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# Mathematical Sciences

The Mathematical Sciences Program is designed to provide broad training in basic mathematics together with some specialization in an area of application of mathematics. Each student must select one of the eight Program Options as a special area:

Because of the somewhat more specific requirements of the Program options, careful planning and frequent consultation with your advisor are essential to ensure timely completion of the program.

The prerequisite to the major in Mathematical Sciences is one of the sequences 215 & 217, 255 & 217, 285 & 217, or 295 & 296. In addition, students must acquire a working knowledge of a high-level computer language (*e.g.* Fortran, C, or C++) at a level equivalent to the completion of EECS 183. Physics 140-141 and 240-241 are required for the Numerical and Applied Analysis and Mathematical Physics options and strongly recommended for the other options. Some of the options have additional requirements as noted below.

The major program must include at least nine courses:** **four basic courses, three courses from one of the Program Options, and** **two additional courses as described below. **At least two of the five (optional and additional) courses must be MATH courses.**

### Basic Courses

The basic courses consist of one from each of the following four groups **completed with a grade of at least C-**:

**Differential Equations:**Math 256, 286, or 316**Discrete Mathematics/Modern Algebra:**Math 312, 412, 465 or 493**Analysis:**Math 351, 354, 450, 451, or 454**Probability:**Math 425 or 525

More advanced students, such as those who have completed Math 396, may substitute higher level courses with the approval of an advisor. All students are strongly encouraged to include in their program one of the more theoretical courses: Math 412, 451, 493, 494, or 525.

### Program Options

A Mathematical Sciences major must choose one of the eight options below and complete at least three of the courses listed under that option or courses for which the courses are prerequisite.This requirement is designed to provide focus and depth to the program and can only be waived by an advisor in favor of a program which provides this depth in some equivalent way. An acceptable program must include some of the more difficult courses. Advice should be sought from an advisor before selecting an option. As an initial guide we give a brief description of the options.

**Discrete and Algorithmic Methods**

Discrete and algorithmic methods are concerned with the analysis of finite structures such as graphs, networks, codes, incidence structures, and combinatorial structures. The rapid growth of this area has been driven largely by its role as the mathematical core of computer science. Typical problems of this field involve optimization, emulation, or probabilistic estimation. Students choosing this option are encouraged to take EECS 280 and 281.

416 Theory of Algorithms | 420 Matrix Algebra II |

475 Elementary Number Theory | 481 Intro. to Mathematical Logic |

561 Linear Programming I | 565 Combin. and Graph Theory |

566 Combinatorial Theory | 567 Intro. to Coding Theory |

575 Intro. to Theory of Numbers | EECS 376 Found. of Computer Sci. |

EECS 477 Intro. to Algorithms | EECS 550 Information Theory |

EECS 574 Comput. Complexity | EECS 586 Design/Anal. Algorithms |

EECS 587 Parallel Computing | IOE 614 Integer Programming |

**Numerical and Applied Analysis**

As computers become more powerful, they are being used to solve increasingly complex problems in science and technology. Examples of such problems include developing high-temperature superconducting materials, determining the structure of a protein from its amino acid sequence, and creating methods to model global climate change. Industrial and government research laboratories require personnel who are trained in applying numerical and analytical techniques to solve such problems. Numerical techniques are algorithms for computer simulation, and analytical techniques may rely on series expansions such as the Taylor or Fourier series expansions. There is a close connection between numerical and analytical techniques. A new analytical technique often leads to more effective numerical algorithms; a good example is the development of wavelets and their applications in signal processing. Students wishing to enter this field must acquire a strong background in mathematics, science, and computing. Students are encouraged to include EECS 283 and MATH 451 in their program, and to also consider doing a minor in another scientific discipline.

Students in this program may choose to pursue a dual major in Informatics (especially the Data Mining and Information Analysis track). This combination is a powerful recommendation to a prospective employer that the student can think quantitatively about information; collect, manage, analyze, and visualize massive datasets; and that the student has both the computational tools and the rigorous analytical methods to reason about information.

354 Fourier Analysis and its Applic. | 404 Intermediate Diff. Equations |

420 Advanced Linear Algebra | 423 Mathematics of Finance |

452 Advanced Calculus II | 454 Bound. Val. Prob. for PDE |

462 Mathematical Models | 463 Math Modeling in Biology |

464 Inverse Problems | 471 Intro. to Numerical Methods |

550 Intro. to Adaptive Systems | 555 Intro. to Complex Variables |

AERO 225 Intro. to Gas Dynamics | EECS 283 Prog. for Sci. and Eng. |

ME 240 Dynam. and Vibrations | PHYS 340/341 Waves, Heat, and Light |

PHYS 401 Intermed. Mechanics | STAT 406 Statistical Computing |

STAT 426 Intro. to Theor. Stat. |

**Operations Research and Modeling**

Mathematical modeling refers generally to the representation of real-world problems in mathematical terms. In some sense this is necessary for any application of mathematics, but the term is used more often to refer to the more recent applications of mathematics to biological, mechanical, and human systems. Analysis of such systems involves complex mathematical descriptions and leads to large problems which can be solved only by use of a computer. Operations Research studies integrated systems including health care, education, manufacturing processes, finance, and transportation. Because the emphasis is on the analysis and operation of systems, practitioners are also qualified to deal with managerial problems. Career opportunities are available in many parts of industry and government. Most students should include Math 561 and Stat 426.

420 Advanced Linear Algebra | 433 Intro. to Differential Geom. |

462 Mathematical Models | 463 Math Modeling in Biology |

561 Linear Programming I | 562 Contin. Optimization Meth. |

CHE 510 Math. Methods in ChemE | IOE 515 Stochastic Processes |

IOE 543 Scheduling | IOE 610 Linear Programming II |

IOE 611 Nonlinear Programming | IOE 612 Network Flows |

IOE 614 Integer Programming | Stat. 426 Intro. to Theor. Stat. |

**Probabilistic Methods**

Probability theory deals with the mathematics of randomness and its applications. It is the basis of mathematical statistics, where the goal is to draw inferences from samples. Non-statistical applications are found in many branches of the social, biological, and physical sciences, as well as in engineering. Because of the intimate connection between probability and statistics, students electing this option will usually include statistics courses in their program and **sometimes have a dual major in Statistics**. Students electing this option must complete Math 525.

Students in this program may choose to pursue a dual major in Informatics (especially the Data Mining and Information Analysis track). This combination is a powerful recommendation to a prospective employer that the student can think quantitatively about information; collect, manage, analyze, and visualize massive datasets; and that the student has both the computational tools and the rigorous analytical methods to reason about information.

423 Mathematics of Finance | 523 Risk Theory |

525 Probability Theory | 526 Discr. State Stoch. Proc. |

547 Biological Sequence Anal. | EECS 502 Stochastic Processes |

Stat. 406 Intro. to Stat. Computing | Stat. 426 Intro. to Theor. Stat. |

Stat. 430 Applied Probability | Stat. 466 Stat. Quality Control |

Stat. 500 Applied Statistics I | Stat. 501 Applied Statistics II |

Stat. 550 Bayesian Decision Anal. |

**Mathematical Economics**

One definition of economics is the study of the optimal allocation of scarce resources. Several mathematical techniques are fundamental to this study: constrained optimization using Lagrange multipliers, n-dimensional calculus, especially the Implicit Function Theorem (dependence of a solution on parameters), dynamics, probability and statistics to deal with inherent uncertainty, game theory to deal with decisions in which the actions of one agent affect the options of others, and proofs for understanding the derivation of economic principles. To ensure coverage of these topics, students choosing the Mathematical Economics option will usually choose Math 351 or 451 as their basic analysis course; Math 423, Stat 426, or Econ 452 as courses from the options list; and Econ 401 and a mathematics course at the 400-level or above as their related courses. A student who completes this option should find opportunities available in business, government, and research organizations which collect, analyze, and model data having economic, social, and political parameters. **Many students in this program pursue a dual major in Economics**; this combination is a powerful recommendation to a prospective employer that the student can think quantitatively, understand complex reasoning, and work with economic models.

420 Advanced Linear Algebra | 423 Mathematics of Finance |

424 Compound Interest & Life Ins. | 452 Advanced Calculus II |

462 Mathematical Models | 471 Intro. to Numerical Methods |

472 Num. Meth. with Fin. App. | 523 Risk Theory |

561 Linear Programming I | 562 Cont. Optimization Meth. |

623 Computational Finance | Econ 409 Game Theory |

Stat. 426 Intro. to Theor. Stat. | ECON 452 Int. Intro to Stats and Econometrics II |

**Control Systems**

Control Systems is a fascinating field which draws upon knowledge in many areas of mathematics. It pervades industry, and its practitioners can be found in such diverse fields as automotive pollution control, avionics, and process control in manufacturing. A control designer will need to interface with the modeling group to develop a mathematical description of the system to be controlled, and perhaps with the testing group to characterize disturbances or other uncertainties affecting the system. The required performance of the system will then be ascertained from the intended use and translated into a set of mathematical specifications for a closed-loop system. At this stage the designer will select from an arsenal of tools for the controller analysis and synthesisâ€”this generally requires a solid foundation in linear algebra, differential equations, real analysis, and probability. Students planning to pursue graduate study in this area are recommended to include Math 451 and EECS 476 in their programs.

354 Fourier Analysis and its Applic. | 420 Advanced Linear Algebra |

451 Advanced Calculus I | 454 Bound. Val. Prob. for PDE |

462 Mathematical Models | 471 Intro. to Numerical Methods |

555 Intro. to Complex Variables | 561 Linear Programming I |

562 Cont. Optimization Meth. | EECS 376 Found. of Computer Sci. |

EECS 460 Control Sys. Anal./Des. | EECS 476 Theor. of Internet App. |

EECS 560 Linear Systems Theory | EECS 561 Digital Control Sys. |

EECS 562 Nonlinear Systems | EECS 565 Linear Feedback |

EECS 567 Intro. to Robotics | Stat. 426 Intro. to Theor. Stat. |

**Mathematical Physics**

Among all of the disciplines which make significant use of mathematics, physics has the longest history. Indeed, several areas of mathematics were developed for the purpose of solving problems in physics. This option allows a student to pursue interests in physics which use undergraduate mathematics. **It is designed to facilitate a concurrent major in Physics.** Every program should include at least two of the Physics courses from the list below. Note that although Physics 401 is prerequisite to several of these, it does not count as one of the option courses.

404 Intermediate Diff. Equation | 433 Intro. to Differential Geom. |

454 Bound. Val. Prob. for PDE | 471 Intro. to Numerical Methods |

555 Intro. to Complex Variables | PHYS 351 Meth. of Theor. Phys. I |

PHYS 405 Intermed. Elec. & Mag. | PHYS 406 Stat. & Thermal Physics |

PHYS 413 Nonlinear Dynamics | PHYS 435 Gravitational Physics |

PHYS 452 Meth. of Theor. Phys. II | PHYS 453 Quantum Mechanics |

** ****Mathematical Biology**

Mathematical Biology is a relatively new area of applied mathematics, and is growing with phenomenal speed. Ever since the advent of high-powered computing, it has become obvious that mathematics can contribute a great deal to biological and medical research; indeed, in many cases, mathematical approaches can answer questions that cannot be addressed by other means, and thus mathematics is often an indispensable tool for biological research. Typical areas of application include such diverse areas as the topology of DNA, genetic algorithms, cell physiology, cancer biology and control strategies, micro-circulation and blood flow, the study of infectious diseases such as AIDS, the biology of populations, neuroscience and the study of the brain, developmental biology and embryology, the study of hormone secretion and endocrine control and bioinformatics. The Mathematical Biology option will thus be appropriate for any student with an interest in Biology or medicine, and a desire to apply the mathematics they learn to current and important biological problems.

An **additional prerequisite** to this major is completion of the Introductory Biology sequence (Bio 171 and 172). Math 463 (Mathematical Biology), and at least one course from Math 404, 452, 454, 462, 471, 558, 563, or 571, must be included in the major program. The last of the three elective courses for this sub-option must be at an advanced level (numbered over 300) and chosen from one of the following areas: Biology, Physiology, Microbiology/Immunology, Neuroscience, Bioinformatics, or Natural Resources and Environment, with the approval of your mathematics advisor. It is recommended that Stat 426 or Stat 510 be a cognate course.

### Advanced Courses

To complete the major program each student should elect two additional advanced courses in Mathematics or a related area. Every student must include, either here or elsewhere in his/her program, a cognate course numbered 300 or above taught outside the department which emphasizes the application of significant mathematical tools (at least at the level of Math 215) in another discipline. In all cases **approval of an advisor is required**. This is a very flexible requirement to accommodate special interests and may be satisfied by a broad range of courses in other departments (generally numbered 300 or above) or by mathematics courses numbered 400 or above.

At least two of the five courses counted towards **Program Options** and **Advanced Courses** must be MATH courses.