# Mathematical Sciences

### Mathematics

Mathematics is essential to science and engineering, and is a fascinating field in its own right. Scientific and engineering problems have often inspired new developments in mathematics, and, conversely, mathematical results have frequently had an impact on business, engineering, the sciences, and technology. At Stevens, we think that an undergraduate program in mathematics should be broad enough to prepare you for a job in industry, while giving you the background to continue your education at the graduate level, should you choose to do so.

**Undergraduate Programs**

### Bachelor of Science in Mathematics

The Bachelor of Science in Mathematics offers a broad background appropriate for students planning to pursue a job in industry, while also offering students the depth and rigor required for graduate studies in mathematics or related fields.

The curriculum satisfies the core Bachelor of Science curriculum that includes certain breadth requirements in mathematics, physics, chemistry, biology, computer science, the humanities and social sciences. In addition to this science core, the student completes twelve upper-level mathematics courses (called technical electives). Most of these technical courses are prescribed by the program but in some cases other courses can be substituted with the approval of the undergraduate advisor. For example, entering freshmen with Advanced Placement (AP) credit should consider enrolling in the seminar course MA 188 along with Calculus II; Discrete Mathematics, MA 134, is a good choice for students wanting to strengthen their grasp of fundamental concepts and for students interested in taking computer science courses beyond the minimum requirement; writing a senior thesis is strongly recommended, especially for students interested in pursuing graduate studies. The program includes two general electives which can be applied toward a minor in another discipline. As early as possible, students should discuss with the advisor how best to use the electives to focus the program on a particular concentration area.

The link to the mathematics curriculum on the right-hand side of this page takes you to a recommended study plan. The courses do not need to be taken in exactly the order listed. See the Department of Mathematics Web page for information on when particular courses are offered. There are additional comments following the study plan regarding recommended electives and possible course substitutions.

#### Minor in Mathematics

A minor in mathematics can be a valuable qualification for students concentrating in other areas. A student wishing to pursue a minor in mathematics must complete a Minor Program Study Plan signed by the Department advisor, and meet the School of Engineering and Science requirements for minor programs.

Required courses for a Minor in Mathematics:

- MA 221 Differential Equations
- MA 227 Multivariable Calculus
- MA 222 Probability and Statistics
- MA 232 Linear Algebra
- MA 234 Complex Variables with Applications
- One elective at 300 or above chosen with the consent of the Department advisor.

The following are prerequisites needed to undertake the minor program:

- MA-121 and MA-122, or MA-115 (Calculus I)
- MA-123 and MA-124, or MA-116 (Calculus II)

### Interdisciplinary Program in Computational Science

Computational Science is an emerging field in which sophisticated computational techniques are used to build models and solve problems related to science and engineering. It complements existing theoretical and experimental approaches and may be thought of as a new mode of scientific inquiry.

At Stevens, undergraduates may study computational science through an interdisciplinary program leading to a Bachelor of Science in Computational Science with a specialization in an area of science or engineering. Typical specializations include:

- Computational Chemistry
- Computational Mechanics
- Computational Oceanography
- Computational Physics
- Computer Vision and Computer Graphics
- Cybersecurity
- Environmental Systems

The program consists of the science curriculum core courses and technical electives. The technical electives are divided between foundation courses in mathematics and computer science, and application courses in the student's area of specialization. An important part of the program is a project or research problem to be done in the senior year. Each student must choose an application area prior to preparing his/her study plan. Each student's study plan reflects his/her interests and aspirations, and is made up by the student, working with an advisor from the application area. Interested students should consult with the undergraduate program director in the Department of Mathematical Sciences for further information.

The link to the computational science curriculum on the right-hand side of this page shows a recommended study plan including the core and foundation courses. Suggested electives for specific application areas are listed below. Courses need not be taken in exactly the order shown in the curriculum.

### Computational Science Application Areas

Application areas correspond to the research interests of the faculty associated with the program and are subject to change. Sample selections of application courses are given below. Note that enrollment in 600-level courses requires special permission.

**Computational Chemistry **

- CH 243 Organic Chemistry I
- CH 322 Theoretical Chemistry
- CH 421 Chemical Dynamics
- CH 498 Chemical Research I
- CH 499 Chemical Research II

**Computational Mechanics **

- E 126 Mechanics of Solids
- CE 345 Modeling and Simulation
- CE 373 Structural Analysis
- MA 498 Senior Research Project I

one of the following:

- CE 613 Matrix Analysis of Structures
- CE 623 Structural Dynamics
- CE 681 Introduction to Finite Element Methods

**Computational Oceanography **

- E 126 Mechanics of Solids
- CE 342 Fluid Mechanics
- OE 526 Computer-Aided Naval Architecture
- OE 648 Numerical Hydrodynamics
- MA 498 Mathematical Research I

**Computational Physics **

- PEP 497 SKIL V
- PEP 498 SKIL VI
- PEP 538 Introduction to Mechanics
- PEP 542 Electromagnetism

one of the following:

- PEP 520 Computational Physics
- PEP 575 Fundamentals of Atmospheric Radiation and Climate

**Computer Vision and Computer Graphics **

- CS 437 Interactive Computer Graphics
- CS 498 Computer Science Research I
- CS 499 Computer Science Research II
- CS 558 Computer Vision
- CS 638 Interactive Computer Graphics II

**Cybersecurity **

- CS 334 Automata and Computation
- CS 498 Computer Science Research I
- CS 499 Computer Science Research II
- CS 668 Foundations of Cryptography
- CS 693 Cryptographic Protocols

**Environmental Systems **

- EN 345 Modeling and Simulation
- EN 375 Environmental Systems
- EN 541 Fate and Transport of Environmental Contaminants
- MA 498 Mathematical Research I

one of the following:

- MA 499 Mathematical Research II
- EN 571 Physiochemical Processes for Environmental Control

#### Graduation Requirements

Refer to the catalog section on Undergraduate Programs for additional procedures and Graduation Requirements that apply to all undergraduate students.

**Graduate Programs**

**Admission Criteria and Application Requirements**

Applications to all graduate programs, degree and certificate, must be prepared and submitted according to the Stevens Office of Graduate Admissions regulations. Instructions and forms may be found on the Graduate Admissions web site (www.stevens.edu/graduate/). Notice that the procedure is different for domestic and international applicants. The following sections describe requirements specific to graduate programs in the Department of Mathematical Sciences.

**Master Degree and Certificate Programs**

Adequate undergraduate preparation for admission to any master degree or certificate program includes analytic geometry and calculus, elementary differential equations, one semester of linear algebra, and one semester of probability or probability and statistics. It is possible to be admitted with the requirement that you make up a deficiency in preparation. Applications to any master degree or certificate program should include

- Two letters of recommendation
- Official transcripts and diplomas. For non-English speaking institutions, these documents must be accompanied by a certified English translation
- GRE General Test scores
- TOEFL score for international students

**Doctoral Program**

Admission to the doctoral program requires the preparation specified above. If your goal is a Ph.D., you should apply directly to the doctoral program and not to a master's program. In order to receive full consideration, applications to the doctoral program should be received by February 15 for admission in the Fall Semester, and October 15 for admission in the spring semester. Because of constraints due to course scheduling, admission for the spring semester is not always feasible and may depend on the student’s preparation. In addition, financial aid is usually not available for students admitted in the spring semester. Applicants requesting financial aid should apply by February 15 and clearly state that such aid is being requested.

Applications to the doctoral program should include the following items, all of which enter into the Graduate Program Committee’s evaluation of applicants:

- A personal statement that, in a succinct manner, describes the student's reasons for pursuing a Ph.D., prior classroom and research experience in mathematics, and current mathematical interests. This should not exceed two pages.
- Official transcripts and diplomas. For non-English-speaking institutions, these documents must be accompanied by a certified English translation.
- Letters of recommendation: at least two; at most, four.
- GRE General Test scores (Math Subject Test recommended).

**Financial Aid**

The department supports a limited number of Ph.D. students through teaching assistantships which entitle the recipients to a salary and a waiver of their tuition costs. Teaching assistants are considered for renewal each year, depending on the student’s progress towards graduation and performance evaluations as a teaching assistant. Save for exceptional cases, teaching assistantships are normally not granted for more than five years and are usually available only for students entering in the fall.

Students who wish to be considered for a teaching assistantship beginning their first year should mention this in their Personal Statement. Students with prior teaching experience are encouraged to submit additional documentation that addresses their teaching skills, such as letters of recommendation, evaluation forms, teaching awards, etc. However, no teaching experience is required for an incoming student to be considered for a teaching assistantship.

### Master of Science - Applied Mathematics

This program provides a background in mathematical techniques which are useful in solving practical problems in science and engineering. You are encouraged to include courses from other departments in your program of study.

The program requires 30 credits (10 courses) of coursework. You may transfer up to one-third of this amount from outside Stevens. If you know the material in one of the required courses, you may substitute another course. Requests for transfer credit or course substitution will need approval from the department and must be accompanied by evidence of the course content. All elective courses must be chosen with the consent of a department advisor.

**Core Courses**

- MA 547 Advanced Calculus I or MA 635 Real Variable I
- MA 552 Linear Algebra
- MA 611 Probability
- MA 615 Numerical Analysis I
- MA 649 Differential Equations I
- MA 681 Functions of a Complex Variable I

**Typical Electives**

- MA 548 Advanced Calculus II
- MA 627 Combinatorial Analysis
- MA 650 Intermediate Partial Differential Equations
- MA 653 Numerical Solutions of Partial Diff. Eqs.
- MA 612 Mathematical Statistics
- MA 620 Intro to Network & Graph Theory
- MA 623 Stochastic Processes
- MA 629 Convex Analysis & Optimization
- MA 635 Real Variables I
- MA 641 Time Series Analysis I
- MA 649 Intermediate Differential Equations
- MA 651 Topology I
- MA 661 Dynamic Programming & Stochastic Optimal Control
- MA 711 Inverse Problems in Science & Engineering
- MA 712 Mathematical Models of Risk
- MA 800 Special Problems in Mathematics (MS)
- CE 519 Advanced Structural Analysis
- CE 601 Theory of Elasticity
- CS 580 The Logic of Program Design
- CS 590 Introduction to Data Structures and Algorithms
- ME 674 Fluid Dynamics
- PEP 520 Computational Physics

### Master of Science - Mathematics

A master’s degree in mathematics requires 30 credits of coursework, including the following core courses:

**Core Courses **

- MA 552 Linear Algebra
- MA 605 Foundations of Algebra I
- MA 611 Probability
- MA 635 Real Variables I
- MA 651 Topology I
- MA 681 Functions of a Complex Variable I

**Typical Electives**

- MA 606 Foundations of Algebra II
- MA 620 Intro to Networks & Graph Theory
- MA 623 Stochastic Processes
- MA 627 Combinatorial Analysis
- MA 629 Convex Analysis & Optimization
- MA 635 Real Variables II
- MA 650 Intermediate Partial Differential Equations
- MA 717 Algebraic Topology
- MA 800 Special Problems in Mathematics (MS)
- MA 810 Special Topics in Mathematics

### Master of Science - Stochastic Systems

This program focuses on analysis and optimal decision-making for complex systems involving uncertain data and risk. The program includes courses in statistics, stochastic processes, stochastic optimization, and stochastic optimal control theory. Applications to financial systems, network design and routing, telecommunication systems, medicine, actuarial mathematics, and other areas are discussed. Students are encouraged to apply the techniques they learn to problems derived from their professional work and interests.

Ten courses are required for the degree; six are core courses. Elective courses are chosen with the consent of the student's academic advisor.

**Core Courses**

- MA 547 Advanced Calculus or MA 635 Real Variables I
- MA 611 Probability
- MA 612 Mathematical Statistics
- MA 623 Stochastic Processes
- MA 629 Convex Analysis and Optimization
- MA 661 Stochastic Optimal Control and Dynamic Programming

**Typical Electives**

- MA 615 Numerical Analysis I
- MA 630 Numerical Methods of Optimization
- MA 632 Game Theory
- MA 635 Real Variables I
- MA 641 Time Series Analysis I
- MA 655 Optimal Control
- MA 662 Stochastic Programming
- MA 720 Multivariate Statistics
- MA 712 Mathematical Models of Risk
- MA 800 Special Problems in Mathematics (MS)
- CS 652 Reliability Theory

**Graduate Certificate Programs**

The Mathematical Science department offers a number of graduate certificate programs. Each program consists of four courses, including one elective chosen with the consent of the departmental advisor. Most courses may be used toward a master's degree, as well as for the certificate.

The admissions requirements for certificate programs are described above. The course requirements are listed below.

**Applied Statistics**

MA 552 Linear Algebra

MA 611 Probability

MA 612 Mathematical Statistics

Typical Electives:

CE 679 Regression and Stochastic Methods

MA 641 Time Series Analysis I

MGT 718 Multivariate Analysis

MGT 730 Design and Analysis of Experiments

**Stochastic Systems**

MA 611 Probability

MA 623 Stochastic Process

MA 629 Convex Analysis and Optimization

Typical Electives:

MA 612 Mathematical Statistics

MA 630 Numerical Methods of Optimization

MA 661 Stochastic Optimal Control and Dynamical Programming

MA 662 Stochastic Programming

**Doctoral Program**

The primary requirement for a doctoral degree in mathematics is that you produce a dissertation containing an original and significant result in mathematics. You will work under the guidance of a faculty advisor who is an expert in your area of research.

Preparation for dissertation work includes both courses in mathematical fundamentals and practice in communicating mathematics orally and in writing. The courses you take will not necessarily include everything you will need to know. As a doctoral student you will be expected to learn some mathematics on your own outside of class. Seminars afford a means to that end. They can be organized informally among students or more formally with a faculty advisor. Seminars of the latter type may be taken for academic credit. Students are encouraged to identify subjects they would like to study and to seek out faculty advisors.

**Degree Requirements**

**Coursework and Credits:**

The doctoral program requires 84 credits beyond the bachelor's degree (54 credits beyond the master's) of which at least 30 credits must be approved doctoral research credits (MA 960). This credit total includes the three- credit "Signature" course PRV961. A prior master's degree may be transferred for up to 30 credits without specific course descriptions with approval of the department and the Dean of Graduate Academics. Up to one-third of additional course credits may be transferred with the approval of the thesis committee and the Dean of Graduate Academics. The grade of "B" (3.0 GPA) or better is required for such courses and such courses may not have been already used to obtain an academic degree.

**General Exam.**This is a written exam covering three subject areas, one of which must be real analysis. The standard choices for the two remaining subjects are algebra and complex analysis. Other subject choices require approval of the Graduate Program Committee. The exam is offered once each semester. The General Exam should be attempted within the first year of completing the masters degree or within the first year of study for students entering with a prior masters degree. One failure of the General Exam is allowed. A second failure will result in the student being dropped from the Ph.D. program. After passing the General Exam a student is officially recognized as a Ph.D. candidate.**Dissertation Advisory Committee.**Within six months of becoming a doctoral candidate, the student needs to select a Research Advisor and agree upon a research topic. The Research Advisor assists with recommending additional members for the student's Dissertation Advisory Committee.See the catalog section on Graduate Programs (General Information) for further details on the requirements and process for approving the Advisory Committee.**Dissertation.**The final and most important step of the Ph.D. program is writing a dissertation of publishable quality. This will embody the results of the student's original research in mathematics, and the dissertation will be presented by the student at a public defense. If the suitably appointed Dissertation Committee approves the defense, the student will be recommended to the Office of Graduate Admissions for the Ph.D. degree.

### Mathematical Sciences Department

Alexei Miasnikov, Director