Vladimir Baranovsky (University of California, Irvine)
X
Let (M, w) be a complex manifold with an holomorphic symplectic form, or an algebraic version of such a pair. Consider also a pair (X, E) consisting of a coisotropic submanifold X in M and a vector bundle E on X. Then (local) sections of E form a module over the ring O of (local) functions on M. Given a deformation quantization O_h of functions, one can ask whether E admits a similar deformation quantization E_h as a module over O_h. For Lagrangian X this is equivalent to existence of a projectively flat connection on E with specific properties. For general coisotropic X only necessary conditions on some characteristic classes are known. The plan is to present some results and conjectures related to the problem, based on recent work of Bogdanova-Vologodsky and a Harish-Chandra bundle approach to quantizations and their Deligne classes, introduced earlier by Bezrukavnikov-Kaledin.
Quantization of symplectic manifolds and vector bundles on coisotropic submanifolds.
I will outline a classification framework for quantum phases within the setting of stabilizer states and quantum cellular automata (QCA). Both admit a precise algebraic formulation—stabilizer codes as modules over Laurent polynomial rings, and QCA as locality-preserving automorphisms of operator algebras—through which topological invariants such as Witt classes and boundary charge modules arise. I will review my recent work classifying Clifford QCA and Pauli stabilizer codes, which fit naturally into the same mathematical framework.
A Case Study in the Classification of Quantum Phase Sponsored by the Meyer Fund
Quantization of symplectic manifolds and vector bundles on coisotropic submanifolds.