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Graduate Study in Differential Equations and Applied Mathematics
Introduction
The name "Applied Mathematics" covers a wide range of mathematical studies overlapping with many areas of mathematics.Dynamical Systems
Dynamics is the study of processes that evolve in time. A differentiable dynamical system arises as the differentiable action of a Lie group on a manifold. Here a Lie group C acts on a manifold X as a homomorphism of C into the group of diffeomorphisms of X. In case C = R, the real numbers, the map R x X -> X induced by this action is called a flow. Differentiable flows define vector fields (as ordinary differential equations). Conversely, solutions of an ordinary differential equation may generate a (global) flow. In the study of dynamical systems as flows, one uses the qualitative theory of differential equations and differential topology. Discrete dynamical systems arise when C = Z, the integers. Here the iterates of a diffeomorphism of X define the orbit of a point x in X and one studies the orbit structure.Problems of research interest to the faculty here at Illinois include the study of particular dynamical systems which arise in control theory, celestial mechanics and nonlinear phenomena in physics. The study of differential equations by dynamical systems methods holds particular interest.
Ordinary Differential Equations
Many physical processes can be described by ordinary differential equations, and, therefore, this subject is important to the study of applied mathematics, as well as being an area of pure mathematics. The study of ordinary differential equations has a long history involving such names as Cauchy, Poincare, and Hilbert, and research in this area continues actively today. Some of the important aspects of the area include:- Existence and uniqueness of solutions satisfying certain conditions.
- Properties of solutions (e.g., How fast or slowly do they grow? How many zeros do they have? What do they look like?)
- Stability theory (e.g., How do small changes in a problem affect the solution?)
- Boundary-value problems, orthogonal functions, and spectral theory.
Partial Differential Equations
Partial differential equations arise as models of physical and biological processes. The language of nature in classical physics and in parts of quantum mechanics is partial differential equations. Despite its long history there was no "theory" of partial differential equations until around 1950 with the introduction of a new language in which to talk about differentiation: the theory of distributions or generalized functions.Since then there has been incredible development in the theory of both linear and nonlinear partial differential equations. The interaction of pde's with differential geometry and topology is important and has also been developed. There is continuing research in all areas.
Some of the problems one solves are:
- existence and uniqueness of solutions
- wellposedness - continuous dependence of solutions on the data and properties of solutions
- inverse problems - determination of coefficients from "experimental" results
- relation of differential equations to physics and geometry
- numerical and asymptotic solutions.
Optimization
An optimization problem involves finding some extremum of a function subject to various side conditions or constraints. In recent decades whole areas, such as calculus of variations, nonlinear programming, flows in networks, discrete optimization, game theory, and still others, have developed to focus study on restricted classes of such problems. For a given type of problem major tasks include uncovering theoretical properties, such as the existence, characterization and stability of solutions, and also developing effective solution methodologies. These activities usually involve significant interaction with one or more other branches or areas of mathematics, such as numerical linear algebra, numerical analysis, convex analysis, nonsmooth analysis, nonlinear analysis, functional analysis, probability, differential equations, graph theory, matroid theory, and combinatorics. More recently, applications are demanding results for various problems of less classical character. Features requiring attention now include absence of linearity, differentiability, single - valuedness, and/or continuity. Modern optimization is increasingly devoted to developing our understanding of how to deal with these more realistic but highly challenging features.Courses
- Math 550 (formerly 443). Ordinary Differential Equations.
- Math 551 (formerly 467). Dynamical Systems Theory. Course is an introduction to the study of dynamical systems. Students who intend to do research in nonlinear dynamics are encouraged to take this course. Specific topics will be chosen to illustrate the theory and use of techniques from global analysis and nonlinear dynamics such as (1) discrete dynamical systems, (2) global theory of ordinary differential equations, (3) Hamiltonian systems, (4) KAM theory, (5) bifurcation and stability, (6) Hopf index theory of vector fields, (7) Morse theory of gradient vector fields, (8) Lyapunov theory, (9) infinite dimensional dynamical systems, (10) structural stability.
- Math 553 (formerly 444). Partial Differential Equations. Basic introduction to the study of partial differential equations; topics include: the Cauchy problem, power-series methods, characteristics, classification, canonical forms, well-posed problems, Riemann's method for hyperbolic equations, the Goursat problem, the wave equation, Sturm-Liouville problems and separation of variables, Fourier series, the heat equation, integral transforms, Laplace's equation, harmonic functions, potential theory, the Dirichlet and Neumann problems, and Green's functions.
- Math 554 (formerly 495). Linear Analysis and Partial Differential Equations.Course will provide students with the basic background in linear analysis associated with partial differential equations. The specific topics chosen will be largely up to the instructor, but will cover such areas as linear partial differential operators, distribution theory and test functions, Fourier transforms, Sobolev spaces, pseudodifferential operators, microlocal analysis, and applications of the above topics.
- Math 556 (formerly 455). Methods of Math Physics I.Course covers several basic mathematical methods of wide use in physics and engineering. Topics will be selected from the following: calculus of variations, Sturm-Liouville theory and eigenvalue problems, Green's functions and generalized functions, Hilbert space techniques.
- Math 579 (formerly 476). Coding Theory. Algebraic and combinatorial construction of efficient error-correcting codes. Description of important decoding algorithms. Codes considered include: Hamming, Reed-Solomon, BCH, Goppa, Golay, Fire, and Convolutional.
- Math 587 (formerly 480). Optimization by Vector Methods. Introduction to normed, Banach and Hilbert spaces; applications of the projection theorem and the Hahn-Banach theorem to problems of minimum norm, least squares estimation, mathematical programming, and optimal control; the Kuhn-Tucker theorem and Pontyragin's maximum principle; and introduction to iterative methods.
- Math 588 (formerly 483). Optimization in Networks. Theory and methods for optimization over directed graphs; path, cuts, flows, and potentials; matching; PERT and CPM; max flow, min path, out-of-kilter, Hungarian, and other algorithms; nonlinear cost functionals; painting theory; and existence and duality
- Math 589 (formerly 484). Conjugate Duality and Optimization. Convex analysis for constrained extremum problems; convex sets, cones, and functions; separation; Fenchel transform; duality correspondences; subdifferential theory; nonlinear programming; sensitivity; and perturbational duality for primal, dual, and Lagrangian problems.
- Math 595. As of Fall 2004, topics courses are sections of
Math 595. Past topics courses include
- Ordinary Differential Equations: Introduction to current research in such areas as stability and asymptotic behavior of solutions, topological dynamics, numerical methods, and boundary value problems and spectral theory of differential operators.
- Analysis
- Combinatorics: Selected topics from graph theory, algebraic coding theory, enumerative analysis, combinatorial design, discrete optimization;
- Optimization: Selected topics in variational inequalities, equilibrium, and complementarity; optimization and nonlinear analysis; nonlinear optimization algorithms; generalized gradients and optimization;
- Theory of Approximation: General approximation theory in normed linear spaces and use of approximations in computing
- Applied Mathematics: Deals with topics in the application of mathematics which are of current research interest. These have recently included topics in nonlinear partial differential equations, soliton theory, chaotic systems, nonlinear programming, game theory, mathematical economics.
| Department
of Mathematics 273 Altgeld Hall, MC-382 1409 W. Green Street, Urbana, IL 61801 USA Telephone: (217) 333-3350 Fax: (217) 333-9576 Email: office@math.uiuc.edu |
College of Liberal Arts and
Sciences Last modified June 18, 2007 |