## Departmental PhD Thesis Exam – Khoa Pham

Monday, July 13, 2020
2:00 p.m.

PhD Candidate:  Khoa Pham
Supervisor:   Joel Kamnitzer
Thesis title:  Multiplication of generalized affine Grassmannian slices and comultiplication of shifted Yangians

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Given a semisimple algebraic group $G$, shifted Yangians are quantizations of certain generalized slices in $G((t^{-1}))$. In this thesis, we work with these generalized slices and the shifted Yangians in the simply-laced case.

Using a presentation of antidominantly shifted Yangians inspired by the work of Levendorskii, we show the existence of a family of comultiplication maps between shifted Yangians. We include a proof that these maps quantize natural multiplications of generalized slices.

On the commutative level, we define a Hamiltonian action on generalized slices, and show a relationship between them via Hamiltonian reduction. This relationship is established by constructing an explicit inverse to a multiplication map between slices.

Finally, we conjecture that the above relationship lifts to the Yangian level. We prove this conjecture for sufficiently dominantly shifted Yangians, and for the $\mathfrak{sl}_2$-case.

A copy of the thesis can be found here: Thesis-Khoa-final