Analysis

Analysis is a broad branch of mathematics that encompasses many fields, generally sharing a basis in calculus. In our Department, research in analysis includes a wide range of fields and subjects, from differential equations to ergodic theory.

Within the field of differential equations, the primary expertise is in partial differential equations (which involve functions of several variables) and their applications to other fields of mathematics and other disciplines. The research of Maxim Braverman includes work on the index theory and determinants of elliptic operators; he has also done interdisciplinary work in superconductivity and hydrodynamics. Chris King works on problems in mathematical physics using a variety of methods, including matrix analysis and convex analysis. Robert McOwen studies elliptic operators on singular and noncompact manifolds, with applications to conformal metrics in differential geometry. Martin Schwarz works in nonlinear analysis: Liouville Theorem for partial differential equations that are completely integrable, Maxwell Higgs, and other nonlinear problems from physical science. Mikhail Shubin works in various areas of analysis (linear and non-linear partial differential equations, geometric analysis, spectral theory of elliptic operators, especially Schrödinger operators), non-commutative geometry and its applications to geometry and topology, mathematical physics and differential geometry. And Peter Topalov studies Hamiltonian partial and ordinary differential equations, dynamical systems, and Riemannian and symplectic geometry.

A delay equation is a version of an ordinary differential equation in which the rate of change of the process described by the equation at any time is allowed to depend on the behavior of the process at earlier times. Solomon Jekel uses topological methods to find closed orbits for dynamical systems on spheres in order to obtain periodic solutions to delay equations.

Ergodic theory is the study of measure-preserving transformations, such as the different ways of mixing two fluids (e.g. gin and vermouth). Stanley Eigen currently works in ergodic theory and some of its connection with tilings and group theory.

Last modified October, 2009