Quantum groups, integrable systems and dualities

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This project lies at the intersection of several fields of mathematics: representation theory, classical and quantum integrable systems, mathematical physics, combinatorics, and enumerative algebraic geometry. Representation theory is the study of symmetries of a vector space such as a three-dimensional Euclidean space (or more generally, an infinite dimensional space) endowed with additional important structures. Such symmetries often arise in families encoded by algebraic objects like groups, Lie algebras, or algebras. One important class of algebras that arise in this way is the class of so-called affinized quantum groups. These algebras have been inspiring active research and interactions between mathematics and physics since the 1980s. The main goals of this project are to resolve important questions intrinsic to the affine nature of affinized quantum groups. The project will enhance our understanding of their internal algebraic structures and establish novel connections to the above fields. In addition, the project will have an educational impact through the training and mentoring of students at various levels from high school to graduate school. In more detail, the project will develop new methods in the study of quantum loop groups with applications to geometric representation theory, integrable systems, mathematical physics, and quantum cluster algebras. Building on recent results, the PI will pursue research in five related areas, with specific goals in each. These research areas are unified by the general notion of duality and the use of the "shuffle algebra approach." The rough plan is as follows: 1. Develop key structures of quantum affine and toroidal algebras; 2. Continue the study of quantized Coulomb branches; 3. Continue work on integrable spin chains; 4. Develop a new approach to finite quantum groups and their integral forms that allows for arbitrary roots of unity, as well as a modular theory, thus generalizing and unifying the classical work of DeConcini-Kac-Procesi; and 5. Provide a rigorous mathematical formulation and proof of the BPS/CFT correspondence for A(n)-type quiver gauge theories in string theory, generalizing the known results for the A(1)-quiver.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.