Mathematics of topological phases of matter: May 1- June 23, 2017

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Organized by: Lukasz Fidkowski, Dan Freed, and Anton Kapustin

May 1 – June 23, 2017

Over the last decade there has been a lot of progress in understanding gapped quantum phases of matter. To a large extent this progress has been achieved by exploiting connections to seemingly unrelated areas of mathematical physics, such as Topological Quantum Field Theory and quantum information science, as well as to algebraic topology. Early examples of nontrivial gapped phases of matter appeared in the study of the quantum Hall effect, but a number of new phases have been identified more recently, such as topological insulators and superconductors. Among all gapped quantum phases (also known as topological phases), Short Range Entangled (SRE) phases are the most manageable ones, and one can hope to give a complete classification of such phases in all dimensions and for all symmetry groups. For free fermionic phases, such a classification has been achieved in the works of Schnyder, Ryu, Furusaki, Ludwig, and Kitaev. In the interacting case it has been proposed that SRE phases can be classified by invertible equivariant TQFTs, or equivalently by homotopy classes of maps of certain spectra (in the sense of algebraic topology). A lot of progress towards a proof of this conjecture has been made in the case of one-dimensional systems using the technology of Matrix Product States. This technique also turned out very useful for finding approximate ground states of general one-dimensional systems with short-range interactions and understanding their properties, such as the area law for the entanglement entropy. Despite all this progress, a mathematically rigorous definition of a (gapped) phase remains elusive. The goal of this program is to bring together mathematicians and physicists in the hope of making the physical results accessible to mathematicians and turning them (the results, not the mathematicians) into mathematical theorems.

Since condensed matter theory is relatively new to this generation of mathematicians engaging with physics, most of whom have experience with field theory and string theory instead, the program will have a significant pedagogical component in addition to its main focus on current research and open problems. There will be several short courses over the course of the program. These are designed to educate mathematicians about relevant aspects of condensed matter physics and physicists about relevant aspects of topology and other topics.

There will also be a mini course held from May 30 – June 2.