Analysis at UF
The analysis group at UF is involved in contemporary
research in a broad range of mathematics with important inter and
intra-discipline connections. Below is a brief synopsis of the activities.
For more information, follow the links to the left to visit individual
faculty and the analysis seminar homepages.
Quantum theory, with an emphasis on algebraic quantum field theory.
This involves using operator algebra theory, among other mathematical tools,
to address conceptual and mathematical problems in quantum field theory,
which itself seeks to describe the fundamental constituents of matter and
their laws of interaction.
Complex Analysis and Number Theory
Complex analysis and its applications to special functions
including Jacobi elliptic functions,
differential equations and number theory.
Measure Theory, Commutative and Noncommutative
Infinite dimensional vector-valued integration and stochastic
analysis, including stochastic integration which is the basis
for much of financial mathematics. One setting for
noncommutative measure theory is within a weak-star closed
subalgebra of a C-star algebra, while another setting
is provided by a (normal) state on a W-star algebra.
Partial Differential Equations and Image Processing
Harmonic maps on manifolds; theories and methods of non-linear PDEs.
The application of PDEs to image processing, particularly medical
such as fMRI. On campus collaborating units include
the Brain Institute, the medical school, and computer science.
Representations of the Weyl and Clifford algebras.
Symplectic geometry has its origins in the Hamiltonian
formulation of classical mechanics.
A symplectic manifold is a differentiable manifolds equipped with a closed,
Operator Theory and Operator Algebras
Single and several variable operator theory, particularly
with connections to analytic functions including interpolation
theorems of Pick type which themselves generalize the
Schwartz Lemma; composition operators; and index theory.
Non-self adjoint operator algebras, and operator systems and
spaces. Self-adjoint operator algebras.
Noncommutative semi-algebraic geometry
A large class of engineering problem take the form system of
dimensionless matrix inequalities; i.e., polynomial inequalities
where the variables are matrices and the only restriction on the
sizes of the matrices involved is that the formulas make sense.
A theory of noncommutative semi-algebraic parallel to the classical
commutative theory is rapidly developing to understand matrix inequalities.
That there are very few convex noncommutative semi-algebraic
sets is one of the most mathematically
striking and practically important differences that
has emerged to date.
Probability, Stochastic Analysis, and Financial Mathematics
Martingales, Probabilistic potential theory, numerical stochastic
integrations. Stochastic integration and financial mathematics.
Applications of probability to image processing.