# Ph.D. in Mathematical Sciences

## Degree Requirements

### Applied Mathematics Track (NJIT)

Students choosing the applied mathematics track must fulfill the requirements for the doctor of philosophy as specified in this catalog. Specific courses of study are planned in consultation with a faculty advisor and are subject to approval. In general, students are encouraged to take courses both in mathematics and in areas of application.

*Seminar:* In addition to the minimum degree credits required, all doctoral students must enroll each semester in MATH 791 Graduate Seminar.

*Courses:* A typical schedule of courses for the first four semesters in Applied Mathematics consists of the following:

Semester I | Term Credits | |
---|---|---|

MATH 599 | Teaching in Mathematics | 3 |

MATH 613 | Advanced Applied Mathematics I: Modeling ^{1} |
3 |

MATH 631 | Linear Algebra ^{2} |
3 |

MATH 645 | Analysis I ^{3} |
3 |

MATH 651 | Methods of Applied Mathematics I ^{1} |
3 |

Term Credits | 15 | |

Semester II | ||

MATH 614 | Numerical Methods I ^{2} |
3 |

MATH 656 | Complex Variables I ^{3} |
3 |

MATH 689 | Advanced Applied Mathematics II: Ordinary Differential Equations | 3 |

MATH 745 | Analysis II ^{3} |
3 |

Term Credits | 12 | |

Semester III | ||

MATH 671 | Asymptotic Methods I | 3 |

MATH 676 | Advanced Ordinary Differential Equations | 3 |

MATH 690 | Advanced Applied Mathematics III: Partial Differential Equations | 3 |

MATH 712 | Numerical Methods II | 3 |

Term Credits | 12 | |

Semester IV | ||

MATH 707 | Advanced Applied Mathematics IV: Special Topics (Advanced Applied Mathematics IV) | 3 |

MATH 713 | Advanced Scientific Computing: Multi-Dimensional Finite-Difference Schemes and Spectral Methods | 3 |

MATH 756 | Complex Variables II | 3 |

Course from Natural Sciences or Engineering relevant to student's interests. | 3 | |

Term Credits | 12 | |

Total Credits | 51 |

^{1} | Helps to prepare for applied mathematics preliminary examination. |

^{2} | Helps to prepare for linear algebra-numerical methods preliminary examination. |

^{3} | Helps to prepare for analysis preliminary examination. |

In addition to these courses, there are advanced courses in:

Code | Title | Credits |
---|---|---|

Mathematical Fluid Dynamics I and Mathematical Fluid Dynamics II | ||

MATH 715 | Mathematical Fluid Dynamics I | 3 |

MATH 716 | Mathematical Fluid Dynamics II | 3 |

Mathematical Biology | ||

MATH 637 | Foundations of Mathematical Biology | 3 |

MATH 672 | Biomathematics I: Biological Waves and Oscillations | 3 |

MATH 673 | Biomathematics II: Pattern Formation in Biological Systems | 3 |

Wave Propagation | ||

MATH 722 | Wave Propagation | 3 |

Asymptotic Methods II | ||

MATH 771 | Asymptotic Methods II | 3 |

Mathematical Modeling II | ||

MATH 639 | Mathematical Modeling II | 3 |

Partial Differential Equations | ||

MATH 675 | Partial Differential Equations | 3 |

Inverse Problems and Global Optimization | ||

MATH 717 | Inverse Problems and Global Optimization | 3 |

Also, there are special topics courses in:

- computational electromagnetics
- computational fluid dynamics
- computational neuroscience
- financial mathematics
- integral equations
- materials science
- microwave processing of materials
- courses in probability and statistics

#### Qualifying Examination

The qualifying examination for the applied mathematics track consists of a preliminary examination in three parts and an oral examination. The three components of the preliminary examination are: Applied Mathematics, Analysis, and Linear Algebra-Numerical Methods. Students must achieve a grade of A in each component to pass the preliminary examination and proceed to the oral examination. Components may be passed at different times. However, a student may attempt each component at most twice and must pass all three components before taking the oral examination. The qualifying examination must be passed by the end of the second year in the program. Typically, two opportunities to take each component are provided each year: Applied Mathematics (January and May), Analysis and Linear Algebra-Numerical Methods (May and August). The oral examination is usually offered in January and May.

Topics for the oral examination are:

- Applied Mathematics, based on the courses MATH 689 Advanced Applied Mathematics II: Ordinary Differential Equations and MATH 690 Advanced Applied Mathematics III: Partial Differential Equations
- choice of two out of the following three:

It should be noted that taking the above courses is not mandatory but students are strongly encouraged to take them before attempting the qualifying examinations. The scope of the qualifying examinations is not limited to the specific list of topics covered in these courses, but these topics are indicative of the overall scope of the examinations.

#### Dissertation Committee

The dissertation committee is an important resource for the doctoral student in the conduct of research for their dissertation. According to the regulations specified in this catalog, doctoral students are required to have a dissertation advisor selected, a dissertation committee formed, and research proposal approved within one year of passage of the qualifying examination.

#### Dissertation Proposal

Doctoral students must prepare a research proposal for approval by their dissertation committee. The student must offer an oral defense of this proposal before the dissertation committee and obtain its approval within one year of passing the qualifying examination. The committee determines if the proposal has an appropriate objective, if there is a reasonable plan to reach that objective, and if the student possesses the knowledge and skills needed to carry out the plan. The dissertation proposal can only be approved by unanimous consent of the committee members.

#### Dissertation Defense

A public oral defense of the dissertation before the dissertation committee is required. All members of the committee must be present for the defense. Success of the defense is determined by a majority vote of the dissertation committee.

### Applied Probability and Statistics Track (NJIT)

Students choosing the applied probability and statistics track must fulfill the requirements for the doctor of philosophy as specified in this catalog. Specific courses of study are planned in consultation with a faculty graduate advisor and are subject to approval. In general, students are encouraged to take courses both in mathematics and in areas of application.

*Seminar:* In addition to the minimum degree credits required, all doctoral students must enroll each semester in MATH 791 Graduate Seminar

*Courses:* A typical schedule of courses for the first four semesters in Applied Probability and Statistics consists of the following:

Semester I | Term Credits | |
---|---|---|

MATH 599 | Teaching in Mathematics | 3 |

MATH 631 | Linear Algebra | 3 |

MATH 644 | Regression Analysis Methods ^{1} |
3 |

MATH 645 | Analysis I ^{2} |
3 |

MATH 662 | Probability Distributions ^{1} |
3 |

Term Credits | 15 | |

Semester II | ||

MATH 665 | Statistical Inference ^{2} |
3 |

MATH 699 | Design and Analysis of Experiments ^{3} |
3 |

MATH 745 | Analysis II ^{2} |
3 |

MATH 768 | Probability Theory ^{3} |
3 |

Term Credits | 12 | |

Semester III | ||

MATH 659 | Survival Analysis | 3 |

MATH 691 | Stochastic Processes with Applications | 3 |

MATH 707 | Advanced Applied Mathematics IV: Special Topics | 3 |

Course in statistics/mathematics/engineering/computing sciences relevant to student's interest | 3 | |

Term Credits | 12 | |

Semester IV | ||

MATH 664 | Methods for Statistical Consulting | 3 |

MATH 698 | Sampling Theory | 3 |

Two Courses in statistics/mathematics/engineering/computer science relevant to student's interest | 6 | |

Term Credits | 12 | |

Total Credits | 51 |

^{1} | Helps to prepare for probability distributions and regression analysis methods preliminary examination. |

^{2} | Helps to prepare for real analysis and statistical inference preliminary examination. |

^{3} | Helps to prepare for probability theory and design and analysis of experiments preliminary examination. |

In addition to these courses, there are advanced courses in:

Code | Title | Credits |
---|---|---|

Time Series Analysis | ||

MATH 646 | Time Series Analysis | 3 |

Clinical Trials Design and Analysis | ||

MATH 654 | Clinical Trials Design and Analysis | 3 |

Statistical Reliability Theory and Applications | ||

MATH 761 | Statistical Reliability Theory and Applications | 3 |

Large Sample Theory and Inference | ||

MATH 786 | Large Sample Theory and Inference | 3 |

Non-Parametric Statistics | ||

MATH 787 | Non-Parametric Statistics | 3 |

#### Qualifying Examination

The qualifying examination for the applied probability and statistics track consists of a preliminary examination in three parts and an oral examination. The three components of the preliminary examination are: Probability Distributions and Regression Analysis Methods, Real Analysis and Statistical Inference, Probability Theory and Design and Analysis of Experiments. Students must achieve a grade of A in each component to pass the preliminary examination and proceed to the oral examination. Components may be passed at different times. However, a student may attempt each component at most twice and must pass all three components before taking the oral examination. The qualifying examination must be passed by the end of the second year in the program. Typically, two opportunities to take each component are provided each year: Probability Distributions and Regression Analysis Methods (January and May), Real Analysis and Statistical Inference and Probability Theory and Design and Analysis of Experiments (May and August). The oral examination is usually offered in January and May.

Topics for the oral examination are:

- Stochastic Processes, based on MATH 691 Stochastic Processes with Applications
- Survival Analysis, based on MATH 659 Survival Analysis
- Generalized Linear Models, based on MATH 707 Advanced Applied Mathematics IV: Special Topics.

It should be noted that taking the above courses is not mandatory but students are strongly encouraged to take them before attempting the qualifying examinations. The scope of the qualifying examinations is not limited to the specific list of topics covered in these courses, but these topics are indicative of the overall scope of the examinations.

#### Dissertation Committee

The dissertation committee is an important resource for the doctoral student in the conduct of research for their dissertation. According to the regulations specified in this catalog, doctoral students are required to have a dissertation advisor selected, a dissertation committee formed, and a research proposal approved within one year of passage of the qualifying examination.

#### Dissertation Proposal

Doctoral students must prepare a research proposal for approval by their dissertation committee. The student must offer an oral defense of this proposal before the dissertation committee and obtain its approval within one year of passing the qualifying examination. The committee determines if the proposal has an appropriate objective, if there is a reasonable plan to reach that objective, and if the student possesses the knowledge and skills needed to carry out the plan. The dissertation proposal can only be approved by unanimous consent of the committee members.

#### Dissertation Defense

A public oral defense of the dissertation before the dissertation committee is required. All members of the committee must be present for the defense. Success of the defense is determined by a majority vote of the dissertation committee.

### Pure Mathematics Track (Rutgers-Newark)

Students interested in the Pure Mathematics track should contact the Department of Mathematics and Computer Science at Rutgers-Newark.