Concentration in Partial Differential Equations

A Joint Endeavor of the Department of Mathematics and the Department of Applied Mathematics and Statistics

Introduction

The Field of Partial Differential Equations (PDEs) is a major link between mathematics and its applications. It began some three hundred years ago as a methodology for quantitative models of various physical phenomena. It has since developed into a major and central part of mathematics and it now has applications to other parts of mathematics, to modern science, and to technology, including engineering, finance, medicine, and computer science. The advent of computers has made the field more important than ever because it facilitates the computation of numerical solutions to equations which before could be discussed only qualitatively.

Researchers at Stony Brook are active in many areas of PDEs.

Description

A combined program in PDEs is offered by the Department of Applied Mathematics and Statistics and the Department of Mathematics for the preparation of students for advanced studies in partial differential equations. The goal of the program is to prepare students during their first two years of graduate study to begin research in areas related to the theory and/or computation of solutions to systems of partial differential equations. The program consists of three basic components: preparatory courses in analysis, introductory courses in the theory of hyperbolic, elliptic, and parabolic partial differential equations, and introductory courses in numerical analysis and numerical solutions of partial differential equations. It also includes advanced courses as listed below. Courses taken in the first year are part of the departmental qualifying exam syllabus. Students who have interests related to but not precisely in partial differential equations may want to take some courses in this program, but not to complete it.

Basic Courses

Basic Theory of Partial Differential Equations:

  • AMS 502 (*) Hyperbolic Equations
  • MAT 546 Elliptic Equations

    • Numerical Methods:

  • AMS 526 (*) Numerical Analysis I
  • AMS 527 (*) Numerical Analysis II
  • AMS 528 (*) Computational Methods in Partial Differential Equations

    • Analysis:

  • AMS 503 (*) Applications of Complex Analysis
  • AMS 504 (*) Foundations of Applied Mathematics
  • MAT 544 (+) Analysis
  • MAT 542 (+) Complex Analysis I
  • MAT 550 (+) Real Analysis I
  • MAT 551 Real Analysis II

    • Advanced Courses

    Applications:

  • AMS 562 Numerical Hydrology
  • AMS 566 Compressible Fluid Dynamics
  • AMS 690 Special Topics in Differential Equations and Applied Analysis

    • Computation:

  • AMS 563 Computational Fluid Dynamics
  • AMS 621 Numerical Solutions of Partial Differential Equations
  • AMS 695 Special Topics in Numerical Analysis and Scientific Computing

    • Theory:

  • AMS 564 Systems of Hyperbolic Conservation Laws and Shock Waves
  • AMS 565 Wave Propagation I
  • MAT 543 Complex Analysis II
  • MAT 568-569 Differential Geometry
  • MAT 632-633 Topics in Partial Differential Equations
  • MAT 674-675 Advanced Topics in Partial Differential Equations
    • (*) Required course as part of the Ph.D. qualifying examinations syllabus for all students in the Department of Applied Mathematics and Statistics Ph.D. program.

    (+) Required course as part of the Ph.D. qualifying examinations syllabus for all students in the Department of Mathematics Ph.D. program.

    Recent Ph.D. Theses Related to PDE (Department of Mathematics)

    Name Thesis Title Ph.D. Year Adviser Position (upon leaving Stony Brook) Employer
    F. Lamontagne Critical Metrics for the L^2-norm of the Curvature Tensor 8/93 Anderson Postdoc. MSRI.
    G. Misiolek Stability of Flows of Ideal Fluids and the Geometry of the Group of Diffeomorphisms 12/92 Ebin Assistant Professor Notre Dame Univ.
    S. Zhou Singular Integral Operators, Contraction Operators and Principal Currents 8/92 Pincus Lecturer SUNY at Stony Brook
    D. Gong L^2 Analytic Torsions, Equivariant Cyclic Cohomology and the Novikov Conjecture 8/92 Pincus Dickson Instructor Univ. of Chicago
    S. Dragomir CR Maps between Strictly Pseudo-convex CR Manifolds, Interpolation Manifolds, and CR Foliations 8/92 Hill Researcher Univ. of Basilicata
    M. Lee Solutions and Hamiltonian Structure for Quasi-geostrophic Flow 8/91 Ebin Research Assistant Univ. of Liverpool
    J. Yu The Euler Equations of an Incompressible Fluid in a High-Dimensional Bounded Region 5/91 Ebin Pension Admin. WY Associates

    Recent Ph.D. Theses Related to PDE (Department of Applied Mathematics and Statistics)

    Name Thesis Title Ph.D. Year Adviser Position (upon leaving Stony Brook) Employer
    R. Holmes The Application of Front Tracking to the Simulation of the Richtmyer-Meshkov Instability 1994 Grove Postdoctoral Fellow (not available)
    S. Rosenthal Mathematical Methods for Realistic Neuron Modeling 1994 Tewarson (not available) (not available)
    D. Coker Exactly Solvable Models, QED, and Porous Media 1993 Lindquist Assistant Professor SUNYIT Utica/Rome
    K.-K. Chang Multi-length Scale Calculations of the Mixing Length Growth in Buckley-Leverett Flow 1993 Lindquist Postdoctoral Fellow Indiana Univ.
    D. Mirkovic Domain Decomposition Approach for a Mixed Finite Element Solution of Elliptic Problems 1993 Lindquist Assistant Professor Univ. of Iowa
    P. da Silva The Role of Surface Tension in Multiphase Flow Regimes 1992 Glimm Postdoctoral Fellow Brunel Univ.
    L. F. Pereira Stochastic Geology and Porous Media Flow: Theory and Simulations 1992 Glimm Postdoctoral Fellow Purdue Univ.
    S. Canic Shock Wave Admissibility for Quadratic Conservation Laws 1992 Plohr Assistant Professor Univ. of Iowa
    Q.-P. Xu The Global Structure of Scale-Invariant Solutions of the Riemann Problem for a Model of Three-Phase Flow in a Porous Medium 1992 Plohr (not available) (not available)
    J. Li Integral Equation Methods for Mixed Boundary Value Problems of Fracture Mechanics 1992 Srivastav (not available) Earth Science Div., Lawrence Berkeley Lab.
    S. Kim Quasi-Newton Algorithms for Solving Large Systems of Nonlinear Equations 1992 Tewarson Assistant Professor Iwai Univ.
    H. Wang Quasi-Gauss-Newton Methods for Solving Nonlinear Algebraic Equations 1992 Tewarson Systems Analyst American Inst. of Phys.
    F. Zhang On the Numerical Methods for Solving Singular Integral Equations 1990 Srivastav Associate Professor Antelope Valley College
    X. Chen Computer Determination of Interior Temperature Distribution from Surface Temperature Measurements in Microwave Processing of Materials 1994 Chen (not available) Landa Electronics, New York
    T-G. Kim Parallel Multigrid Algorithms for Inverse Problems of General 3D Elastic Wave Equation 1993 Chen Analyst Citibank Corp.
    T. Baus A Solution of the Electromagnetic Inverse Scattering Problem Utilizing the Generalized Pulse Spectrum Technique 1990 Chen Research Scientist Naval Underwater Systems, New London, CT
    J. Zhu On the Application of GPST Algorithm to History Matching Three Dimensional Multiphase Reservoir Models 1990 Chen Assistant Professor Mississippi State Univ.