# Applied and Computational Mathematics Master of science degree

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## Overview

Create innovative computing solutions, mathematical models, and dynamic systems to solve problems in industries such as engineering, biology, and more.

Sophisticated mathematical tools are increasingly used to solve problems in management science, engineering, biology, financial portfolio planning, facilities planning, control of dynamic systems, and design of composite materials. The goal is to find computing solutions to real-world problems. The applied and computational mathematics master’s degree refines your capabilities in applying mathematical models and methods to study a range of problems, with an emphasis on developing and implementing computing solutions.

The ideas of applied mathematics pervade several applications in a variety of businesses and industries as well as the government. Sophisticated mathematical tools are increasingly used to develop new models, modify existing ones, and analyze system performance. This includes applications of mathematics to problems in management science, biology, portfolio planning, facilities planning, control of dynamic systems, and design of composite materials. The goal is to find computable solutions to real-world problems arising from these types of situations.

The masters of science degree in applied and computational mathematics provide students with the capability to apply mathematical models and methods to study various problems that arise in industry and business, with an emphasis on developing computable solutions that can be implemented. The program offers concentrations in discrete mathematics, dynamical systems, and scientific computing. Electives may be selected from the graduate course offerings in the School of Mathematical Sciences or from other graduate programs, with approval from the graduate program director. Students have the option to complete a thesis, which includes the presentation of original ideas and solutions to a specific mathematical problem. The proposal for the thesis work and the results must be presented and defended before the advisory committee.

#### Nature of work

Mathematicians use mathematical theory, computational techniques, algorithms, and the latest computer technology to solve economic, scientific, engineering, physics, and business problems. The work of mathematicians falls into two broad classes — theoretical (pure) mathematics and applied mathematics. These classes, however, often overlap. Applied mathematicians start with a practical problem, envision its separate elements, and then reduce the elements to mathematical variables. They often use computers to analyze relationships among the variables, and they solve complex problems by developing models with alternative solutions.

#### Types of mathematics

Most often the work involving applied mathematics is done by persons whose titles are other than mathematician, including engineer, economist, analyst (e.g. operations research), physicist, cryptanalyst (codes), actuary, teacher, market researcher, and financial advisor.

Many mathematicians work for federal or state agencies. The Department of Defense accounts for about 81 percent of the mathematicians employed by the federal government. In the private sector, mathematicians are employed by scientific research and development services, software publishers, insurance companies, and in aerospace or pharmaceutical manufacturing.

#### Cooperative education

Cooperative education enables students to alternate periods of study on campus with periods of full-time, paid professional employment. Students may pursue a co-op position after their first semester. Co-op is optional for this program.

#### Industries

• Government (Local, State, Federal)

• Insurance

• Investment/Portfolio Management

• Defense

• Scientific and Technical Consulting

• Biotech and Life Sciences

• Telecommunications

#### Typical Job Titles

 Engineer Economist Analyst (e.g. Operations Research) Physicist Cryptanalyst (codes) Actuary Teacher Market Researcher Financial Advisor

## Curriculum

#### Applied and Computational Mathematics (thesis option), MS degree, typical course sequence

Course Sem. Cr. Hrs.
First Year
Choose four of the following core courses: 12
MATH-601
Methods of Applied Mathematics
This course is an introduction to classical techniques used in applied mathematics. Models arising in physics and engineering are introduced. Topics include dimensional analysis, scaling techniques, regular and singular perturbation theory, and calculus of variations.

MATH-602
Numerical Analysis I
This course covers numerical techniques for the solution of nonlinear equations, interpolation, differentiation, integration, and matrix algebra.

MATH-605
Stochastic Processes
This course is an introduction to stochastic processes and their various applications. It covers the development of basic properties and applications of Poisson processes and Markov chains in discrete and continuous time. Extensive use is made of conditional probability and conditional expectation. Further topics such as renewal processes, reliability and Brownian motion may be discussed as time allows.

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.

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.

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.

MATH-606
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
MATH-607
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

Electives
6
Second Year
MATH-790
Research & Thesis
Masters-level research by the candidate on an appropriate topic as arranged between the candidate and the research advisor.
7

Elective
3
Total Semester Credit Hours
30

#### Applied and Computational Mathematics (project option), MS degree, typical course sequence

Course Sem. Cr. Hrs.
First Year
Choose four of the following core courses:
12
MATH-601
Methods of Applied Mathematics
This course is an introduction to classical techniques used in applied mathematics. Models arising in physics and engineering are introduced. Topics include dimensional analysis, scaling techniques, regular and singular perturbation theory, and calculus of variations.

MATH-602
Numerical Analysis I
This course covers numerical techniques for the solution of nonlinear equations, interpolation, differentiation, integration, and matrix algebra.

MATH-605
Stochastic Processes
This course is an introduction to stochastic processes and their various applications. It covers the development of basic properties and applications of Poisson processes and Markov chains in discrete and continuous time. Extensive use is made of conditional probability and conditional expectation. Further topics such as renewal processes, reliability and Brownian motion may be discussed as time allows.

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.

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.

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.

MATH-606
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
MATH-607
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

Electives
6
Second Year
MATH-790
Research & Thesis
Masters-level research by the candidate on an appropriate topic as arranged between the candidate and the research advisor.
4

Electives
6
Total Semester Credit Hours
30

To be considered for admission to the MS program in applied and computational mathematics, candidates must fulfill the following requirements:

• Hold a baccalaureate degree (or equivalent) from an accredited university or college in mathematics or a related field.
• Submit official transcripts (in English) of all previously completed undergraduate and graduate course work.
• Have knowledge of a programming language.
• Have a minimum cumulative GPA of 3.0 (or equivalent).
• Submit a personal statement of educational objectives.
• Submit 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 79 (internet-based) is required. A minimum IELTS score of 6.5 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.

Although Graduate Record Examination (GRE) scores are not required, submitting them may enhance a candidate's acceptance into the program.

A student may also be granted conditional admission and be required to complete bridge courses selected from among RIT’s existing undergraduate courses, as prescribed by the student’s adviser. Until these requirements are met, the candidate is considered a nonmatriculated student. The graduate program director evaluates the student’s qualifications to determine eligibility for conditional and provisional admission.