# Analysis

Functions are representations of relations between sets, and in particular are useful for representing the changing states of a system: the velocity of a projectile, the frequencies present in a sound signal, the color of a pixel in a digital image, or the prices of a portfolio of stocks. The mathematical field of analysis seeks to formulate methods to analyze quantitatively the change exhibited by the outputs of functions with respect to their inputs, as a way of distilling important information about the underlying systems---such as the way stock prices change over time.

Historically, the development of calculus, with its application to Newtonian physics, was a very successful development in analysis. Today, analysis forms the foundation of several highly active areas of mathematics, with powerful applications throughout mathematics as well as in the natural sciences and industry. The Duke math department includes several faculty who work on various topics within analysis or use the tools of analysis to study various applications.

Harmonic analysis seeks to decompose functions into their component "harmonics" or waves; that such a decomposition is possible for broad classes of functions was initially discovered by Fourier in the early 1800's, and continues to have applications in a vast swath of mathematics, ranging from image compression methods via "wavelets" to counting integral solutions to Diophantine equations via the Hardy-Littlewood "circle method."

Complex analysis studies the behavior of functions on the complex plane, and has deep connections to number theory, and in particular to the distribution of prime numbers, via the Riemann zeta function.

Real analysis develops a rigorous theory of integration which extends the familiar notions of calculus to a broader class of functions, and in particular provides a foundation for many concepts in probability.

Stochastic analysis seeks to understand the behavior of differential equations and dynamical systems whose parameters or inputs are random and unpredictable; this has many applications in the sciences and engineering and economics.

Applying analytic methods to partial differential equations (PDE's) allows researchers to study the evolution of a system that is changing in a manner governed by precise constraints (the differential equations in the name); PDE's may be used to represent many important problems in the physical world, relating for example to diffusion of heat, fluid flow, or quantum mechanics.

## Faculty

#### William K. Allard

###### Professor Emeritus of Mathematics

Keywords in this area

Geometric Measure Theory
Multiresolution Geometrical Analysis
Image Processing

#### J. Thomas Beale

###### Professor Emeritus of Mathematics

Keywords in this area

computation of singular and nearly singular integrals, motion of fluid interfaces, equations of incompressible flow, maximum norm estimates for finite difference methods, convergence of numerical methods for fluid flow

#### Robert Bryant

###### Phillip Griffiths Professor of Mathematics

Keywords in this area

differential geometry, exterior differential systems, complex geometry

#### Robert Calderbank

###### Charles S. Sydnor Distinguished Professor of Computer Science

Keywords in this area

detection and estimation, discrete harmonic analysis

#### Ingrid Daubechies

###### James B. Duke Distinguished Professor of Mathematics and Electrical and Computer Engineering

Keywords in this area

wavelets, inverse problems

#### Jayce Robert Getz

###### Associate Professor of Mathematics

Keywords in this area

Automorphic representations, Trace formulae

#### Heekyoung Hahn

###### Assistant Research Professor of Mathematics

Keywords in this area

Laplacian eigenvalues and relative Weyl law

###### Professor of Mathematics

Keywords in this area

Geometric measure theory, especially regularity theory and singularities of calibrated currents. Nonlinear systems of elliptic PDEs. Analysis on noncompact complete spaces and on incomplete or singular spaces.

#### Gregory Joseph Herschlag

###### Assistant Research Professor of Mathematics

Keywords in this area

fluids flow across dynamic channels

#### Alexander A. Kiselev

###### William T. Laprade Distinguished Professor of Mathematics

Keywords in this area

Fourier analysis, functional analysis

#### Jianfeng Lu

###### Associate Professor of Mathematics

Keywords in this area

electronic structure models, calculus of variations, semiclassical analysis

#### Jonathan Christopher Mattingly

###### James B. Duke Distinguished Professor

Keywords in this area

Stochastic Analysis, Malliavin Calculus, Ergodic Theory

#### James H. Nolen

###### Associate Professor of Mathematics

Keywords in this area

partial differential equations, probability, asymptotic analysis, homogenization

#### William L. Pardon

###### Professor of Mathematics

Keywords in this area

Singular spaces

#### Lillian Beatrix Pierce

###### Nicholas J. and Theresa M. Leonardy Associate Professor

Keywords in this area

oscillatory integrals, Carleson operators, discrete operators

Other research areas
Analysis Number Theory

#### Mark A. Stern

###### Professor of Mathematics

Keywords in this area

geometric analysis, elliptic partial differential equations

#### Thomas P. Witelski

###### Professor in the Department of Mathematics

Keywords in this area

perturbation methods

#### Hau-Tieng Wu

###### Associate Professor of Mathematics

Keywords in this area

harmonic analysis, wavelet analysis, time-frequency analysis