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Applied Mathematics M.S. Program

This is a two-year professional degree program designed to prepare students in the mathematical sciences for a career in contemporary industry or business. The students receive thorough training in applied mathematics and scientific computing, exposure to mathematics-related subjects in science and engineering, and experience in a group project. The program's graduates have been successful in securing desirable positions with companies ranging from small, local firms to large, international corporations.

Director: Markos Katsoulakis

Current Students:



Program goals and structure

The Master's Degree Program in Applied Mathematics is specially designed to prepare graduates for a successful career in today's industrial/business world. Accordingly, the program is structured into the following three components:

The graduate courses in the Department concentrate on Analytical Methods, Numerical Methods, and Probability/Statistics. These two-semester courses sequences give the student a thorough background in advanced applied mathematics.

The elective courses outside the Department are determined depending on each student's interests and preparation. In recent years, they have been chosen from Computer Science, Engineering (Industrial, Mechanical, Electrical), Physics, and Management Science. These courses expose the student to the use of practical mathematical tools by scientists and engineers.

The group project is the most novel component of this program. It is intended to emulate industrial teamwork on a large, technical problem. Through the combined efforts and diverse talents of the group members, a mathematical model is developed, a computer code is implemented, and a final report is written. In the process, the students learn how to start solving a new and hard problem, how to make a professional presentation of their work, and how to collaborate effectively with their coworkers.



Student experiences and employment

How is student life in the program? A comradery develops naturally among the students through their common coursework and the group project class, which meets weekly as a seminar and requires joint work outside class. There are around ten students in the program in any given year. This size allows the students and the faculty to interact easily and frequently. Also, the second-year students often share their experiences and contacts with the first-year students.

Where do the graduates go? While a few find the program useful for developing their mathematics prior to pursuing other advanced degrees, most graduates find jobs in industry. Typically, these jobs are in high-technology firms, often falling under the label of software development. Some recent graduates are employed by large, well-known companies: DEC, GTE, Hewlett-Packard, MIT Lincoln Labs, Pfizer. Others work for smaller, local firms, such as Artios (Ludlow, MA) and Amherst Process Instruments (Hadley, MA).

How do the graduates fare on the job market? It appears that many employers prefer to hire candidates having strong mathematical training together with good programming skills, rather than those with other, more specialized, degrees. And, indeed, all the recent graduates from the program have secured good jobs upon completion of the program. Many have received several attractive offers.

What do the recent graduates have to say? Feedback from our graduates underscores the value of program, and the group project in particular, as preparation for the workplace. Here are a few examples of their comments:

A 1993 grad now at Lincoln Labs writes, "One thing I would like to mention is that even though I may be using more computer skills than math, I feel that my math background has helped a lot. When I was hired, my boss told me that she preferred someone with a math background who could program than a computer science major. I think that was the key for me. Also, the applied math group project is a great idea because it teaches you to work as a group and prepares you for the "real world"."

One of our 1994 alumni, who worked for a while at Fuji Capital Markets Corporation before returning to school, recalls, "Well I remember some words from my boss at FCMC : He told me that, 'A good background in mathematics and programming makes you a very valuable person for an Investment bank or a Capital Markets firm like Fuji Capital Markets. Usually people are either one or the other. If you combine both and have good conversational skills, that's exactly what employers are looking for.' And that is exactly what we tried to learn in the Master's Program : get a thorough math background, learn to program and communicate."

An alumna from 1995, now working at GTE, states, "I think the best selling point is the project. Not so much in terms of what the project is about, but rather the fact that you're working in a group where you have to deal with people not getting their portion of the project done, and project management issues. The team aspect is emphasized a lot at GTE and also I think in other companies. The C coding is also very important; even little things like the RCS configuration management tool are good things to talk about (We're using one now called ClearCase, and tho' I didn't know exactly how to use it, at least I knew the principles behind why we needed such a thing.)."

A 1994 graduate reports, "I am currently a Statistician/Quality Engineer for the Hewlett-Packard Company. The Applied Math Program at UMASS was great for me. I found it very flexible. I was able to choose many classes that built on my previous engineering degree. The faculty at UMASS are also very supportive and are available for the students. I never had a problem trying to get advice on any matter. In addition, the faculty realizes that students will be looking for a job after the program and they give the students many opportunities to make contacts and explore various professional paths."

A 1990 alumna who later received her PhD in meteorology writes, "The best thing about the applied math program is the solid theoretical background it gave me and the 'hands-on' application of that knowledge in a project. I also enjoyed enormously the course I took outside the department in fluid dynamics, which set the foundation of my current research work."



Group projects

Each year a group project is completed by all of the current students. In a sense, this project class is the organizing experience for the students in the degree program. In addition, it serves as the thesis component of the M.S. degree. The second-year students are expected to take a leadership role in the project, along with the two faculty members who guide it. The first-year students gradually acquire the skills (as modelers, coders and communicators) that they will use during the next year, when they lead the project. The class meets as a weekly seminar throughout the academic year, although most of the real work occurs outside the classroom.

The projects from recent years are described briefly below.

1992-93: Spectral computations in fluid dynamics
The behavior of a two-dimensional viscous fluid was simulated by a direct numerical computation using a pseudospectral method. First, some simpler one-dimensional codes were written for the Burgers and Korteweg-DeVries equations, and some wave interaction phenomena governed by these equations were studied. Then, the full code for a Navier-Stokes flow in two dimensions was implemented, and various vortex interactions were displayed.


1993-94: Optics analysis
The design of a lens system was tackled using a direct numerical approach based on ray-tracing for the geometrical optics. Optical properties (focussing, magnification) of various instruments (simple telescopes, microscopes) were examied by computing the three-dimensional pencils of rays, without the classical paraxial approximation. Then the aberrations (spherical, coma, astigmatism, ...) were quantified numerically, and an optimization code was used to vary the lens system parameters so as to minimize a given aberration.

Images from this project: 1   


1994-95: Models of convective turbulent diffusion
The steady-state concentration field of a pollutant introduced into a flowing, turbulent atmosphere was analyzed. A finite-difference method (alternating direction implicit) was implemented to solve the variable-coefficient diffusion equation in three dimensions, under a parabolic approximation in which the downstream variable is time-like. The plume formed by a source was computed and displayed graphically for various sheared wind-flow conditions.

Images from this project: 1   


1995-96: Acoustic radiation and propagation
The sound field of a planar generator of general shape and/or mode was calculated by using a surface integral representation of the solution to the governing Helmholtz equation. Comparisons were made with some classical formulas available either in simple, symmetric cases, or in asymptotic regimes. Interesting interference patterns in the sound intensity nearby the radiator were detailed over a range of frequencies and generator characteristics. The directivity of these sound generators was also studied.

Images from this project: 1   


1996-97: Monte-Carlo simulation of turbulent atmospheric diffusion
The physical and mathematical diffusion of particles through a turbulent velocity field was calculated via two methods, a Random Eddy Model and Fourier Spectrum Model. The Random Eddy algorithm simulates a lattice of Rankine vortices; the Fourier Spectrum code utilizes a sum of sine and cosine terms to approximate the stream function of the turbulent velocity field. Spatial correlation experiments were performed to ensure appropriate behavior for the moving particles, as well as parameter choices. Simulations of particle emanation from a smoke stack were also performed.

Images from this project: 1   


1997-98: Macroscopic modelling of traffic flow
Traffic flow is modelled through a hydrodynamic analogy, and the resulting nonlinear hyperbolic partial differential equation is solved numerically. In addition, on-ramps, off-ramps, and bottlenecks are modelled, and these complexities also are implemented in the computer simulation program. Then, in order to model a two lane highway, the concept of lane changing is examined. A model of lane changing from the literature is discussed, and it is argued that this formulation is incorrect. Moreover, a modified lane changing model is presented, and its validity is supported by the results of several simulations, again performed through the numerical solution of the governing differential equation. Finally, in order to illustrate the interrelationship between the effects of ramps and bottlenecks and the process of lane changing, results are presented for simulations which model ramps and bottlenecks along a two lane highway.

Images from this project: 1   


1998-99: Quasi-geostrophic turbulence modelling using pseudospectral methods
The objective of this project is to develop a mathematical model for forcasting atmospheric pressure patterns. The common assumption is made that the atmosphere can be modelled as an incompressible fluid. The laws governing atmospheric pressure changes are then described using a Navier-Stokes equation in a rotating coordinant frame. Solutions to this nonlinear partial differential equation are obtained numerically, by means of pseudospectral method.

Images from this project: 1   2   3   4   5   


1999-00: Modeling and visualizing human movement via mechanics and optimal control


Images from this project: 1   


2000-01: Dynamic control of a multilink mechanical system


Java applets from this project: Multilink system (without desire motion)   Chin up   Stand up   


2001-02: Artificial Neural Networks


2002-03: Modeling of the Kidney and Lungs
The regulation of sodium chloride in the kidney is modeled. Each kidney contains over one million nephrons, the basic functional unit of the kidney. Each nephron regulates the composition of sodium chloride amongst other things. The transport of sodium chloride in the loop of Henle, which is part of the nephron, is modeled with a partial differential equation. Using analysis, the partial differential equation is studied to understand when stable and unstable solutions might occur. The equation is solved numerically using the Lax-Wendroff method. The computed solutions exhibit oscillations in the sodium concentration in time as is predicted by the analysis. This is observed in rats and humans.

The group also worked with a research pulmonologist at Bay State Medical Center looking at the amount of carbon dioxide exhaled in healthy patents and patents with asthma versus time. The group tested different methods for removing the noise from the data(smoothing the data). The group proposed several good methods that the pulmonologist could use in his work.


2003-04: Traffic Flow with Cellular Automata and Kinetic Models
The group looked at different models for simulating traffic flow. The primary model was one based on cellular automata. This model uses a finite set of vehicles with a finite set of rules governing their interaction. The results gave very realistic results. The group showed that one can predict the mean velocity of a collection of vehicles depending on the density. One and two lanes were modeled along with stop lights and ramps.

The group also derived a kinetic model, one between the microscopic cellular automata and the macroscopic partial differential equation. The kinetic model produced solutions similar to the cellular automata which the differential equation is not capable of.


2004-05: Pattern Formation, Tumor Growth and Turing Instability
The spots and stripes which occur on plants and animals is modeled by the group. The model is based on Turing instability or diffusion driven instability. A coupled pair of partial differential equations are used to model the pattern formation. These equations are studied analytically to understand when instabilities will occur. The equations are then solved with a finite difference method numerically in different domains with varying parameters producing spots, stripes and combinations of the two.

A simple model of tumor growth is also proposed. The model is based on the tumor releasing a chemical(TAF) and using this chemical to recruit blood vessels to proliferate in its direction and eventually vascularize the tumor. The model involved a partial differential equation for the chemical and one for the blood vessels. The equations were solved with a finite difference method numerically. The solutions were similar to what is observed.


2005-06: The Google Search Engine and the Mechanics of Human Locomotion
The group worked on two projects. The first project was to model the search engine Google. After people type keywords in Google, it prepares a list of websites associated with those keywords. By applying the power method in numerical analysis, the students were able to simulate the page-rank processing of a network. They wrote a program, called a webcrawler, that crawls the internet site by site to determine how sites are interconnected. This created a network of 60,000 sites containing the Department of Mathematics and Statistics and its connected sites. The students then applied the page-rank algorithm to this network. The students also applied the same algorithm to other topics such as developing a ranking system of US airports that would help determine which airports are the most important according to the numbers of passengers. Once again using the power method and applying it to an actual data set obtained from the Bureau of Transportation, the students concluded that Dallas/Fort Worth International Airport is the most important airport in the US.

The second project focused on running. Undertaking this project involved input from many academic areas. The students first had to understand the physiology of the leg as well as the mechanics of how the leg moves and interacts with various forces during running. As they learned, the running process can be broken down into two phases. The stance phase is the period of time when the foot is still in contact with the ground, and the flight phase is the period of time when the foot and the body are in the air. Each phase is governed by a different set of equations derived from Newton's laws of motion. The stance phase is described by three second-order differential equations while the flight phase is described by the equations for common projectile motion found in physics. The students solved the coupled differential equations for the two phases numerically in order to simulate running.



Applications and admissions

Those wishing to be considered for Fall admission should submit all application materials to the Graduate Admissions Office during the preceding Spring. Applications are reviewed beginning on February 1, with precedence given to those before that date. Later applications are considered provided that openings are available. Applicants are encouraged to visit in person, if possible, to meet the faculty and students in the program.

All applicants are expected to have a strong undergraduate preparation in mathematics, including advanced calculus, linear algebra, and differential equations. Some exposure to computer science and/or scientific computing is also desirable, as is some knowledge of another area of science or engineering. A Bachelor's Degree in Mathematics, however, is not necessary. Students with undergraduate majors in Physics or Engineering, for instance, and with sufficient mathematical background, are encourage to apply.

The program is able to offer a tuition waiver and a stipend to a limited number of students upon admission. This financial support takes the form of a teaching assistantship in the department. The duties of the students in the Master's Degree Program are usually restricted to grading or consulting for an undergraduate course, although instructing in an elementary course is also possible.

For additional information, contact the Program Director Markos Katsoulakis.



Possible interdisciplinary courses

Computer science

CmpSci 515: Introduction to Computer and Network Security
This course provides an introduction to the principles and practice of system and network security. A focus on both fundamentals and practical information will be stressed. The three key topics of this course are cryptography, system security, and network security. Subtopics include ciphers, hashes, key exchange, security services (integrity, availability, confidentiality, etc.), security attacks, vulnerabilities, exploits, countermeasures. Students will make extensive use of a lab for experimenting with security countermeasures. Grades will be determined by class participation, lab work, homework, and exams. Prerequisites include 377 and 453 (or 591E) and a familiarity with Unix. Co-taught with Chris Misra and Jake Cunningham of OIT. 3 credits.

CmpSci 551: Three-Dimensional Animation and Digital Editing
This seminar is dedicated to the production of high quality 3-dimensional computer animation using graphics technology. For example, color 3-D objects are defined and manipulated, digitized images created and altered, and photo-realistic effects and animated sequences produced. Techniques are used to bend and twist shapes around objects or lines, to provide a variety of light and texture, and to trace over images including digitized pictures. The course is directed at production of an informative and approachable ten minute 3-dimensional animated piece. Using computer-generated graphical analogies as well as cartoon caricature, the video is designed to educate and entertain. The class does not have lab facilities for all students interested in this material and thus we limit the class to students who do well on the first assignment. This assignment will be graded and returned to students before the end of the Add/Drop period. Students are cordially invited to attend the first class, the first Tuesday/Thursday of the semester. At that time we will explain the course, what is expected of students and the entry condition. 3 credits.

CmpSci 552: Interactive Multimedia Production
This course explores the potential of high quality interactive authoring tools to develop presentation and training systems. Programming languages within professional presentation and editing packages will be used to create systems capable of presenting graphics, animation, text, sound and music, based on the users requests. Students will learn how to define and manipulate classical techniques such as storyboarding, staging, and interactivity. The course will concentrate on state-of-the-art multimedia composition and presentation techniques and developing small individual projects. The class does not have lab facilities for all students interested in this material and thus we limit the class to students who do well on the first assignment. This assignment will be graded and returned to students before the end of the Add/Drop period. Students are cordially invited to attend the first class, the first Tuesday/Thursday of the semester. At that time we will explain the course, what is expected of students and the entry condition. Prerequisite: CmpSci 551 (591x) - 3D Computer Animation and Digital Editing. Permission of the instructor required; contact: Beverly Woolf 545-4265. 3 credits.

CmpSci 553: Interactive Web Animation
This course teaches basic animation for the Web, interactivity, color theory, design, action scripting, and transitions. Students maintain their own web sites and submit projects every 2 weeks in Flash. Individual as well as, a final project are required. Knowledge of basic Web development, e.g., HTML, Java Script. Prerequisite: CmpSci 391F; CmpSci 551 and CmpSci 552 preferred. Permission of instructor is required. 3 credits.

CmpSci 575: Combinatorics and Graph Theory
This course is a basic introduction to combinatorics and graph theory for advanced undergraduates in computer science, mathematics, engineering and science. Topics covered include: elements of graph theory; Euler and Hamiltonian circuits; graph coloring; matching; basic counting methods; generating functions; recurrences; inclusion-exclusion; and Polya's theory of counting. Prerequisites: mathematical maturity; calculus; linear algebra; strong performance in some discrete mathematics class, such as CmpSci250 or MATH 455. Modern Algebra - MATH 411 - is helpful but not required. 3 credits.

CmpSci 585: Introduction to Natural Language Processing
The field of natural language processing is concerned with practical and theoretical issues that arise in getting computers to perform various tasks with human languages. In this introductory course you will learn about automated techniques for parsing English sentences, tagging words according to their part-of-speech, encoding spelling rules, modeling language semantics, discovering the evolutionary tree of various Roman-alphabet languages, learning to accurately filter junk email, clustering news articles by topic, and extracting from the Web a database of business people who graduated from UMass. Our work will be a combination of learning new algorithms, discussing linguistics, and programming useful systems that operate on real data. Whether you are interested in the intersection between the humanities and computer science, or you want a job at Google, this introductory course will help you on your way. CmpSci 383 is recommend, but not required with permission of the instructor. 3 credits.

CmpSci 591T: Seminar - Introduction to Algorithmics
This course will introduce the methodology and "culture" of algorithmic reasoning and techniques of analysis. The emphasis will be on understanding rather than just learning. The specific topics to be covered and the method of covering them will be geared to the needs of the enrollees. 3 credits.

CmpSci 611: Advanced algorithms
Principles underlying the design and analysis of efficient algorithms. Topics to be covered include: divide-and-conquer algorithms, graph algorithms, matroids and greedy algorithms, randomized algorithms, NP-completeness, approximation algorithms, linear programming. Prerequisites: The mathematical maturity expected of incoming Computer Science graduate students, knowledge of algorithms at the level of CmpSci 311. 3 credits.


Mechanical and industrial engineering

MIE 586 - Quantitative Decision Making
Survey in operations research. Introduction to models and procedures for quantitative analyses of decision problems. Topics include linear programming and extensions, integer programming. Required for IE graduate students who lack operations research exposure.

MIE 605 - Finite Element Analysis
The underlying mathematical theory behind the finite element method and its application to the solution of problems from solid mechanics. Includes a term project involving the application of the finite element method to a realistic and sufficiently complex engineering problem selected by the student and approved by the instructor; requires the use of a commercial finite element code.

MIE 644 - Applied Data Analysis
The basics of data acquisition and analysis, pattern classification, system identification, neural network modeling, and fuzzy systems. Essential to students whose thesis projects involve experimentation and data analysis.

MIE 684 - Stochastic Processes In Industrial Engineering
Introduction to the theory of stochastic processes with emphasis on Markov chains, Poisson processes, markovian queues and networks, and computational techniques in Jackson networks. Applications include stochastic models of production systems, reliability and maintenance, and inventory control.

MIE 707 - Viscous Fluids
Exact solutions to Navier-Stokes flow and laminar boundary layer flow. Introduction to transition and turbulent boundary layers, and turbulence modeling. Boundary layer stability analysis using pertubation methods.


Civil and environmental enginerring

CEE 511 - Traffic Engineering
Fundamental principles of traffic flow and intersection traffic operations including traffic data collection methods, traffic control devices, traffic signal design, and analysis techniques. Emphasizes quantitative and computerized techniques for designing and optimizing intersection signalization. Several traffic engineering software packages used.

CEE 548 - Finite Element Method
Application of numerical methods to solution of problems of structural mechanics. Finite difference techniques and other methods for solution of problems in the vibration, stability, and equilibrium of structural elements.

CEE 605 - Finite Element Analysis
Introduction to finite element method in engineering science. Derivation of element equations by physical, variational, and residual methods. Associated computer coding techniques and numerical methods. Applications.


Management

Sch-Mgmt 640 - Financial Analysis and Decisions
Basic concepts, principles, and practices involved in financing businesses and in maintaining efficient operation of the firm. Framework for analyzing savings-investment and other financial decisions. Both theory and techniques applicable to financial problem solving.

Sch-Mgmt 641 - Financial Management
Internal financial problems of firms: capital budgeting, cost of capital, dividend policy, rate of return, and financial aspects of growth. Readings and case-studies.

Sch-Mgmt 745 Financial Models
Analytical approach to financial management. Emphasis on theoretical topics of financial decision making. Through use of mathematical, statistical, and computer simulation methods, various financial decision making models are made.

Sch-Mgmt 747 - Theory of Financial Markets
In-depth study of portfolio analysis and stochastic processes in security markets. Emphasis on quantitative solution techniques and testing procedures.

Sch-Mgmt 871 - Micro Theory Of Finance
Optimum financial policies and decisions of nonfinancial firms. Theory of competition and optimum asset management of financial firms.




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