There are many works on geometric representation theory of quiver varieties and their relation to quantum loop algebras and Yangians. Recently, I have been interested in their variants, where quiver varieties are replaced by σ-quiver varieties, the fixed point loci of involutions on quiver varieties. I will explain my recent work on geometric representation theory of σ-quiver varieties and twisted Yangian, focusing on the special case of cotangent bundles of l-step isotropic flag varieties.

We present our solutions to two long standing open problems, one from probability theory formulated by Malyshev in 1970 and another one from a crossroad of geometry and dynamics, going back to Darboux in 1879. The Malyshev problem is of finding effective, explicit necessary and sufficient conditions in the closed form to characterize all random walks in the quarter plane with a finite group of the random walk of order 2n, for all n ≥ 2. Previously known results covered the cases n = 2, 3, and 4. We also describe all n-periodic Darboux transformations for four-bar link problems for all n ≥ 2, thus completely solving the Darboux problem, that he solved for n = 2, and which was recently extended to n = 3. The talk is based on a joint work with Milena Radnovic (arXiv:2512.21976).