Theory of normal forms of differential operators and some of its applications

Abstract: The theory of normal forms appeared in a series of my works with co‑authors as a convenient tool for studying quantum integrable systems (in a broad sense) — by this I mean rings of commuting operators of various natures. The normal forms of differential operators mentioned in the title are operators of a special type in a certain complete non‑commutative ring \hat{D}_n^{sym} (not the well‑known ring of formal pseudo‑differential operators!). They are obtained by conjugating the operators under study by an invertible operator (a generalised Schur operator), which is constructed from a chosen operator in the commutative ring.

In my talk, I will give an overview of recent results obtained by myself and with co‑authors that follow from the theory of normal forms: an alternative classification of commutative rings of ordinary differential operators and the associated description of torsion‑free sheaves with zero cohomology on irreducible projective curves; a constructive one‑to‑one correspondence between solutions of the string equation and commuting ordinary differential operators; and the resulting proof of the Dixmier conjecture for the first Weyl algebra.

quantum algebrarings and algebras

Audience: researchers in the topic

European Non-Associative Algebra Seminar