(Operator Algebras) Elliott's program to classify simple separable nuclear C*-algebras via K-theory and traces is essentially complete module the necessary regularity assumptions of Z-stability and the Universal coefficient theorem. My aim is to move beyond the classifiable case and tackle the obstructions which appear when studying normal operators acting on separable Hilbert spaces in the framework of a unital C*-algebra of stable rank one. Whether it require more delicate invariants such as the Cuntz semigroup, or specialized tools in algebraic topology, I study the fine structures which help distinguish any two well-behaved algebras.
References:
Cantier, Laurent. "Towards a classification of unitary elements of C*-algebras." arXiv preprint arXiv:2304.01761 (2023).
An, Qingnan, George Elliott, and Zhichao Liu. "Classification of homomorphisms from C(\Omega) to a C*-algebra." arXiv preprint arXiv:2408.16657 (2024).
(Data Assimilation and Plasma Physics) The 2-field Hasegawa-Wakatani model has been studied extensively in order to understand resistive drift wave turbulence and electrostatic potential of toroidally confined highly magnetized plasma. My goal is to improve the current models with the help of data assimilation and stochastic filtering theory.
References:
Andrew J. Majda, Di Qi, Antoine J. Cerfon; A flux-balanced fluid model for collisional plasma edge turbulence: Model derivation and basic physical features. Phys. Plasmas 1 October 2018; 25 (10): 102307.