Abstract: Integrable systems first came to prominence as a geometric abstraction of structure in classical mechanics. Despite these origins, integrable systems have been found to interact meaningfully with pure mathematics. Modern examples include the role such systems play in toric geometry, the Langlands program, mirror symmetry, and quantum cohomology. At the same time, Mishchenko-Fomenko systems represent another celebrated paradigm of integrable systems in pure mathematics. They exhibit the kind of Lie-theoretic symmetry that allows difficult geometric problems to be posed and solved entirely in algebraic terms. Techniques in commutative algebra thereby yield new examples of integrable systems, as well as deeper insights into existing ones.

I will give a non-technical overview of the themes mentioned above. Some emphasis will be placed on my contributions to the subject, and on directions for future research.

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