Applied Mathematics (PhD)
Graduate School
Graduate Field
Program Description
The graduate program in applied mathematics is based on a solid foundation in pure mathematics, which includes the fundamentals of algebra and analysis. It involves a grounding in the methods of applied mathematics and studies of scientific areas in which significant applications of mathematics are made. The field has a broadly based interdepartmental faculty that can direct student programs in a large number of areas of the mathematical sciences.
Many specialized or interdisciplinary programs can be designed for individual students, including, for example, a variety of possibilities in biomathematics.
The dissertation is normally a mathematical contribution toward the solution of a problem arising outside mathematics.
Concentrations
- Applied mathematics
Program Information
- Instruction Mode: In Person
- Location: Ithaca, NY
- Minimum Credits for Degree: 144
Program Requirements
- Minimum Semesters for Degree: 8
Graduate School Milestones
- Responsible Conduct of Research Training: Required
- Open Researcher and Contributor ID (ORCID): Required
- Student Progress Reviews (SPR) begin: Second Year
- Examination for admission to candidacy (A Exam): Spring or summer of third year
- The A-Exam must be scheduled before the beginning of the seventh semester unless a special petition is filed. The A-Exam should be scheduled well in advance of the exam and the appropriate form (Schedule of Examination form) must be submitted to the Graduate School at least 7 calendar days ahead. CAM graduate students are eligible for a Non-Thesis Masters Degree upon completion of the A-Exam. In order for this to be awarded, the Committee Chair must check the relevant box on the Results of Examination Form. The form must be turned into the Graduate School within 3 days after the exam.
- Defense of Dissertation (B Exam): Spring or summer of fifth year
- The B-Exam is scheduled with the same form as the A-Exam and must be submitted to the Grad School at least 7 calendar days ahead.
Field Specific Milestones
- End of semester audit for enrolled students
Course Requirements
- Course requirements are determined by the student’s Special Committee.
- Enrollment in GRAD research course or the equivalent field specific research course is expected of all students.
The following are required for admission to candidacy for the Ph.D., or to obtain an M.S. degree in Applied Mathematics:
- Prerequisite to the graduate program are familiarity with analysis and algebra at the advanced undergraduate level (e.g., MATH 4130-MATH 4140 and MATH 4330-MATH 4340). Students lacking either prerequisite (which may be determined by their special committee chair and/or the CAM Director), should take the appropriate courses within their first two years of study. No more than two of these courses can count towards meeting other CAM degree requirements.
- Students are required to take at least eight courses in mathematics and its applications that are approved by their special committee, at least 4 of which must be numbered 6000 or above. Suggested areas for these courses are given in the list of Focal Areas for Applied Mathematics.
- The courses taken to satisfy item (2) must include an advanced course in computational methods (focal area (a)). In order to achieve breadth in Applied Mathematics, courses from at least three other Focal Areas should normally be included. Should a course be listed under more than one focal area, then it will count towards only one such area as chosen by the student’s Special Committee.
- Students are required to have minors in Mathematics and in another field relevant to their doctoral research. Note that the course requirements listed above may suffice to satisfy the requirements for a graduate minor in Mathematics.
Exceptions to these requirements can sometimes be made, if approved in advance by the student’s Special Committee and the CAM Director. All requirement courses must be taken for a letter grade.
Focal Areas for Applied Mathematics
The seven major Focal Areas for the field of applied mathematics are listed below, along with examples of recommended courses for completing the requirements in each focal area. In addition to the listed example courses, appropriate courses containing substantial mathematical content, offered by any department, may be taken to satisfy field requirements in mathematics and its applications, subject to approval by the student’s Special Committee.
All coursework that is part of graduate curriculum must be taken at the 5000 level or higher, other courses will not count toward graduate degrees.
A. Computational Methods
| Code | Title | Hours |
|---|---|---|
| CEE 5745 | Inverse Problems: Theory and Applications | 3 |
| CEE 6300 | Spectral Methods for Incompressible Fluid Flows | 4 |
| CEE 6720 | ||
| CS 6210 | Matrix Computations | 3 |
| CS 6220 | Data-Sparse Matrix Computations | 3 |
| CS 6241 | Numerical Methods for Data Science | 3 |
| MAE 6230 | Computational Fluid Dynamics | 4 |
| MATH 5250 | Numerical Analysis and Differential Equations | 4 |
| MATH 4260 | Numerical Analysis: Linear and Nonlinear Problems 1 | 4 |
- 1
MATH 4260/CS 4220 has not yet been designated with a 5000 level course number and may not be offered to graduate students for some time.
B. Mathematical Analysis
| Code | Title | Hours |
|---|---|---|
| MATH 6110 | Real Analysis 1 | 4 |
| MATH 6120 | Complex Analysis 1 | 4 |
| MATH 6210 | Measure Theory and Lebesgue Integration | 3 |
| MATH 6220 | Applied Functional Analysis 2 | 3 |
| MATH 7130 | Functional Analysis 2 | 3 |
C. Differential Equations and Dynamical Systems
| Code | Title | Hours |
|---|---|---|
| CEE 5735 | Mathematical Modeling of Natural and Engineered Systems | 3 |
| CHEME 7530 | ||
| MAE 5790 | Nonlinear Dynamics and Chaos | 3 |
| MAE 6010 | Foundations of Fluid Mechanics I | 4 |
| MAE 6110 | Foundations of Solid Mechanics I | 3 |
| MAE 6330 | ||
| MAE 6840 | ||
| MATH 6260 | Dynamical Systems | 3 |
| MATH 6180 | ||
| MATH 6150 | Partial Differential Equations | 3 |
| MATH 6160 | Partial Differential Equations | 3 |
| MATH 6230 | Differential Games and Optimal Control | 4 |
| MATH 6280 | ||
| MATH 6520 | Differentiable Manifolds (crosslisted) | 4 |
D. Stochastic Methods (Probability, Stochastic Processes, Statistics, Machine Learning, Signal and Image Processing, etc.)
| Code | Title | Hours |
|---|---|---|
| BTRY 7180 | 1 | |
| CS 6780 | Advanced Machine Learning | 4 |
| CS 6783 | Machine Learning Theory | 4 |
| CS 6784 | Advanced Topics in Machine Learning | 4 |
| CS 6788 | Advanced Topic Modeling | 3 |
| ECE 5555 | ||
| ECE 5620 | Fundamentals of Data Compression | 3 |
| ECE 5630 | ||
| MATH 6710 | Probability Theory I | 3 |
| MATH 6720 | Probability Theory II | 3 |
| MATH 6730 | Mathematical Statistics I | 3 |
| MATH 6740 | Mathematical Statistics II | 3 |
| MATH 7740 | Statistical Learning Theory | 3 |
| ORIE 6500 | Applied Stochastic Processes | 4 |
| ORIE 6510 | Probability | 4 |
| ORIE 6540 | ||
| ORIE 6570 | ||
| ORIE 6580 | Simulation | 3 |
| ORIE 6700 | Statistical Principles | 4 |
| ORIE 6710 | ||
| ORIE 6720 | ||
| ORIE 6750 | 3 | |
| ORIE 6780 | Bayesian Statistics and Data Analysis | 3 |
| STSCI 6520 | Statistical Computing I | 4 |
| STSCI 7170 | Theory of Linear Models | 3 |
- 1
Most 6000-level BTRY courses not suitable.
E. Optimization and Discrete Mathematics
| Code | Title | Hours |
|---|---|---|
| MATH 5410 | Introduction to Combinatorics I | 4 |
| MATH 4420 | Introduction to Combinatorics II 1 | 4 |
| MATH 6230 | Differential Games and Optimal Control | 4 |
| ORIE 6300 | Mathematical Programming I | 4 |
| ORIE 6310 | ||
| ORIE 6320 | ||
| ORIE 6325 | ||
| ORIE 6327 | ||
| ORIE 6328 | Convex Analysis | 3 |
| ORIE 6330 | 3 | |
| ORIE 6334 | Combinatorial Optimization | 3 |
| ORIE 6335 |
- 1
MATH 4420 has not yet been designated with a 5000 level course number and may not be offered to graduate students for some time.
F. Algorithms and Complexity
| Code | Title | Hours |
|---|---|---|
| CS 4814 | Introduction to Computational Complexity 1 | 3 |
| CS 6810 | Theory of Computing | 4 |
| CS 6820 | Analysis of Algorithms | 4 |
| CS 6840 | Algorithmic Game Theory | 4 |
| ORIE 6350 |
- 1
CS 4814 has not yet been designated with a 5000 level course number and may not be offered to graduate students for some time.
G. Algebra and Logic
| Code | Title | Hours |
|---|---|---|
| CS 6117 | Category Theory for Computer Scientists | 4 |
| CS 6764 | Reasoning about Knowledge | 4 |
| CS 6766 | Reasoning about Uncertainty | 4 |
| CS 6860 | ||
| MATH 6310 | Algebra | 4 |
| MATH 6320 | Algebra | 4 |
| MATH 6340 | Commutative Algebra with Applications in Algebraic Geometry | 3 |
| MATH 6390 | Lie Groups and Lie Algebras | 3 |
| MATH 6810 | Logic | 3 |
| MATH 7880 |
University Graduation Requirements
Requirements for All Students
In order to receive a Cornell degree, a student must satisfy academic and non-academic requirements.
Academic Requirements
A student’s college determines degree requirements such as residency, number of credits, distribution of credits, and grade averages. It is the student’s responsibility to be aware of the specific major, degree, distribution, college, and graduation requirements for completing their chosen program of study. See the individual requirements listed by each college or school or contact the college registrar’s office for more information.
Non-academic Requirements
Conduct Matters. Students must satisfy any outstanding sanctions, penalties or remedies imposed or agreed to under the Student Code of Conduct (Code) or Policy 6.4. Where a formal complaint under the Code or Policy 6.4 is pending, the University will withhold awarding a degree otherwise earned until the adjudication process set forth in those procedures is complete, including the satisfaction of any sanctions, penalties or remedies imposed.
Financial Obligations. Outstanding financial obligations will not impact the awarding of a degree otherwise earned or a student’s ability to access their official transcript. However, the University may withhold issuing a diploma until any outstanding financial obligations owing to the University are satisfied.
Learning Outcomes
A graduate student in Applied Mathematics is expected to demonstrate both mastery of knowledge in mathematics and its applications, and ability to create new mathematical knowledge and innovative ways to apply mathematical tools to important problems in science, industry and society.
Each student is expected to demonstrate the following proficiencies.
- Make substantial original contributions to applied mathematics:
- Identify new important and promising research problems
- Think independently, critically and creatively
- Complete research work by bringing it to the stage where it can be published and be used by the others - Maintain ability to acquire new knowledge by keeping up with the new developments in the field through professional publications and professional meetings.
- Ability to communicate effectively research findings and plans:
- Present results in the format of technical papers and have them published in professional journals and conference proceedings
- Explain complex ideas to peers in technical presentations; being aware of funding opportunities and ability to write effective research proposals and obtain research funding - Dedication to advancing science through effective teaching, advising, mentoring and service to professional community.
- Awareness of the ethical standards in the field, and ability to maintain and advance these standards.