This seminar talk by Peter Koroteev explores the geometric foundations of integrable systems, tracing their evolution from classical mechanics to modern quantum theory. Discover how algebraic geometry and representation theory have become essential tools for understanding integrable systems, with recent breakthroughs allowing these connections to be expressed in purely geometric terms. Learn about a novel geometric structure called an "oper" that unifies the phase spaces of many-body integrable systems and quantum spin chain spectra. The presentation establishes connections across mathematical physics, including representation theory, cluster algebras, quantum cohomology, and quantum hydrodynamics. Koroteev, who transitioned from theoretical physics to mathematics, specializes in the intersection of enumerative algebraic geometry, geometric representation theory, and integrable systems, with recent interests extending to number theory. The 39-minute BIMSA Member Seminar includes downloadable slides and video resources for deeper exploration of these mathematical concepts.