Colorado State University   Mathematical

Quantum Groups and Differential Equations

By  Sachin Gautam
From  Department of Mathematics
Columbia University
When  Tuesday, February 3, 2015
2:00 pm
Where  Weber 223

Quantum groups are known to be natural receptacles for monodromy of Fuchsian connections, since the pioneering works of Drinfeld and Kohno. A conjectural variant of this principle for trigonometric Casimir connections was formulated by Toledano Laredo. This monodromy conjecture aims at relating two distinct classes of representations of the affine braid groups associated with a simple Lie algebra g. The first one arises as the monodromy of a system of differential equations with coefficients from the Yangian Yh(g). The second is given by quantum Weyl group operators of the quantum loop algebra Uq(Lg). This conjecture led to a more general principle that hints at deep connections between quantum cohomology of symplectic resolutions, representation theory of quantum algebras and the geometry of derived categories.

In this talk I will present a precise formulation of the monodromy conjecture and report the progress towards its proof. The new perspective is achieved through understanding the concrete relation between the finite-dimensional representations of Yangians and quantum loop algebras.

Gerhard Dangelmayr