Abstract. We study quantizations of transverse slices to Schubert varieties in the affine Grassmannian. The quantization is constructed using quantum groups called shifted Yangians -these are subalgebras of the Yangian we introduce which generalize the Brundan-Kleshchev shifted Yangian to arbitrary type. Building on ideas of Gerasimov-Kharchev-Lebedev-Oblezin, we prove that a quotient of the shifted Yangian quantizes a scheme supported on the transverse slices, and we formulate a conjectural description of the defining ideal of these slices which implies that the scheme is reduced. This conjecture also implies the conjectural quantization of the Zastava spaces for P GL n of Finkelberg-Rybnykov.

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Abstract. We study a coproduct in type A quantum open Toda lattice in terms of a coproduct in the shifted Yangian of sl2. At the classical level this corresponds to the multiplication of scattering matrices of euclidean SU (2) monopoles. We also study coproducts for shifted Yangians for any simply-laced Lie algebra. In my youth I have multiplied too many matrices, so now I try to avoid it if there is a way around.

Truncated shifted Yangians are a family of algebras which are natural quantizations of slices in the affine Grassmannian. We study the highest weight representations of these algebras. In particular, we conjecture that the possible highest weights for these algebras are described by product monomial crystals, certain natural subcrystals of Nakajima's monomials. We prove this conjecture in type A. We also place our results in the context of symplectic duality and prove a conjecture of Hikita in this situation.

We generalize the mathematical definition of Coulomb branches of 3-dimensional N = 4 SUSY quiver gauge theories in [Nak16, BFN18, BFN16] to the cases with symmetrizers. We obtain generalized affine Grassmannian slices of type BCF G as examples of the construction, and their deformation quantizations via truncated shifted Yangians. Finally, we study modules over these quantizations and relate them to the lower triangular part of the quantized enveloping algebra of type ADE.