Modern Analysis

 

Geometric Analysis

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, sub-Riemannian geometry, as well as hyperbolic and quasihyperbolic geometry.

The group's research has been funded by the National Science Foundation and the Simons Foundation. The group holds regular weekly research seminars.

Faculty:

Faculty members in Geometric Analysis have collaborations with:

  • Helsinki University of Technology, Finland
  • Linkoping University, Sweden
  • Polish Academy of Sciences (IM PAN)
  • Steklov Mathematical Institute, Russia
  • St. Petersburg State University, Russia
  • Universidad Complutense de Madrid, Spain
  • University of Cambridge, England
  • University of Ferrara, Italy
  • University of Jyväskylä, Finland
  • University of Trento, Italy
  • Okinawa Institute of Science and Technology Graduate University, Japan

Partial Differential Equations

Research interests in the PDE group include: nonlinear dispersive equations and traveling waves; integrable systems; elliptic, parabolic and hyperbolic equations and coupled systems; boundary value problems for nonlinear equations; harmonic and geometric analysis. These problems have a wide range of applications to water and electromagnetic wave propagation, physical, biological and ecological models, diffusion-reactions, and control theory and optimization. Current members of the group have research support from the National Science Foundation and Simons Foundation, and organize conferences with NSF support.

Faculty:

  • Bilman, Deniz: Nonlinear Waves, Integrable Systems, and Dispersive Partial Differential Equations
  • Buckingham, Robert: Nonlinear Wave Equations
  • Goldberg, Michael: Dispersive Equations and Harmonic Analysis
  • Korman, Phil: Elliptic Partial Differential Equations
  • Zhang, Bingyu: Control Theory of Partial Differential Equations, Nonlinear Dispersive Waves, and Applied Harmonic Analysis

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).

Faculty:

Operator Algebras

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.

The group's research has been funded by the Simons Foundation.

Faculty:

  • Peligrad, Costel: Operator Algebras, Harmonic Analysis, Groups and Quantum Groups Actions