The National Science Foundation has awarded a three year grant to Professor Valerio Toledano Laredo to fund his research on ”Monodromy Theorems, Affine Quantum Groups, and Meromorphic Tensor Categories”, starting in August 2015.

Professor Toledano Laredo will study the relationship between the quantum groups corresponding to the plane, the sphere and the torus. These have been traditionally thought to sit on distinct rungs in a ladder of increasing complexity, but recent research by the PI has uncovered the surprising fact that these quantum groups are in fact equivalent. Such equivalences are very interesting in themselves, but also allow to relate in a very precise manner the appearance of braid groups in very disparate Mathematical and Physical contexts. A related theme of the project is the computation of the monodromy of the rational affine, trigonometric, and elliptic Casimir connection in terms of quantum Weyl group operators in a way reminiscent of the Kohno-Drinfeld quantum group. The proposal has potential implications for the representation theory of Yangians, quantum loop algebras and elliptic quantum groups, quantum integrable systems and enumerative geometry. Professor Toledano Laredo will conduct his research together with a group of graduate students and postdocs.