Applied and Computational Math

Academic Year 2023 – 2024

General Information

Address
Fine Hall, Washington Road
Phone

Program Offerings:

  • Ph.D.

Department for program:

Director of Graduate Studies:

Graduate Program Administrator:

Overview

The Program in Applied and Computational Mathematics offers a select group of highly qualified students the opportunity to obtain a thorough knowledge of branches of mathematics indispensable to science and engineering applications, including numerical analysis and other computational methods.

Apply

Application deadline
December 15, 11:59 p.m. Eastern Standard Time (This deadline is for applications for enrollment beginning in fall 2024)
Program length
4 years
Fee
$75
GRE
General Test - optional/not required; subject tests in Mathematics, Physics, or a related field - optional/not required.

Program Offerings

Program Offering: Ph.D.

Courses

Students enroll in courses based on research topics that they choose in consultation with faculty.  Typically, students take regular or reading courses with their advisers in the three topic areas of their choice, completing the regular exams and course work for these courses.  

Students must choose three areas out of a list of the following six topic areas in which to take courses and be examined:

  • Asymptotics, analysis, numerical analysis, and signal processing;
  • Discrete mathematics, combinatorics, algorithms, computational geometry, and graphics;
  • Mechanics and field theories (including computational physics/chemistry/biology);
  • Optimization (including linear and nonlinear programming and control theory);
  • Partial differential equations and ordinary differential equations (including dynamical systems);  
  • Stochastic modeling, probability, statistics, and information theory

Additional topics may be considered with prior approval by the director of graduate studies.

The student should choose specific topics by the end of October. In consultation with the student, the director of graduate studies appoints a set of advisers from among the faculty and associated faculty. The adviser in each topic meets regularly with the student, monitors progress, and assigns additional reading material. Advisers are usually program faculty or associated faculty, but faculty members from other departments may serve as advisers with approval.

Additional pre-generals requirements

At the end of the first year, students will take a preliminary exam, consisting of a joint interview by their three first-year topic advisers. Each student should decide with their first-year advisers which courses are relevant for the examination areas.

Students should assess their level of preparation for the preliminary examination by reviewing homework and examinations from the previous year’s work.  Students who fail the preliminary examination may take the examination a second time with the support of the first-year advisers.

General exam

Before being admitted to the third year of study, students must pass the general examination. The general examination, or generals, is designed as a sequence of interviews with assigned professors covering three applied mathematics areas. The generals culminate in a seminar on a research topic, usually delivered toward the end of the fourth term.  A student who completes all program requirements (coursework, preliminary exams, with no incompletes) but fails the general examination may take it a second time. Students who fail the general examination a second time will have their degree candidacy terminated.

Qualifying for the M.A.

The Master of Arts degree is normally an incidental degree on the way to full Ph.D. candidacy but may also be awarded to students who, for various reasons, leave the Ph.D. program. Students who have successfully completed all courses undertaken during their graduate study, have satisfactorily resolved all incompletes (if any), and have passed the preliminary exam may be awarded an M.A. degree.  Upon learning the program’s determination of their candidacy to receive the M.A., students apply for the master's degree online through the advanced degree application system. 

Dissertation and FPO

The doctoral dissertation must consist of either a mathematical contribution to some field of science or engineering or the development or analysis of mathematical or computational methods useful for, inspired by, or relevant to science or engineering.

The Ph.D. is awarded after the candidate’s doctoral dissertation has been accepted and the final public oral examination sustained.

Faculty

  • Director

    • Peter Constantin
  • Director of Graduate Studies

    • Maria Chudnovsky
  • Director of Undergraduate Program

    • Ramon van Handel
  • Executive Committee

    • Noga M. Alon, Mathematics
    • René A. Carmona, Oper Res and Financial Eng
    • Emily Ann Carter, Mechanical & Aerospace Eng
    • Maria Chudnovsky, Mathematics
    • Peter Constantin, Mathematics
    • Paul Seymour, Mathematics
    • Amit Singer, Mathematics
    • Howard A. Stone, Mechanical & Aerospace Eng
    • Romain Teyssier, Astrophysical Sciences
    • Jeroen Tromp, Geosciences
    • Ramon van Handel, Oper Res and Financial Eng
  • Associated Faculty

    • Ryan P. Adams, Computer Science
    • Amir Ali Ahmadi, Oper Res and Financial Eng
    • Michael Aizenman, Physics
    • Yacine Aït-Sahalia, Economics
    • William Bialek, Physics
    • Mark Braverman, Computer Science
    • Carlos D. Brody, Princeton Neuroscience Inst
    • Adam S. Burrows, Astrophysical Sciences
    • Roberto Car, Chemistry
    • Bernard Chazelle, Computer Science
    • Jianqing Fan, Oper Res and Financial Eng
    • Jason W. Fleischer, Electrical & Comp Engineering
    • Mikko P. Haataja, Mechanical & Aerospace Eng
    • Gregory W. Hammett, PPPL Theory
    • Isaac M. Held, Atmospheric & Oceanic Sciences
    • Sergiu Klainerman, Mathematics
    • Naomi E. Leonard, Mechanical & Aerospace Eng
    • Simon A. Levin, Ecology & Evolutionary Biology
    • Luigi Martinelli, Mechanical & Aerospace Eng
    • William A. Massey, Oper Res and Financial Eng
    • Assaf Naor, Mathematics
    • H. Vincent Poor, Electrical & Comp Engineering
    • Frans Pretorius, Physics
    • Herschel A. Rabitz, Chemistry
    • Peter J. Ramadge, Electrical & Comp Engineering
    • Jennifer Rexford, Computer Science
    • Clarence W. Rowley, Mechanical & Aerospace Eng
    • Szymon M. Rusinkiewicz, Computer Science
    • Mykhaylo Shkolnikov, Oper Res and Financial Eng
    • Frederik J. Simons, Geosciences
    • Jaswinder P. Singh, Computer Science
    • Ronnie Sircar, Oper Res and Financial Eng
    • Mete Soner, Oper Res and Financial Eng
    • John D. Storey, Integrative Genomics
    • Sankaran Sundaresan, Chemical and Biological Eng
    • Robert E. Tarjan, Computer Science
    • Corina E. Tarnita, Ecology & Evolutionary Biology
    • Salvatore Torquato, Chemistry
    • Olga G. Troyanskaya, Computer Science
    • Robert J. Vanderbei, Oper Res and Financial Eng
  • Professor

    • Noga M. Alon
    • Maria Chudnovsky
    • Peter Constantin
    • Amit Singer
    • Romain Teyssier
    • Jeroen Tromp
  • Associate Professor

    • Ramon van Handel
  • Lecturer

    • Henry F. Schreiner
  • Visiting Professor

    • Ehud Yariv

For a full list of faculty members and fellows please visit the department or program website.

Permanent Courses

Courses listed below are graduate-level courses that have been approved by the program’s faculty as well as the Curriculum Subcommittee of the Faculty Committee on the Graduate School as permanent course offerings. Permanent courses may be offered by the department or program on an ongoing basis, depending on curricular needs, scheduling requirements, and student interest. Not listed below are undergraduate courses and one-time-only graduate courses, which may be found for a specific term through the Registrar’s website. Also not listed are graduate-level independent reading and research courses, which may be approved by the Graduate School for individual students.

AOS 576 - Current Topics in Dynamic Meteorology (also APC 576)

An introduction to topics of current interest in the dynamics of large-scale atmospheric flow. Possible topics include wave-mean flow interaction and nonacceleration theorems, critical levels, quasigeostrophic instabilities, topographically and thermally forced stationary waves, theories for stratospheric sudden warmings and the quasi-biennial oscillation of the equatorial stratosphere, and quasi-geostrophic turbulence.

APC 503 - Analytical Techniques in Differential Equations (also AST 557)

Local analysis of solutions to linear and nonlinear differential and difference equations. Asymptotic methods, asymptotic analysis of integrals, perturbation theory, summation methods, boundary layer theory, WKB theory, and multiple scale theory. Prerequisite: MAE 306 or equivalent.

APC 523 - Numerical Algorithms for Scientific Computing (also AST 523/CSE 523/MAE 507)

A broad introduction to scientific computation using examples drawn from astrophysics. From computer science, practical topics including processor architecture, parallel systems, structured programming, and scientific visualization will be presented in tutorial style. Basic principles of numerical analysis, including sources of error, stability, and convergence of algorithms. The theory and implementation of techniques for linear and nonlinear systems of equations, ordinary and partial differential equations will be demonstrated with problems in stellar structure and evolution, stellar and galactic dynamics, and cosmology.

APC 524 - Software Engineering for Scientific Computing (also AST 506/CSE 524/MAE 506)

The goal of this course is to teach basic tools and principles of writing good code, in the context of scientific computing. Specific topics include an overview of relevant compiled and interpreted languages, build tools and source managers, design patterns, design of interfaces, debugging and testing, profiling and improving performance, portability, and an introduction to parallel computing in both shared memory and distributed memory environments. The focus is on writing code that is easy to maintain and share with others. Students will develop these skills through a series of programming assignments and a group project.

APC 599 - Summer Extramural Research Project

A summer research project, designed in conjunction with the student's advisor, APC, and an industrial, NGO, or government sponsor, that will provide practical experience relevant to the student's research area. Start date no earlier than June 1; end date no later than Labor Day. A final paper and sponsor evaluation is required.

AST 559 - Turbulence and Nonlinear Processes in Fluids and Plasmas (also APC 539)

A comprehensive introduction to the theory of nonlinear phenomena in fluids and plasmas, with an emphasis on turbulence and transport. Experimental phenomenology; fundamental equations, including Navier-Stokes, Vlasov, and gyrokinetic; numerical simulation techniques, including pseudo-spectral and particle-in-cell methods; coherent structures; transition to turbulence; statistical closures, including the wave kinetic equation and direct-interaction approximation; PDF methods and intermittency; variational techiques. Applications from neutral fluids, fusion plasmas, and astrophysics.

CBE 502 - Mathematical Methods of Engineering Analysis II (also APC 502)

Linear ordinary differential equations (systems of first-order equations, method of Frobenius, two-point boundary-value problems); spectrum and Green's function; matched asymptotic expansions; partial differential equations (classification, characteristics, uniqueness, separation of variables, transform methods, similarity); and Green's function for the Poisson, heat, and wave equations, with applications to selected problems in chemical, civil, and mechanical engineering.

CBE 554 - Topics in Computational Nonlinear Dynamics (also APC 544)

The numerical solution of partial differential equations (finite element and spectral methods); computational linear algebra; direct and interactive solutions and continuation methods; and stability of the steady states and eigen problems. Time-dependent solutions for large systems of ODEs; computation and stability analysis of limit cycles; Lyapunov exponents of chaotic solutions are explored. Vectorization and FORTRAN code optimization for supercomputers as well as elements of symbolic computation are studied.

MAE 501 - Mathematical Methods of Engineering Analysis I (also APC 501/CBE 509)

Methods of mathematical analysis for the solution of problems in physics and engineering. Topics include an introduction to functional analysis, Sturm-Liouville theory, Green's functions for the solution of ordinary differential equations and Poisson's equation, and the calculus of variations.

MAE 502 - Mathematical Methods of Engineering Analysis II (also APC 506)

A complementary presentation of theory, analytical methods, and numerical methods. The objective is to impart a set of capabilities commonly used in the research areas represented in the Department. Standard computational packages will be made available in the courses, and assignments will be designed to use them. An extension of MAE 501.

MAE 541 - Applied Dynamical Systems (also APC 571)

Phase-plane methods and single-degree-of-freedom nonlinear oscillators; invariant manifolds, local and global analysis, structural stability and bifurcation, center manifolds, and normal forms; averaging and perturbation methods, forced oscillations, homoclinic orbits, and chaos; and Melnikov's method, the Smale horseshoe, symbolic dynamics, and strange attractors. Offered in alternate years.

MAT 522 - Introduction to PDE (also APC 522)

The course is a basic introductory graduate course in partial differential equations. Topics include: Laplacian, properties of harmonic functions, boundary value problems, wave equation, heat equation, Schrodinger equation, hyperbolic conservation laws, Hamilton-Jacobi equations, Fokker-Planck equations, basic function spaces and inequalities, regularity theory for second order elliptic linear PDE, De Giorgi method, basic harmonic analysis methods, linear evolution equations, existence, uniqueness and regularity results for classes of nonlinear PDE with applications to equations of nonlinear and statistical physics.

MAT 572 - Topics in Combinatorial Optimization (also APC 572)

This course covers current topics in combinatorial optimization. More specific topic details are provided when the course is offered.

MAT 585 - Mathematical Analysis of Massive Data Sets (also APC 520)

This course focuses on spectral methods useful in the analysis of big data sets. Spectral methods involve the construction of matrices (or linear operators) directly from the data and the computation of a few leading eigenvectors and eigenvalues for information extraction. Examples include the singular value decomposition and the closely related principal component analysis; the PageRank algorithm of Google for ranking web sites; and spectral clustering methods that use eigenvectors of the graph Laplacian.

MAT 586 - Computational Methods in Cryo-Electron Microscopy (also APC 511/MOL 511/QCB 513)

This course focuses on computational methods in cryo-EM, including three-dimensional ab-initio modelling, structure refinement, resolving structural variability of heterogeneous populations, particle picking, model validation, and resolution determination. Special emphasis is given to methods that play a significant role in many other data science applications. These comprise of key elements of statistical inference, image processing, and linear and non-linear dimensionality reduction. The software packages RELION and ASPIRE are routinely used for class demonstration on both simulated and publicly available experimental datasets.

MSE 515 - Random Heterogeneous Materials (also APC 515/CHM 559)

Foams, composites, porous media, and biological media are all examples of random heterogeneous materials. The relationship between the macroscopic (transport, mechanical, electromagnetic and chemical) properties and microstructure of random media is formulated. Topics include correlation functions; percolation theory; fractal concepts; sphere packings; Monte Carlo techniques; and image analysis; homogenization theory; effective-medium theories; cluster and perturbation expansions; variational bounding techniques; topology optimization methods; and cross-property relations. Biological and cosmological applications will be discussed.

ORF 550 - Topics in Probability (also APC 550)

An introduction to nonasymptotic methods for the study of random structures in high dimension that arise in probability, statistics, computer science, and mathematics. Emphasis is on developing a common set of tools that has proved to be useful in different areas. Topics may include: concentration of measure; functional, transportation cost, martingale inequalities; isoperimetry; Markov semigroups, mixing times, random fields; hypercontractivity; thresholds and influences; Stein's method; suprema of random processes; Gaussian and Rademacher inequalities; generic chaining; entropy and combinatorial dimensions; selected applications.