# Research in mathematical analysis

Mathematical analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions. These theories are usually studied in the context of real and complex numbers and functions. *Source: Wikipedia*

## Professors with interests in analysis

## Research areas in mathematical analysis

### Complex analysis

The research interests of this group include the theories of both one and several complex variables. Of particular interest are boundary values and mapping properties of analytic functions of one complex variable; applications of functional analysis to complex function theory, such as multivariable operator theory and Hp-theory; and classification of singularities of complex spaces.

### Functional analysis

The research interests of the functional analysis group include abstract harmonic analysis on topological groups (multipliers, spaces of functions, Banach algebra structures, ergodic theory, analysis on topological groups); operator algebras (C*-algebras, analytic flows, connections with K-theory and algebraic topology); Banach spaces (classification up to isomorphism, operator spaces, probability theory on Banach spaces); and spaces of analytic functions (Hp spaces, rational approximation on various domains).

### Harmonic analysis and geometric analysis

The research interests of this group include harmonic analysis and their applications to linear and nonlinear partial differential equations, geometric analysis on Riemannian manifolds, and probability theory.

Among them are: covering theorems, normed inequalities for singular and fractional integrals, Hardy space theory; geometric inequalities such as Poincare and Sobolev inequalities in Euclidean space, on nilpotent Lie groups and Riemannian manifolds; group and measure algebras and algebras of completely bounded multilinear maps, unitary representations of various classes of locally compact groups; analysis on the Heisenberg and nilpotent Lie groups and generalizations to metric spaces; sharp constants for geometric inequalities (e.g. Moser-Trudinger and Sobolev inequalities) and their applications to conformal geometry; subelliptic partial differential equations on Carnot-Carathedory spaces; and nonlinear partial differential equations (e.g., the Monge-Ampere equation and the prescribing curvature problems on manifolds) and geometric measure theory (e.g. the Alexandrov-Fenchel inequalities for convex bodies).

### Real analysis

Research by this group deals with such subjects as measure theory (geometric measure theory and the Plateau problem, surface area theory, and abstract Weierstrass integrals involved in the calculus of variations); Fourier analysis; integral geometry; and functions of one and several variables in the classical vein.