Quantized Integrable Systems, Normal Forms, and Variation of Hodge Structures

Explore a complex mathematical lecture on quantized integrable systems, normal forms, and variations of Hodge structures delivered by Alexander Soibelman at the M-Seminar, Kansas State University. Delve into the classical Birkhoff theorem and its applications in complex symplectic manifolds. Discover how this theorem relates to the transformation of functions near Morse critical points through formal symplectomorphisms. Learn about the quantum analog of the Birkhoff normal form and its role in computing Schrödinger operator eigenvalues. Examine the geometric approach to obtaining the Birkhoff normal form of a quantum Hamiltonian, and understand its connections to the quantization of integrable systems and formal deformations of variations of Hodge structures. Gain insights into ongoing research conducted in collaboration with Maxim Kontsevich during this 1 hour and 20 minute presentation from October 13, 2022.