The UC Irvine Department of Mathematics has been awarded a highly competitive Graduate Assistance in Areas of National Need grant. The GAANN program provides "fellowships through academic departments of institutions of higher education to assist graduate students of superior ability who demonstrate financial need and plan to pursue the highest degree available in their course of study at the institution."
A quantum Banach space (operator space for short) is a linear subspace of Hilbert space operators together with its induced matrix norm structure. It is said to be a quantum operator algebra (operator algebra for short) if it is closed under multiplication.
In a joint work with Matt Neal, a necessary and sufficient condition is given for a operator space to support a multiplication making it quantum isomorphic to a unital operator algebra. The condition involves only the holomorphic structure of the matrix spaces with entries from the operator space. The proof involves an algebraic structure which is equivalent to the holomorphic structure.
Based on a paper joint with David R. Adams, we will address quasi-continuities of Morrey potentials and their applications to fine properties of weak solutions of two p-Laplace systems: (p,q)-type harmonic map and Lane-Emden systems, whence getting that any local singular set of the minimizing p-harmonic maps from a bounded domain to the unit sphere is discrete.
I will present a systematic approach to develop models for biofilm and solvent mixture. Models at various time and length scales will be introduced.
Quorum sensing, cell adhesion, and other cellular movement due to chemotaxis and haptotaxis will be discussed. Biofilm growth and interaction with the ambient flow
in an infinitely long channel and a finite length flow chamber will be simulated using a 3-D numerical simulation tool developed based on the models. The role of antimicrobial agents
will be investigated as well.
