Abstract: In the recent papers by Braverman-Finkelberg-Nakajima a mathematical construction of the Coulomb branches of 3d N=4 quiver gauge theories was proposed, whose quantization is conjecturally described via the so-called shifted Yangians and shifted quantum affine algebras.

The goal of this talk is to explain how both of these shifted algebras provide a new insight towards integrable systems via the RTT realization. In particular, the study of Bethe subalgebras associated to the antidominantly shifted Yangians of sl(n) provides an interesting plethora of integrable systems generalizing the famous Toda and DST systems. As another interesting application, the shifted quantum affine algebras in the simplest case of sl(2) give rise to a new family of 3^{n-2} q-Toda systems of sl(n), generalizing the well-known one due to Etingof and Sevostyanov. Time permitted, I will also explain how one can generalize the latter construction to produce exactly 3^{rk(g)-1} modified q-Toda systems for any semisimple Lie algebra g.

These talks are based on the joint works with M. Finkelberg, R. Gonin and a current project with R. Frassek, V. Pestun.