# Mathematical Sciences

## Faculty

**B. Vernescu**, Professor and Head; Ph.D., Institute of Mathaematics, Bucharest, Romania, 1989; partial differential equations, phase transitions and free boundaries, viscous flow in porous media, asymptotic methods and homogenization.

**J. Abraham**, Actuarial Mathematics Coordinator; Fellow, Society of Actuaries, 1991; B.S., University of Iowa, 1980.

**M. Blais**, Coordinator of Professional Science Master’s Programs; Ph.D., Cornell University, 2005; mathematical finance.

**D. D. Berkey**, Professor and President; Ph.D., University of Cincinnati, 1974; applied mathematics, differential equations, optimal control.

**P. R. Christopher**, Professor;. Ph.D., Clark University, 1982; graph theory, group theory, algebraic graph theory, combinatorics, linear algebra.

**S. Dai**, Visiting Assistant Professor;. Ph.D., University of Maryland, 2005; Partial differential equations, pattern formation and domain coarsening in materials science, phase transitions and thin liquid films, free boundary problems, numerical analysis.

**P. W. Davis**, Professor; Ph.D., Rensselaer Polytechnic Institute, 1970; unit commitment, optimal power flow, economic dispatch, state estimation, other control and measurement problems for electric power networks.

**W. Farr**, Associate Professor; Ph.D., University of Minnesota 1986; ordinary and partial differential equations, dynamical systems, local bifurcation theory with symmetry and its application to problems involving chemical reactions or fluid mechanics (or a combination of both).

**J. D. Fehribach**, Associate Professor; Ph.D., Duke University, 1985; partial differential equations and scientific computing, free and moving boundary problems (crystal growth), nonequilibrium thermodynamics and averaging (molten carbonate fuel cells).

**J. Goulet**, Coordinator, Master of Mathematics for Educators Program; Ph.D., Rensselaer Polytechnic Institute, 1976; applications of linear algebra, cross departmental course development, project development, K-12 relations with colleges, mathematics of digital and analog sound and music.

**A. C. Heinricher**, Professor; Ph.D., Carnegie Mellon University, 1986; applied probability, stochastic processes and optimal control theory.

**M. Humi**, Professor; Ph.D., Weizmann Institute of Science, 1969; mathematical physics, applied mathematics and modeling, Lie groups, differential equations, numerical analysis, turbulence and chaos.

**C. J. Larsen**, Associate Professor; Ph.D., Carnegie Mellon University, 1996; variational problems from applications such as optimal design, fracture mechanics, and image segmentation, calculus of variations, partial differential equations, geometric measure theory, analysis of free boundaries and free discontinuity sets.

**R. Y. Lui**, Professor; Ph.D., University of Minnesota, 1981; mathematical biology, partial differential equations.

**K. A. Lurie**, Professor; Ph.D. (1964), D.Sc. (1972), A. F. Ioffe Physical-Technical Institute, Academy of Sciences of the USSR, Russia; control theory for distributed parameter systems, optimization and nonconvex variational calculus, optimal design.

**W. J. Martin**, Associate Professor; Ph.D., University of Waterloo, 1992; algebraic combinatorics, applied combinatorics.

**U. Mosco**, H. J. Gay Professor; Libera Docenza, University of Rome, 1967; partial differential equations, convex analysis, optimal control, variational calculus, fractals.

**B. Nandram**, Professor; Ph.D., University of Iowa, 1989; survey sampling theory and methods, Bayes and empirical Bayes theory and methods, categorical data analysis.

**J. D. Petruccelli**, Professor; Ph.D., Purdue University, 1978; time series (nonlinear models), optimal topping (best choice problems), statistics.

**M. Sarkis**, Associate Professor; Ph.D., Courant Institute of Mathematical Sciences, 1994; domain decomposition methods, numerical analysis, parallel computing, computational fluid dynamics, preconditioned iterative methods for linear and non-linear problems, numerical partial differential equations, mixed and non-conforming finite methods, overlapping non-matching grids, mortar finite elements, eigenvalue solvers, aeroelasticity, porous media reservoir modeling.

**H. Sayit**, Assistant Professor; Ph.D., Cornell University, 2005; stochastic optimization, stochastic differential equations, statistical estimation and inference, financial mathematics, computational finance.

**B. Servatius**, Professor; Ph.D., Syracuse University, 1987; combinatorics, matroid and graph theory, structural topology, geometry, history and philosophy of mathematics.

**D. Tang**, Professor; Ph.D., University of Wisconsin, 1988; biofluids, biosolids, blood flow, mathematical modeling, numerical methods, scientific computing, nonlinear analysis, computational fluid dynamics.

**B. S. Tilley**, Associate Professor; Ph.D., Northwestern University, 1994; free-boundary problems in continuum mechanics, interfacial fluid dynamics, viscous flows, partial differential equations, mathematical modeling, asymptotic methods.

**D. Vermes**, Associate Professor; Ph.D., University of Szeged, Hungary, 1975; optimal stochastic control theory, nonsmooth analysis, stochastic processes with discontinuous dynamics, adaptive optimal control in medical decision making, massively parallel data analysis and simulation, portfolio risk management, financial mathematics.

**D. Volkov**, Assistant Professor; Ph.D., Rutgers University, 2001; electromagnetic waves, inverse problems, wave propagation in waveguides and in periodic structures, electrified fluid jets.

**H. F. Walker**, Professor; Ph.D., Courant Institute of Mathematical Sciences, New York University, 1970; numerical analysis, especially numerical solution of large-scale linear and nonlinear systems, unconstrained optimization, applications to ordinary and partial differential equations and statistical estimation, computational and applied mathematics.

**S. Weekes**, Associate Professor and Associate Department Head; Ph.D., University of Michigan, 1995; numerical analysis, computational fluid dynamics, porous media flow, hyperbolic conservation laws, shock capturing schemes.

**Z. Wu**, Assistant Professor; Ph.D., Yale University, 2009; Biostatistics, high-dimensional model selection, linear and generalized linear modeling, statistical genetics, bioinformatics.

**V. Yakovlev**, Research Associate Professor; Ph.D., Institute of Radio Engineering and Electronics, Russian Academy of Sciences, 1991; antennas for MW and MMW communications, electromagnetic fields in transmission lines and along media interfaces, control and optimization of electromagnetic and temperature fields in microwave thermal processing, issues in modeling of microwave heating, computational electromagnetics with neural networks, numerical methods, algorithms and CAD tools for RF, MW and MMW components and subsystems.

### Emeritus

**G. C. Branche**, Professor

**E. R. Buell**, Professor

**V. Connolly**, Professor

**W. J. Hardell**, Professor

**J. J. Malone**, Professor

**B. C. McQuarrie**, Professor

**W. B. Miller**, Professor

## Research Interests

Active areas of research in the Mathematical Sciences Department include applied and computational mathematics, industrial mathematics, applied statistics, scientific computing, numerical analysis, ordinary and partial differential equations, non-linear analysis, electric power systems, control theory, optimal design, composite materials, homogenization, computational fluid dynamics, biofluids, dynamical systems, free and moving boundary problems, porous media modeling, turbulence and chaos, mathematical physics, mathematical biology, operations research, linear and nonlinear programming, discrete mathematics, graph theory, group theory, linear algebra, combinatorics, applied probability, stochastic processes, time series analysis, Bayesian statistics, Bayesian computation, survey research methodology, categorical data analysis, Monte Carlo methodology, statistical computing, survival analysis and model selection.

## Programs of Study

The Mathematical Sciences Department offers four programs leading to the degree of master of science, a combined B.S./ Master’s program, a program leading to the degree of master of mathematics for educators, and a program leading to the degree of doctor of philosophy.

### Master of Science in Applied Mathematics Program

This program gives students a broad background in mathematics, placing an emphasis on areas with the highest demand in applications: numerical methods and scientific computation, mathematical modeling, discrete mathematics, mathematical materials science, optimization and operations research. In addition to these advanced areas of specialization, students are encouraged to acquire breadth by choosing elective courses in other fields that complement their studies in applied mathematics. Students have a choice of completing their master’s thesis or project in cooperation with one of the department’s established industrial partners. The program provides a suitable foundation for the pursuit of a Ph.D. degree in applied mathematics or a related field, or for a career in industry immediately after graduation.

### Master of Science in Applied Statistics Program

This program gives graduates the knowledge and experience to tackle problems of statistical design, analysis and control likely to be encountered in business, industry or academia. The program is designed to acquaint students with the theory underlying modern statistical methods, to provide breadth in diverse areas of statistics and to give students practical experience through extensive application of statistical theory to real problems.

Through the selection of elective courses, the student may choose a program with an industrial emphasis or one with a more theoretical emphasis.

### Professional Master of Science in Financial Mathematics Program

This program offers an efficient, practice-oriented track to prepare students for quantitative careers in the financial industry, including banks, insurance companies, and investment and securities firms. The program gives students a solid background and sufficient breadth in the mathematical and statistical foundations needed to understand the cutting edge techniques of today and to keep up with future developments in this rapidly evolving area over the span of their careers. It also equips students with expertise in quantitative financial modeling and the computational methods and skills that are used to implement the models. The mathematical knowledge is complemented by studies in financial management, information technology and/or computer science.

The bridge from the academic environment to the professional workplace is provided by a professional master’s project that involves the solution of a concrete, real-world problem directly originating in the financial industry. Students are encouraged to complete summer internships at financial firms. The department may help students to find suitable financial internships through the industrial connections of faculty affiliated with the Center for Industrial Mathematics and Statistics. Graduates of the program are expected to start or advance their professional careers in such areas as financial product development and pricing, risk management, investment decision support and portfolio management.

### Professional Master of Science in Industrial Mathematics Program

This is a practice-oriented program that prepares students for successful careers in industry. The graduates are expected to be generalized problem-solvers, capable of moving from task to task within an organization. In industry, mathematicians need not only the standard mathematical and statistical modeling and computational tools, but also knowledge within other areas of science or engineering. This program aims at developing the analytical, modeling and computational skills needed by mathematicians who work in industrial environments. It also provides the breadth required by industrial multidisciplinary team environments through courses in one area of science or engineering, e.g., physics, computer science, mechanical engineering, and electrical and computer engineering.

The connection between academic training and industrial experience is provided by an industrial professional master’s project that involves the solution of a concrete, real-world problem originating in industry. The department, through the industrial connections of the faculty affiliated with the Center for Industrial Mathematics and Statistics, may help students identify and select suitable industrial internships. Graduates of the program are expected to start or advance their professional careers in industry.

### Master of Mathematics for Educators

This is an evening program designed primarily for secondary school mathematics teachers. Courses offer a solid foundation in areas such as geometry, algebra, modeling, discrete math and statistics, while also including the study of modern applications. Additionally, students develop materials, based on coursework, which may be used in their classes. Technology is introduced when possible to give students exposure for future consideration. Examples include Geometer’s Sketchpad; Maple for algebra, calculus and graphics; Matlab for analysis of sound and music; and the TI CBL for motion and heat.

### Doctor of Philosophy in Mathematical Sciences Program

The goal of this program is to produce active and creative problem solvers, capable of contributing in academic and industrial environments. One distinguishing feature of this program is a Ph.D. project to be completed under the guidance of an external sponsor, e.g., from industry or a national research center. The intention of this project is to connect theoretical knowledge with relevant applications and to improve skills in applying and communicating mathematics.

### Combined B.S./Master’s Program

This program allows a student to work concurrently toward bachelor and master of science degrees in applied mathematics, applied statistics, financial mathematics and industrial mathematics.

## Admission Requirements

A bachelor’s degree is required for admission to all M.S. programs. A basic knowledge of undergraduate analysis, linear algebra and differential equations is assumed for applicants to the master’s programs in applied mathematics and industrial mathematics. A strong background in mathematics, which should include courses in undergraduate analysis and linear algebra, is assumed for applicants to the master’s program in financial mathematics. Typically, an entering student in the master of science in applied statistics program will have an undergraduate major in the mathematical sciences, engineering or a physical science; however, individuals with other backgrounds will be considered. In any case, an applicant will need a strong background in mathematics, which should include courses in undergraduate analysis and probability. Students with serious deficiencies may be required to correct them on a noncredit basis.

Candidates for the master of mathematics for educators degree must have a bachelor’s degree and must possess a background equivalent to at least a minor in mathematics, including calculus, linear algebra, and statistics. Students are encouraged to enroll in courses on an ad hoc basis without official program admission. However, (at most) four such courses may be taken prior to admission.

## Degree Requirements

### For the M.S. in Applied Mathematics

The master’s program in applied mathematics is a 30-credit-hour program. The student’s program must include at least seven MA numbered courses other than 501 or 511. Among these must be MA 503, MA 510, and either MA 535 or MA 530. In addition, students are required to complete a Capstone Experience, which can be satisfied by one of the following options:

- A six credit master’s thesis.
- A three to six credit master’s project.
- A three credit master’s practicum.
- A three credit research review report or research proposal.
- A master’s exam.

The master’s thesis is an original piece of mathematical research work which focuses on advancing the state of the mathematical art. The master’s project consists of a creative application of mathematics to a real-world problem. It focuses on problem definition and solution using mathematical tools. The master’s practicum requires a student to demonstrate the integration of advanced mathematical concepts and methods into professional practice. This could be done through a summer internship in industry or an applied research laboratory.

The remaining courses may be chosen from the graduate offerings of the Mathematical Sciences Department. Upper-level undergraduate mathematics courses or a two-course graduate sequence in another department may be taken for graduate credit, subject to the approval of the departmental Graduate Committee. Candidates are required to successfully complete the graduate seminar MA 560.

### For the M.S. in Applied Statistics

The master’s program in applied statistics is a 30-credit-hour program. Courses taken must include MA 540, MA 541, MA 546, MA 547, 3 credits of MA 559 and at least three additional departmental statistics offerings: MA 509 and courses numbered 542 through 556. Students who can demonstrate a legitimate conflict in scheduling MA 559 will be assigned an alternative activity by the Mathematical Science Department Graduate Committee. In addition the student must complete a Capstone Experience, which can be satisfied by one of the following options:

- A six credit master’s thesis.
- A three to six credit master’s project.
- A three credit master’s practicum.
- A three credit research review report or research proposal.
- A master’s exam.

Upper-level undergraduate courses may be taken for graduate credit subject to the approval of the departmental Graduate Committee.

### For the M.S. in Financial Mathematics

The professional M.S. Degree Program in Financial Mathematics is a 30-credit hour program. The curriculum consists of the following components:

**1. 6 credits from required foundation courses:**

- MA 503 Analysis I or MA529 Stochastic Processes
- MA 540 Probability and Mathematical Statistics I

**2. 12 credits from core financial mathematics courses:**

- MA 571 Financial Mathematics I
- MA 572 Financial Mathematics II
- MA 573 Computational Methods of Financial Mathematics
- MA 574 Portfolio Valuation and Risk Management
- MA 575 Market and Credit Risk Management

**3. 3 credits chosen from Mathematical Sciences graduate courses MA 502-590. **

- BS/MS students can count undergraduate credits MA 4213 Risk Theory, MA 4235 Mathematical Optimization, MA 4237 Probabilistic Methods in Operations Research, MA 4473 Partial Differential Equations, MA 4632 Probability and Mathematical Statistics II towards electives

**4. 6 credit block in one of the following complementary areas outside of the Mathematical Sciences Department: Financial Management, Information Technology, or Computer Science.**

- Students with a degree or substantial work experience in one of the above complementary areas can substitute them with other courses subject to prior approval by the graduate committee
- BS/MS students can count suitable undergraduate courses towards the complementary area requirement according the number of credits of the corresponding graduate courses
- 2 of the complementary area credits can be earned by taking MA579 Financial Programming Workshop

**5. 3 graduate credits for a project originating in the financial industry**

**6. MA562A and MA562B Professional Master’s Seminar** (for no credit)

### For the M.S. in Industrial Mathematics

The professional master’s degree program in industrial mathematics is a 30-credit-hour program. Students must complete four foundation courses: MA 503, MA 510 and two courses out of MA 508, MA 509, MA 529 and MA 530. Students must also complete a 12-credit-hour module composed of two courses within the department and a sequence of two courses from one graduate program outside the Mathematical Sciences Department. The department offers a wide selection of modules to suit students’ interest and expertise.

In addition, students are required to complete a 3-credit-hour elective from the Mathematical Sciences Department and a 3-credit-hour master’s project on a problem originating from industry. Candidates are required to successfully complete the Professional Master’s Seminars MA 562A and MA 562B. The Plan of Study and the project topic require prior approval by the departmental Graduate Committee.

#### Examples of Modules for the M.S. Degree in Industrial Mathematics

The courses comprising the 12-credit module should form a coherent sequence that provides exposure to an area outside of mathematics and statistics, providing at the same time the mathematical tools required by that particular area. Examples of typical modules are:

- Dynamics and control module—MA 512, MA 540, ME 522 and ME 523 or ME 527;
- Materials module—MA 512, MA 526, and ME 531;
- Fluid dynamics module—MA 512, MA 526, ME 511 and ME 512 or ME 513;
- Biomedical engineering module—MA 512, MA 526, BE/ME 554 and BE/ME 558;
- Machine learning module—MA 540, MA 541, CS 509 and CS 539;
- Cryptography module—MA 533, MA 514, CS 503 and ECE 578.

### For the Combined B.S./ Master’s Programs in Applied Mathematics and Applied Statistics

A maximum of four courses may be counted toward both the undergraduate and graduate degrees. All of these courses must be 4000-level or above, and at least one must be a graduate course. Three of them must be beyond the 7 units of mathematics required for the B.S. degree. Additionally, students are advised that all requirements of a particular master’s program must be satisfied in order to receive the degree, and these courses should be selected accordingly.

Acceptance into the program means that the candidate is qualified for graduate school and signifies approval of the four courses to be counted for credit toward both degrees. However, in order to obtain both undergraduate and graduate credit for these courses, grades of B or better have to be obtained.

### For the Master of Mathematics for Educators (M.M.E.)

Candidates for the master of mathematics for educators must successfully complete 30 credit hours of graduate study, including a 6-credit-hour project (see MME 592, MME 594, MME 596). This project will typically consist of a classroom study within the context of a secondary mathematics course and will be advised by faculty in the Mathematical Sciences Department. Typically, a student will enroll in 4 credit hours per semester during the fall and spring, with the remaining credit hours taken in the summer.

Students may complete the degree in as little as slightly over two years by taking two courses per semester, 3 semesters per year, and doing a project. However, the program can accommodate other completion schedules as well. The MME degree may be used to satisfy the Massachusetts Professional License requirement, provided the person holds an Initial License.

### For the Ph.D.

The course of study leading to the doctor of philosophy in mathematical sciences requires the completion of at least 90 credit hours beyond the bachelor’s degree or at least 60 credit hours beyond the master’s degree, as follows:

General Courses (credited for students with master’s degrees) - 30 credits

Research Preparation Phase - 24-30 credits

- Research-Related Courses or Independent Studies - 9-18 credits
- Ph.D. Project 1-9 credits
- Extra-Departmental Studies 6 credits

Dissertation Research - at least 30 credits

A brief description of other Ph.D. program requirements follows below. For further details, students are advised to consult the document *Ph.D. Program Requirements and Administrative Rules for the Department of Mathematical Sciences*, available from the departmental graduate secretary.

Within a full-time student’s first semester of study (second semester for part-time students), a Plan of Study leading to the Ph.D. degree must be submitted to the departmental Graduate Committee for review and approval. The Plan of Study may subsequently be modified with review by the departmental Graduate Committee.

#### Extra-Departmental Studies Requirement

A student must complete at least six semester hours of courses, 500 level or higher, in WPI departments other than the Mathematical Sciences Department.

#### General Comprehensive Examination

A student must pass the general comprehensive examination (GCE) in order to become a Ph.D. candidate. The purpose of the GCE is to determine whether a student possesses the fundamental knowledge and skills necessary for study and research at the Ph.D. level. It is a written examination normally offered twice a year, once in January and once in August. A full-time student must make the first attempt within one year (two years for part-time students) of entering the Ph.D. program. Students entering with master’s degrees are encouraged to take the GCE as early as they can.

#### Mathematical Sciences Ph.D. Project

A student must complete a Ph.D. project involving a problem originating with a sponsor external to the department. The purposes of the project are to broaden perspectives on the relevance and applications of mathematics and to improve skills in communicating mathematics and formulating and solving mathematical problems. Students are encouraged to work with industrial sponsors on problems involving applications of the mathematical sciences. Each Ph.D. project requires prior approval by the project advisor, the external sponsor, and the departmental Graduate Committee.

#### Ph.D. Preliminary Examination

Successful completion of the preliminary examination is required before a student can register for dissertation research credits. The purpose of the preliminary examination is to determine whether a student’s understanding of advanced areas of mathematics is adequate to conduct independent research and successfully complete a dissertation. The preliminary examination consists of both written and oral parts. A full-time student must make the first attempt by the end of his or her third year (sixth year for part-time students) in the Ph.D. program.

#### Ph.D. Dissertation

The Ph.D. dissertation is a significant work of original research conducted under the supervision of a dissertation advisor, who is normally a member of the departmental faculty. The dissertation advisor chairs the student’s dissertation committee, which consists of at least five members, including one recognized expert external to the department, and which must be approved by the departmental Graduate Committee. At least six months prior to completion of the dissertation, a student must submit a written dissertation proposal and present a public seminar on the research plan described in the proposal. The proposal must be approved by the dissertation committee. Upon completion of the dissertation and other program requirements, the student presents the dissertation to the dissertation committee and to the general community in a public oral defense. The dissertation committee determines whether the dissertation is acceptable.

## Mathematical Sciences Computer Facilities

The Mathematical Sciences Department makes up-to-date computing equipment available for use by students in its programs.

Current facilities include a mixed environment of approximately 85 Windows, Linux/Unix and Macintosh workstations utilizing the latest in single- and dual-processor 32 and 64 bit technology. Access is available to our supercomputer, a 16 CPU SGI Altix 350. The Mathematical Sciences Department also has 3 state-of-the-art computer labs, one each dedicated to the Calculus, Statistics, and Financial Mathematics programs.

The department is continually adding new resources to give our faculty and students the tools they need as they advance in their research and studies.

### Center for Industrial Mathematics and Statistics (CIMS)

The Center for Industrial Mathematics and Statistics was established in 1997 to foster partnerships between the university and industry, business and government in mathematics and statistics research.

The problems facing business and industry are growing ever more complex, and their solutions often involve sophisticated mathematics. The faculty members and students associated with CIMS have the expertise to address today’s complex problems and provide solutions that use relevant mathematics and statistics.

The Center offers undergraduates and graduate students the opportunity to gain real-world experience in the corporate world through projects and internships that make them more competitive in today’s job market. In addition, it helps companies address their needs for mathematical solutions and enhances their technological competitiveness.

The industrial projects in mathematics and statistics offered by CIMS provide a unique education for successful careers in industry, business and higher education.