to contribute to this rapidly evolving field, due to discipline-specific language barriers. Through a combination of pedagogical lectures and discussions of recent research topics, this interdisciplinary summer school will break these language barriers by introducing graduate students and postdoctoral fellows to the modern mathematical physics of quantum matter, with a focus on topological and geometrical aspects.
Please consult our Registration page to apply.
This event is sponsored by the Pacific Institute for the Mathematical Sciences (PIMS), and is an activity of the PIMS Collaborative Research Group on Quantum Topology and its Applications.
(Organizers: Lindsay LeBlanc, Joseph Maciejko, and Steven Rayan)