Partial Differential Equations (PDEs)
PDEs, created to describe the mechanical behavior of objects such as vibrating strings and blowing winds, has developed into a subject that interacts with many branches of mathematics, such as differential geometry, complex analysis, and representation theory. Modern techniques in the analysis of PDEs include harmonic analysis, functional analysis, microlocal analysis, topological methods, and symplectic geometry, to name a few. Members of our PDE group are active in studies of linear and nonlinear wave motion, spectral theory and scattering theory, analysis on singular spaces, and inverse problems.
Starting with the classical connections between exponential and trigonometric functions, complex analysis has enriched function theory with the extra structure inherent in extension of functions to the complex domain. analytic continuation has similarly enriched harmonic analysis, representation theory, and differential geometry. analysts in our department do research in several complex variables, almost complex structures, iteration of meromorphic functions, distortion theorems of holomorphic curves, composition operators, and numerical conformal mappings.
Dynamical Systems & Ergodic Theory
Dynamical systems are systems in motion, often governed by differential equations, but also perhaps by other continuous or discrete formulae. Ergodic theory is the study of statistical properties of that motion. Originally created to connect thermodynamics to statistical mechanics, it has been extended to connect with many branches of mathematics, including differential geometry and number theory. Members of our ergodic theory group do research on complex dynamics, nonsingular and measure preserving ergodic theory, symbolic dynamics, tiling dynamical systems, ergodic theorems, return times theorems, multiple recurrence, nonconventional ergodic averages, connections with analysis and probability theory, Lie group actions, and foliations.
Harmonic analysis & Operator Theory
Analysis of functions into simpler components, such as complex exponentials, wavelets, etc., and synthesis of operators on function spaces, form two parts of the modern functional analytic approach. Members of our department do research on topics including connections with singular integral operators and PDE, eigenfunction expansions and convergence questions, noncommutative harmonic analysis, operator theory on Hilbert spaces, subnormal and composition operators, operator algebras, operator K-theory, and the ergodic theory of operators.