## Departmental PhD Thesis Title – Alex Weekes

Everyone is welcome to attend.  Refreshments will be served in the Math Lounge before the exam.

Friday, May 6, 2016
11:10 a.m.
BA6183

PhD Candidate:  Alex Weekes
Supervisor:  Joel Kamnitzer
Thesis title:  Highest weights for truncated shifted Yangians

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Abstract:

Truncated shifted Yangians are a family of algebras which are conjectured to quantize slices to Schubert varieties in the affine Grassmannian. In this thesis we study the highest weight theory of these algebras, and explore connections with Nakajima quiver varieties and their cohomology. We give a conjectural parametrization of the set of highest weights in terms of product monomial crystals, which are related to Nakajima’s monomial crystal. In type A we prove this conjecture.

Our main tool in describing the set of highest weights is the B–algebra, which is a non-commutative generalization of the notion of torus fixed-point subscheme. We give a conjectural presentation for this algebra based on calculations using Yangians, and show how this presentation admits a natural geometric interpretation in terms of the equivariant cohomology of quiver varieties. We conjecture that this gives an explicit presentation for the equivariant cohomology ring of the Nakajima quiver variety of a finite ADE quiver, and show that this conjecture could be deduced from a special case. We give a proof of this conjecture in type A.

This work can be thought of in the context of symplectic duality. In our case, slices to Schubert varieties in the affine Grassmannian are expected to be symplectic dual to Nakajima quiver varieties. The relationship between B–algebras and equivariant cohomology is part of a general conjecture of Nakajima for symplectic dual varieties. These ideas represent a first approximation to expected connections between the category $\mathcal O$’s for a symplectic dual pair of varieties.

A copy of the thesis can be found here: Alex Weekes – thesis