Semifinite and purely infinite von Neumann algebras, C*-algebras of real rank zero, K-theory of C*-algebras, classical and Ko-valued Fredholm Index, structure of multiplier algebras, homotopy of the unitary group of C*-algebras, groups and quantum group actions on C*-algebras, spectra, invariant derivations, cross products, Kadison-Singer extension problem, ideals, traces and commutators on Hilbert spaces, arithmetic means and majorization theory, wavelets and frames.
- Herb Halpern: von Neumann algebras
- Victor Kaftal: operator algebras, operator theory
- Costel Peligrad: operator algebras
- Gary Weiss: operator algebras, operator theory, harmonic analysis
- Shuang Zhang: operator theory K-theory, homotopy theory, Fredholm index theory
A primary area of focus for the modern analysis group is Geometric Analysis. This area includes research in geometric function theory on Riemann surfaces, quasiconformal mappings in R^n and in metric spaces, geometry of domains in metric spaces and connections to Gromov hyperbolicity, analysis and potential theory on metric spaces, geometric measure theory, and generalization of Riemannian and conformal structures, as well as hyperbolic and quasihyperbolic geometry.
The group's research has been funded by the National Science Foundation. The group holds regular weekly research seminars.
- David Herron: geometric function theory, quasi-conformal mappings, hyperbolic geometry, potential theory
- Andrew Lorent: geometric measure theory, calculus of variations
- Nages Shanmugalingam: analysis on metric spaces, potential theory, geometric function theory
- Leonid Slavin: Euclidean harmonic analysis; quasi-conformal mappings
- Gareth Speight: geometric measure theory, analysis on metric spaces, sub-Riemannian geometry
Faculty members in Geometric Analysis have collaborations with:
- Helsinki University of Technology, Finland
- University of Jyväskylä, Finland
- Universidad Complutense de Madrid, Spain
- University of Ferrara, Italy
- University of Cambridge, England
- Steklov Mathematical Institute, Russia
- St. Petersburg State University, Russia
Partial Differential Equations
Research interests in the PDE group include: elliptic, parabolic and hyperbolic equations and coupled systems; boundary value problems for nonlinear equations and systems; control theory and optimization; nonlinear dispersive and traveling waves; obstacle problems; harmonic and geometric analysis. Methods involve: bifurcations; fixed point indices; semigroups; upper-lower solutions; radial symmetry and many others. Applications concern: biological and ecological models; diffusion-reactions; computational finance; fluid and plasma dynamics; fission reactor dynamics; water and electromagnetic waves etc. The group is very active in current publications. In the past, members of the group had been supported by National Science Foundation, Department of Energy and Air Force. A. Leung and S. Stojanovic have written reference books in their specialties. P. Korman and B. Zhang are editors of several research journals.
- Michael Goldberg: dispersive equations and harmonic analysis
- Phil Korman: elliptic partial differential equations
- Srdjan Stojanovic: computational finance and nonlinear partial differential equations
- Bing-Yu Zhang: control theory, non-linear waves, harmonic analysis
- Robert Buckingham: nonlinear wave equations
Dynamical Systems & Ordinary Differential Equations
Dynamical systems is the study of processes that evolve in time; it includes "chaos theory," Newtonian mechanics, and the modern theory of ordinary differential equations (ODEs). The Mathematical Sciences Department at UC has a long history of excellence in dynamical systems, particularly in the areas of celestial mechanics and bifurcation theory. Present areas of research include: celestial mechanics, smooth foliations, bifurcations, averaging methods for ODEs, Hamiltonian mechanics, fixed point theory, applications of ODE techniques to PDEs and vice-versa, and global invariants and integrability of ODEs.
UC's dynamics group is also involved in a number of collaborative projects in other disciplines and with researchers at other institutions. These range from problems of control and optimization (UC College of Engineering and Georgia Tech), to computer-assisted computation (UC College of Engineering and Univ. of Pamplona, Spain), beam stability in particle accelerators (Univ. of New Mexico, Cornell Univ. and DESY-Hamburg), and problems in the kinetic theory of multi-particle systems (Univ. of Paris).
- Phil Korman: differential equations