# Mathematical Modeling Ph.D. - Curriculum

## Mathematical Modeling Ph.D.

### Mathematical Modeling, Ph.D. degree, typical course sequence

Course | Sem. Cr. Hrs. | |
---|---|---|

First Year |
||

MATH-602 | Numerical Analysis I This course covers numerical techniques for the solution of nonlinear equations, interpolation, differentiation, integration, and matrix algebra. (Prerequisites: MATH-411 or equivalent course and graduate standing.) Lecture 3 (Fall). |
3 |

MATH-606 | Graduate Seminar I The course prepares students to engage in activities necessary for independent mathematical research and introduces students to a broad range of active interdisciplinary programs related to applied mathematics. (This course is restricted to students in the ACMTH-MS or MATHML-PHD programs.) Lecture 2 (Fall). |
1 |

MATH-607 | Graduate Seminar II This course is a continuation of Graduate Seminar I. It prepares students to engage in activities necessary for independent mathematical research and introduces them to a broad range of active interdisciplinary programs related to applied mathematics. (Prerequisite: MATH-606 or equivalent course or students in the ACMTH-MS or MATHML-PHD programs.) Lecture 2 (Spring). |
1 |

MATH-622 | Mathematical Modeling I This course will introduce graduate students to the logical methodology of mathematical modeling. They will learn how to use an application field problem as a standard for defining equations that can be used to solve that problem, how to establish a nested hierarchy of models for an application field problem in order to clarify the problem’s context and facilitate its solution. Students will also learn how mathematical theory, closed-form solutions for special cases, and computational methods should be integrated into the modeling process in order to provide insight into application fields and solutions to particular problems. Students will study principles of model verification and validation, parameter identification and parameter sensitivity and their roles in mathematical modeling. In addition, students will be introduced to particular mathematical models of various types: stochastic models, PDE models, dynamical system models, graph-theoretic models, algebraic models, and perhaps other types of models. They will use these models to exemplify the broad principles and methods that they will learn in this course, and they will use these models to build up a stock of models that they can call upon as examples of good modeling practice. (This course is restricted to students in the ACMTH-MS or MATHML-PHD programs.) Lecture 3 (Fall). |
3 |

MATH-722 | Mathematical Modeling II This course will continue to expose students to the logical methodology of mathematical modeling. It will also provide them with numerous examples of mathematical models from various fields. (Prerequisite: MATH-622 or equivalent course.) Lecture 3 (Spring). |
3 |

MATH Concentration Courses |
6 | |

MATH Elective |
3 | |

Second Year |
||

MATH-751 | High-performance Computing For Mathematical Modeling Students in this course will study high-performance computing as a tool for solving problems related to mathematical modeling. Two primary objectives will be to gain experience in understanding the advantages and limitations of different hardware and software options for a diverse array of modeling approaches and to develop a library of example codes. The course will include extensive hands-on computational (programming) assignments. Students will be expected to have a prior understanding of basic techniques for solving mathematical problems numerically. (Prerequisite: MATH-602 or equivalent course.) Lecture 3 (Spring). |
3 |

MATH-790 | Research & Thesis Masters-level research by the candidate on an appropriate topic as arranged between the candidate and the research advisor. (This course is restricted to students in the ACMTH-MS or MATHML-PHD programs.) Thesis (Fall, Spring, Summer). |
6 |

MATH Concentration Course |
3 | |

MATH Electives |
6 | |

Third Year |
||

MATH-790 | Research & Thesis Masters-level research by the candidate on an appropriate topic as arranged between the candidate and the research advisor. (This course is restricted to students in the ACMTH-MS or MATHML-PHD programs.) Thesis (Fall, Spring, Summer). |
10 |

Fourth Year |
||

MATH-790 | Research & Thesis Masters-level research by the candidate on an appropriate topic as arranged between the candidate and the research advisor. (This course is restricted to students in the ACMTH-MS or MATHML-PHD programs.) Thesis (Fall, Spring, Summer). |
6 |

Fifth Year |
||

MATH-790 | Research & Thesis |
6 |

Total Semester Credit Hours |
60 |

### Concentrations

#### Applied Inverse Problems

Course | Sem. Cr. Hrs. | |
---|---|---|

MATH-625 | Applied Inverse Problems Most models in applied and social sciences are formulated using the broad spectrum of linear and nonlinear partial differential equations involving parameters characterizing specific physical characteristics of the underlying model. Inverse problems seek to determine such parameters from the measured data and have many applications in medicine, economics, and engineering. This course will provide a thorough introduction to inverse problems and will equip students with skills for solving them. The topics of the course include existence results, discretization, optimization formulation, and computational methods. (Prerequisites: MATH-431 or equivalent course or graduate student standing.) Lecture 3 (Fall). |
3 |

MATH-633 | Measure Theory of Elements and Functional Analysis This course will provide a general introduction to Lebesgue measure as applied to the real numbers, real-valued functions of a real variable, and the Lebesgue integral of such functions. It also covers topics in functional analysis relevant to application of measure theory to real-world problems. Students will be expected to read and understand proofs, and to demonstrate their understanding of topics by writing their own proofs of various facts. (Prerequisites: Graduate student standing in COS, GCCIS or KGCOE or B+ or better in MATH 432 or equivalent course.) Lecture 3 (Fall). |
3 |

MATH-741 | Partial Differential Equations I This course uses methods of applied mathematics in the solution of problems in physics and engineering. Models such as heat flow and vibrating strings will be formulated from physical principles. Characteristics methods, maximum principles, Green's functions, D'Alembert formulas, weak solutions and distributions will be studied. (Prerequisites: MATH-231 or equivalent course or graduate student standing in ACMTH-MS or MATHML-PHD programs.) Lecture 3 (Spring). |
3 |

#### Biomedical Mathematics

Course | Sem. Cr. Hrs. | |
---|---|---|

MATH-631 | Dynamical Systems This course is a study of dynamical systems theory. Basic definitions of dynamical systems are followed by a study of maps and time series. Stability theory of solutions of differential equations is studied. Asymptotic behavior of solutions is investigated through limit sets, attractors, Poincaré–Bendixson theory, and index theory. The notion of local bifurcation is introduced and investigated. Chaotic systems are studied. (Prerequisites: (MATH-231 and (MATH 241 or MATH-241H)) or equivalent courses or graduate standing in ACMTH-MS or MATHML-PHD programs.) Lecture 3 (Fall). |
3 |

MATH-702 | Numerical Analysis II The course covers the solutions of linear systems by direct and iterative methods, numerical methods for computing eigenvalues, theoretical and numerical methods for unconstrained and constrained optimization, and Monte-Carlo simulation. (Prerequisite: MATH-602 or equivalent course and graduate standing.) Lecture 3 (Spring). |
3 |

MATH-761 | Mathematical Biology This course introduces areas of biological sciences in which mathematics can be used to capture essential interactions within a system. Different modeling approaches to various biological and physiological phenomena are developed (e.g., population and cell growth, spread of disease, epidemiology, biological fluid dynamics, nutrient transport, biochemical reactions, tumor growth, genetics). The emphasis is on the use of mathematics to unify related concepts. (Graduate Science) Lecture 3 (Fall). |
3 |

#### Discrete Mathematics

Course | Sem. Cr. Hrs. | |
---|---|---|

CSCI-665 | Foundations of Algorithms This course provides an introduction to the design and analysis of algorithms. It covers a variety of classical algorithms and their complexity and will equip students with the intellectual tools to design, analyze, implement, and evaluate their own algorithms. Note: students who take CSCI-261 or CSCI-264 may not take CSCI-665 for credit. (Prerequisites: (CSCI-603 and CSCI-605 and CSCI-661 with grades of B or better) or ((CSCI-243 or SWEN-262) and (CSCI-262 or CSCI-263)) or equivalent courses. This course is restricted to COMPSCI-MS, COMPSCI-BS/MS, or COMPIS-PHD students.) Lec/Lab 3 (Fall, Spring). |
3 |

MATH-645 | Graph Theory This course introduces the fundamental concepts of graph theory. Topics to be studied include graph isomorphism, trees, network flows, connectivity in graphs, matchings, graph colorings, and planar graphs. Applications such as traffic routing and scheduling problems will be considered. (This course is restricted to students with graduate standing in the College of Science or Graduate Computing and Information Sciences.) Lecture 3 (Fall). |
3 |

MATH-646 | Combinatorics This course introduces the fundamental concepts of combinatorics. Topics to be studied include counting techniques, binomial coefficients, generating functions, partitions, the inclusion-exclusion principle and partition theory. (This course is restricted to students in the ACMTH-MS or MATHML-PHD programs.) Lecture 3 (Spring). |
3 |

#### Dynamical Systems and Fluid Dynamics

Course | Sem. Cr. Hrs. | |
---|---|---|

MATH-631 | Dynamical Systems This course is a study of dynamical systems theory. Basic definitions of dynamical systems are followed by a study of maps and time series. Stability theory of solutions of differential equations is studied. Asymptotic behavior of solutions is investigated through limit sets, attractors, Poincaré–Bendixson theory, and index theory. The notion of local bifurcation is introduced and investigated. Chaotic systems are studied. (Prerequisites: (MATH-231 and (MATH 241 or MATH-241H)) or equivalent courses or graduate standing in ACMTH-MS or MATHML-PHD programs.) Lecture 3 (Fall). |
3 |

MATH-741 | Partial Differential Equations I This course uses methods of applied mathematics in the solution of problems in physics and engineering. Models such as heat flow and vibrating strings will be formulated from physical principles. Characteristics methods, maximum principles, Green's functions, D'Alembert formulas, weak solutions and distributions will be studied. (Prerequisites: MATH-231 or equivalent course or graduate student standing in ACMTH-MS or MATHML-PHD programs.) Lecture 3 (Spring). |
3 |

MATH-831 | Mathematical Fluid Dynamics The study of the dynamics of fluids is a central theme of modern applied mathematics. It is used to model a vast range of physical phenomena and plays a vital role in science and engineering. This course provides an introduction to the basic ideas of fluid dynamics, with an emphasis on rigorous treatment of fundamentals and the mathematical developments and issues. The course focuses on the background and motivation for recent mathematical and numerical work on the Euler and Navier-Stokes equations, and presents a mathematically intensive investigation of various models equations of fluid dynamics. (Prerequisite: MATH-741 or equivalent course.) Lecture 3 (Fall, Spring, Summer). |
3 |

#### Geometry, Relativity and Gravitation

Course | Sem. Cr. Hrs. | |
---|---|---|

ASTP-660 | Introduction to Relativity and Gravitation This course is the first in a two-course sequence that introduces Einstein’s theory of General Relativity as a tool in modern astrophysics. The course will cover various aspects of both Special and General Relativity, with applications to situations in which strong gravitational fields play a critical role, such as black holes and gravitational radiation. Topics include differential geometry, curved spacetime, gravitational waves, and the Schwarzschild black hole. The target audience is graduate students in the astrophysics, physics, and mathematical modeling (geometry and gravitation) programs. (This course is restricted to students in the ASTP-MS, ASTP-PHD, MATHML-PHD and PHYS-MS programs.) Lecture 3 (Fall). |
3 |

ASTP-861 | Advanced Relativity and Gravitation This course is the second in a two-course sequence that introduces Einstein’s theory of General Relativity as a tool in modern astrophysics. The course will cover various aspects of General Relativity, with applications to situations in which strong gravitational fields play a critical role, such as black holes and gravitational radiation. Topics include advanced differential geometry, generic black holes, energy production in black-hole physics, black-hole dynamics, neutron stars, and methods for solving the Einstein equations. The target audience is graduate students in the astrophysics, physics, and mathematical modeling (geometry and gravitation) programs. (Prerequisite: ASTP-660 or equivalent course.) Lecture 3 (Spring). |
3 |

MATH-702 | Numerical Analysis II The course covers the solutions of linear systems by direct and iterative methods, numerical methods for computing eigenvalues, theoretical and numerical methods for unconstrained and constrained optimization, and Monte-Carlo simulation. (Prerequisite: MATH-602 or equivalent course and graduate standing.) Lecture 3 (Spring). |
3 |