The physics of quantum information and matter has deep connections to a wide range of mathematical disciplines. In recent years, the development of new mathematical concepts and technical tools has provided a rigorous language for describing quantum physics while also opening new directions within mathematics itself.

This winter school aims to offer a mathematically focused introduction to several key ideas underpinning contemporary research in quantum information and quantum matter theory.

The program consists of four courses covering quantum information, quantum error correction, topological matter, and topological band theory. Quantum information theory provides a framework for understanding quantum states as resources, emphasizing ideas like entanglement, channels, and complexity. Quantum error correction refines this perspective by introducing algebraic and topological structures that protect quantum information against noise, leading to connections with coding theory, homological algebra, and topology. The study of topological phases extends these ideas to many-body systems, where phases of matter are characterized by global, topological invariants and patterns of long-range entanglement, often described using topological field theory and modular tensor categories. Complementing this, topological band theory analyzes single-particle systems with tools from differential geometry and K-theory.

Together, these courses illustrate how quantum physics motivates rich mathematical structures while also addressing questions of practical importance, from fault-tolerant quantum computation to the classification of phases of matter. By emphasizing the mathematical formulation of these ideas, the school aims to equip participants with a conceptual framework and technical background necessary to engage with current research at the intersection of mathematics and quantum science.

The school is aimed at advanced undergraduate and graduate students, specialized in both math and theoretical physics, who are familiar with the basics of linear algebra. In advance of the school, students will be equipped with detailed prerequisites for the courses and a catch-up reading list.