# Ph.D. in Mathematics, Specializing in Applied Math

### Table of Contents

- Overview of Applied Mathematics at the Courant Institute
- PhD Study in Applied Mathematics
- Core Courses
- Regular Courses
- Topics Courses
- Courses for Breadth
- Sample Programs
- Faculty

### Overview of Applied Mathematics at the Courant Institute

Applied mathematics has long had a central role at the Courant Institute, and roughly half of all our PhD's in Mathematics are in some applied field. There are a large number of applied fields that are the subject of research. These include:

- Atmosphere and Ocean Science
- Biology, including biophysics, biological fluid dynamics, theoretical neuroscience, physiology, cellular biomechanics
- Computational Science, including computational fluid dynamics, adaptive mesh algorithms, analysis-based fast methods, computational electromagnetics, optimization, methods for stochastic systems.
- Data Science
- Financial Mathematics
- Fluid Dynamics, including geophysical flows, biophysical flows, fluid-structure interactions, complex fluids.
- Materials Science, including micromagnetics, surface growth, variational methods,
- Stochastic Processes, including statistical mechanics, Monte-Carlo methods, rare events, molecular dynamics

### PhD study in Applied Mathematics

PhD training in applied mathematics at Courant focuses on a broad and deep mathematical background, techniques of applied mathematics, computational methods and specific application areas.

There are three levels of classes. Introductory or core classes are offered every year and should be of interest to most students heading toward applied areas. More specialized classes are offered regularly every other year. A typical student would choose among these depending on her or his interests. Advanced topics classes discuss areas of current research interest to specific faculty. These change from year to year.

The sample programs below illustrate some ways in which the students can structure their study. It is recommended that students with applied interests take most core classes by the end of their first year, as allowed by other interests or constraints. Beyond that, students will begin to choose regular and topics classes depending on their developing interests.

### Core Courses, offered every year

These classes should be of general interest to students of applied mathematics. We recommend that students interested in any applied area take all or most of these during their first year.

*MATH-GA 2702-001 PDE I**MATH-GA 2702-001 Fluid Mechanics**MATH-GA 2010-001 Numerical Methods I**MATH-GA 2704-001 Applied Stochastic Processes**MATH-GA 2902-001 Stochastic Calculus**MATH-GA 2701-001 Methods of Applied Mathematics**MATH-GA 2020-001 Numerical Methods II**MATH-GA 2111-001 Linear Algebra (one term)*

### Regular Courses, offered every year or every other year

These classes should also be of general interest to students of applied mathematics, though they are more specialized than the core courses. Students interested in any applied area should choose among them depending on their interests, and take them during their first two years of study.

*MATH-GA 2012-002 Convex and Non-smooth Optimization**MATH-GA 2012-001 Finite Element Methods**MATH-GA 2012-001 Monte Carlo Methods**MATH-GA 2840-002 Data Analysis**MATH-GA 2710-001 Mechanics**MATH-GA 2830-004 Complex and Biological Fluids**MATH-GA 2852-001 Stochastic Problems in Biology**MATH-GA 2855-001 Mathematical Physiology**MATH-GA 2011-001 Approximation Theory and Practice**MATH-GA 2945-002 Spectral Methods for ODEs and PDEs**MATH-GA 3001-001 Geophysical Fluid Dynamics**MATH-GA 2012-002 Immersed Boundary Methods**MATH-GA 2902-001 Stochastic Calculus**MATH-GA 2012-003 High-performance Computing**MATH-GA 2862-001 Methods for fluid-solid interaction*

### Topics Courses

These classes change from year to year, and students take them depending on their interests and fields of study. The following are suggested elective courses in mathematics:

- Advanced Topics in Applied Mathematics. This has included advanced courses in signal processing, fluid dynamics, stochastic processes, data analysis, dynamics systems, and others.
- Advanced Topics in Numerical Analysis. This has included advanced courses in analysis-based fast methods, homogenization methods, computational methods for fluids and for electromagnetics, optimization, and finite-element methods.
- Advanced Topics in Mathematical Biology. This has included advanced courses in neuroscience, biophysics, physiology, population dynamics, and networks.
- Advanced Topics in Atmosphere-Ocean Science. This has included advanced courses in climate change, ice-sheet dynamics, geophysical experiment.
- Advanced Topics in Fluid Dynamics. This includes turbulence, waves, complex fluids, biological fluids, and others.

### Courses for Breadth

Throughout their studies, students are encouraged to take classes beyond those directly linked to their chosen areas of research. These may include core classes in mathematics and computer science, in applied fields different from their specialization, as well as classes from other departments outside of Courant.

### Sample Programs

The only absolute requirements in the PhD program in Mathematics are that the student pass the written and oral exams, take a minimum number of classes, and write a thesis. Incoming students should take the classes they need to prepare for the written exam, including Linear Algebra and/or Complex Variables as needed. Beyond that, students with applied interests should choose core applied mathematics classes. There are more core classes than are possible, so a student will have to postpone one or more of them until the second year. Several sample programs for the first two years of study are shown below:

#### Program I

Year I - Fall Term | Year I - Spring Term |
---|---|

Linear Algebra | Applied Stochastic Processes |

PDE I | Asymptotic Analysis |

Fluid Mechanics | Continuum mechanics |

Numerical Methods I | Numerical Methods II |

Year II - Fall Term | Year II - Spring Term |
---|---|

Neurophysiology and Neuronal Networks | Data Analysis |

Complex fluids | Mathematical Physiology |

Real Variables | Geophysical Fluid Dynamics |

Computational Fluid Dynamics | Nonlinear Optimization |

#### Program II

Year I - Fall Term | Year I - Spring Term |
---|---|

Linear Algebra | Stochastic Calculus |

PDE I | Methods of Applied Mathematics |

Complex Variables | PDE II |

Numerical Methods I | Numerical Methods II |

Year II - Fall Term | Year II - Spring Term |
---|---|

Approximation Theory and Practice | Advanced Topics Class |

Fluid Mechanics | Applied Stochastic Processes |

Ordinary Differential Equations | High Performance Computing |

Monte Carlo Methods | Nonlinear Optimization |

### Faculty

A list of the current research interests of individual faculty is available on the Math research page.