Mathematical Modeling Doctor of philosophy (Ph.D.) degree
Mathematical Modeling
Doctor of philosophy (Ph.D.) degree
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 Mathematical Modeling Ph.D.
Overview
Mathematical modeling is the process of developing mathematical descriptions, or models, of realworld systems. These models can be linear or nonlinear, discrete or continuous, deterministic or stochastic, and static or dynamic, and they enable investigating, analyzing, and predicting the behavior of systems in a wide variety of fields. Through extensive study and research, graduates of this program will have the expertise not only to use the tools of mathematical modeling in various application settings, but also to contribute in creative and innovative ways to the solution of complex interdisciplinary problems and to communicate effectively with domain experts in various fields.
Plan of study
The degree requires at least 60 credit hours of course work and research. The curriculum consists of three required core courses, three required concentration foundation courses, a course in scientific computing and highperformance computing (HPC), three elective courses focused on the student’s chosen research concentration, and a doctoral dissertation. Elective courses are available from within the School of Mathematical Sciences as well as from other graduate programs at RIT, which can provide applicationspecific courses of interest for particular research projects. A minimum of 30 credits hours of course work is required. In addition to courses, at least 30 credit hours of research, including the Graduate Research Seminar, and an interdisciplinary internship outside of RIT are required.
Students develop a plan of study in consultation with an application domain advisory committee. This committee consists of the program director, one of the concentration leads, and an expert from an application domain related to the student’s research interest. The committee ensures that all students have a roadmap for completing their degree based on their background and research interests. The plan of study may be revised as needed.
Qualifying examinations
All students must pass two qualifying examinations to determine whether they have sufficient knowledge of modeling principles, mathematics, and computational methods to conduct doctoral research. Students must pass the examinations in order to continue in the Ph.D. program.
The first exam is based on the Numerical Analysis I (MATH602) and Mathematical Modeling I, II (MATH622, 722). The second exam is based on the student's concentration foundation courses and additional material deemed appropriate by the committee and consists of a short research project.
Dissertation research advisor and committee
A dissertation research advisor is selected from the program faculty based on the student's research interests, faculty research interest, and discussions with the program director. Once a student has chosen a dissertation advisor, the student, in consultation with the advisor, forms a dissertation committee consisting of at least four members, including the dissertation advisor. The committee includes the dissertation advisor, one other member of the mathematical modeling program faculty, and an external chair appointed by the dean of graduate education. The external chair must be a tenured member of the RIT faculty who is not a current member of the mathematical modeling program faculty. The fourth committee member must not be a member of the RIT faculty and may be a professional affiliated with industry or with another institution; the program director must approve this committee member.
The main duties of the dissertation committee are administering both the candidacy exam and final dissertation defense. In addition, the dissertation committee assists students in planning and conducting their dissertation research and provides guidance during the writing of the dissertation.
Admission to candidacy
When a student has developed an indepth understanding of their dissertation research topic, the dissertation committee administers an examination to determine if the student will be admitted to candidacy for the doctoral degree. The purpose of the examination is to ensure that the student has the necessary background knowledge, command of the problem, and intellectual maturity to carry out the specific doctorallevel research project. The examination may include a review of the literature, preliminary research results, and proposed research directions for the completed dissertation. Requirements for the candidacy exam include both a written dissertation proposal and the presentation of an oral defense of the proposal. This examination must be completed at least one year before the student can graduate.
Dissertation defense and final examination
The dissertation defense and final examination may be scheduled after the dissertation has been written and distributed to the dissertation committee and the committee has consented to administer the final examination. Copies of the dissertation must be distributed to all members of the dissertation committee at least four weeks prior to the final examination. The dissertation defense consists of an oral presentation of the dissertation research, which is open to the public. This public presentation must be scheduled and publicly advertised at least four weeks prior to the examination. After the presentation, questions will be fielded from the attending audience and the final examination, which consists of a private questioning of the candidate by the dissertation committee, will ensue. After the questioning, the dissertation committee immediately deliberates and thereafter notifies the candidate and the mathematical modeling graduate director of the result of the examination.
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Curriculum
Mathematical Modeling, Ph.D. degree, typical course sequence
Course  Sem. Cr. Hrs.  

First Year  
MATH602 
Numerical Analysis I
This course covers numerical techniques for the solution of nonlinear equations, interpolation, differentiation, integration, and matrix algebra.

3 
MATH606 
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.

1 
MATH607 
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.

1 
MATH622 
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, closedform 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, graphtheoretic 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.

3 
MATH722 
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.

3 
Concentration Foundation Courses 
6  
Elective 
3  
Second Year  
MATH789 
Highperformance Computing For Mathematical Modeling
This is a masterlevel course on a topic that is not part of the formal curriculum. This course is structured as an ordinary course and has specific prerequisites, contact hours, and examination procedures.

3 
MATH790 
Research & Thesis
Masterslevel research by the candidate on an appropriate topic as arranged between the candidate and the research advisor.

6 
Concentration Foundation Course 
3  
Electives 
6  
Third Year  
MATH790 
Research & Thesis
Masterslevel research by the candidate on an appropriate topic as arranged between the candidate and the research advisor.

10 
Fourth Year  
MATH790 
Research & Thesis
Masterslevel research by the candidate on an appropriate topic as arranged between the candidate and the research advisor.

6 
Fifth Year  
MATH790 
Research & Thesis
Masterslevel research by the candidate on an appropriate topic as arranged between the candidate and the research advisor.

6 
Total Semester Credit Hours  60 
Concentrations
Applied Inverse Problems
Course  Sem. Cr. Hrs.  

MATH625 
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.

3 
MATH633 
Measure Theory of Elements and Functional Analysis
This course will provide a general introduction to Lebesgue measure as applied to the real numbers, realvalued 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 realworld 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.

3 
MATH741 
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.

3 
Biomedical Mathematics
Course  Sem. Cr. Hrs.  

MATH631 
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, PoincareBendixson theory, and index theory. The notion of local bifurcation is introduced and investigated. Chaotic systems are studied.

3 
MATH702 
Numerical Analysis II
This course covers the solutions of initial value problems and boundary value problems, spectral techniques, simulation methods, optimization and techniques employed in modern scientific computing.

3 
MATH761 
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.

3 
Discrete Mathematics
Course  Sem. Cr. Hrs.  

CSCI665 
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 CSCI261 or CSCI264 may not take CSCI665 for credit.

3 
MATH645 
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.

3 
MATH646 
Combinatorics
This course introduces the fundamental concepts of combinatorics. Topics to be studied include counting techniques, binomial coefficients, generating functions, partitions, the inclusionexclusion principle and partition theory.

3 
Dynamical Systems and Fluid Dynamics
Course  Sem. Cr. Hrs.  

MATH631 
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, PoincareBendixson theory, and index theory. The notion of local bifurcation is introduced and investigated. Chaotic systems are studied.

3 
MATH741 
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.

3 
MATH831 
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 NavierStokes equations, and presents a mathematically intensive investigation of various models equations of fluid dynamics.

3 
Geometry, Relativity and Gravitation
Course  Sem. Cr. Hrs.  

ASTP760 
Introduction to Relativity and Gravitation
This course is the first in a twocourse 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.

3 
ASTP861 
Advanced Relativity and Gravitation
This course is the second in a twocourse 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 blackhole physics, blackhole dynamics, introductory cosmology, and methods for solving the Einstein equations.

3 
MATH702 
Numerical Analysis II
This course covers the solutions of initial value problems and boundary value problems, spectral techniques, simulation methods, optimization and techniques employed in modern scientific computing.

3 
Admission Requirements
To be considered for admission to the Ph.D. program in mathematical modeling, candidates must fulfill the following requirements:
 Complete a graduate application.
 Hold a baccalaureate degree (or equivalent) from an accredited university or college.
 Submit official transcripts (in English) of all previously completed undergraduate and graduate course work.
 Have a minimum cumulative GPA of 3.0 (or equivalent) in a primary field of study.
 Submit scores from the GRE.
 Submit a personal statement of educational objectives and research interests.
 Submit a current resume or curriculum vitae.
 Submit a minimum of two letters of recommendation from academic or professional sources.
 International applicants whose native language is not English must submit scores from the TOEFL, IELTS, or PTE. A minimum TOEFL score of 100 (internetbased) is required. A minimum IELTS score of 7.0 is required. The English language test score requirement is waived for native speakers of English or for those submitting transcripts from degrees earned at American institutions.
Mathematical modeling encompasses a wide variety of scientific disciplines, and candidates from diverse backgrounds are encouraged to apply. If applicants have not taken expected foundational course work, the program director may require the student to successfully complete foundational courses prior to matriculating into the Ph.D. program. Typical foundation course work includes calculus through multivariable and vector calculus, differential equations, linear algebra, probability and statistics, one course in computer programming, and at least one course in real analysis, numerical analysis, or upperlevel discrete mathematics.
Learn about admissions and financial aid
Additional Info
Financial aid, scholarships, and assistantships
Graduate assistantships and tuition remission scholarships are available to qualified students. Applicants seeking financial assistance must submit all application documents to the Office of Graduate and Parttime Enrollment. Please contact the office for current application materials and deadlines. Students whose native language is not English are advised to obtain as high a TOEFL or IELTS score as possible if they wish to apply for a teaching or research assistantship. These candidates also are encouraged to take the Test of Spoken English in order to be considered for financial assistance.
Residency
All students in the program must spend at least two consecutive semesters (summer excluded) as resident fulltime students to be eligible to receive the doctoral degree.
Maximum time limitations
University policy requires that doctoral programs be completed within seven years of the date of the student passing the qualifying exam. All candidates must maintain continuous enrollment during the research phase of the program. Such enrollment is not limited by the maximum number of research credits that apply to the degree.