# Mathematical and Computational Science

## Contacts

Office: Sequoia Hall, 390 Serra Mall

Mail Code: 94305-4065

Phone: (650) 723-2620

Email: helnnn@stanford.edu

Web Site: http://stanford.edu/group/mathcompsci

Courses offered by Mathematical and Computational Science program are listed under the subject code MCS on the Stanford Bulletin's ExploreCourses website.

This interdepartmental interschool undergraduate program provides a major for students interested in the mathematical and computational sciences, or in the use of mathematical ideas and analysis in problems in the social or management sciences. It provides a core of mathematics basic to all the mathematical sciences and an introduction to concepts and techniques of computation, optimal decision making, probabilistic modeling, and statistical inference. It also provides an opportunity for elective work in any of Stanford's mathematical science disciplines.

The program uses the faculty and courses of the departments of Computer Science, Management Science and Engineering, Mathematics, and Statistics. It prepares students for graduate study or employment in the mathematical and computational sciences or in those areas of applied mathematics which center around the use of computers and are concerned with the problems of the social and management sciences. A biology option is offered for students interested in applications of mathematics, statistics, and computer science to the biological sciences (bioinformatics, computational biology, statistical genetics, neurosciences); and in a similar spirit, an engineering option.

## Undergraduate Mission Statement for Mathematical and Computational Science

The mission of the Mathematical and Computational Science Program is to provide students with a core of mathematics basic to all the mathematical sciences and an introduction to concepts and techniques of computation, optimal decision making, probabilistic modeling and statistical inference. The program is interdisciplinary in its focus, and students are required to complete course work in mathematics, computer science, statistics, and management science and engineering. A computational biology track is available for students interested in biomedical applications. The program prepares students for careers in academic, financial and government settings as well as for study in graduate or professional schools.

## Learning Outcomes

The program expects undergraduate majors to be able to demonstrate the following learning outcomes. These learning outcomes are used in evaluating students and the department's undergraduate program. Students are expected to be able to demonstrate:

- understanding of principles and tools of statistics.
- command of optimization and its applications and the ability to analyze and interpret problems from various disciplines.
- an understanding of computer applications emphasizing modern software engineering principles.
- an understanding of multivariate calculus, linear algebra, and algebraic and geometric proofs.

## Bachelor of Science in Mathematical and Computational Science

The requirement for the bachelor's degree, beyond the University's basic requirements, is an approved course program of a minimum of 72 units, distributed as follows:

Units | ||
---|---|---|

Mathematics (MATH) (28) | ||

MATH 41 | Calculus ^{*} | 5 |

MATH 42 | Calculus ^{*} | 5 |

Select one of the following: | 5 | |

Linear Algebra and Differential Calculus of Several Variables | ||

Honors Multivariable Mathematics | ||

Select one of the following: | 5 | |

Integral Calculus of Several Variables | ||

Honors Multivariable Mathematics | ||

Select one of the following: | 5 | |

Ordinary Differential Equations with Linear Algebra | ||

Honors Multivariable Mathematics | ||

Select one of the following: | 3 | |

Applied Matrix Theory | ||

Linear Algebra and Matrix Theory | ||

Computer Science (CS) (17-24) | ||

CS 103 | Mathematical Foundations of Computing | 5 |

Select one of the following: | 5-10 | |

Programming Abstractions (Accelerated) | ||

or both | ||

Programming Methodology | ||

Programming Abstractions | ||

Select two of the following: | 7-9 | |

Introduction to Scientific Computing | ||

Computer Organization and Systems | ||

Introduction to Automata and Complexity Theory | ||

Design and Analysis of Algorithms | ||

Computers, Ethics, and Public Policy | ||

Management Science and Engineering (MS&E) (7-11) | ||

Linear and Nonlinear Optimization | ||

Stochastic Modeling | ||

Or select three of the following: | 7-11 | |

Introduction to Optimization | ||

Introduction to Stochastic Modeling | ||

Linear and Nonlinear Optimization | ||

Stochastic Modeling | ||

Stochastic Control | ||

Statistics (STATS) (11) | ||

STATS 116 | Theory of Probability | 5 |

STATS 200 | Introduction to Statistical Inference | 3 |

Select one of the following: | 3 | |

Introduction to Applied Statistics | ||

Introduction to Regression Models and Analysis of Variance |

### Writing in the Major Requirement (3-4 units)

The University requires students to complete at least one approved writing-intensive course in each of their majors. See the Hume Center for Writing ad Speaking web site for a full description of the WIM requirement.

Units | ||
---|---|---|

Choose one from the following to fulfill the WIM requirement: | 3-4 units | |

Applied Group Theory | ||

Applied Number Theory and Field Theory | ||

Groups and Rings | ||

Fundamental Concepts of Analysis | ||

Computers, Ethics and Public Policy | ||

Statistical Methods in Computational Genetics |

### Mathematical and Computational Science Electives (9 Units)

Three courses in mathematical and computational science, 100-level or above, at least 3 units each.

Units | ||
---|---|---|

Choose three courses from the following: | 9-15 | |

Advanced Topics in Econometrics | ||

Causal Inference and Program Evaluation | ||

Introduction to Financial Economics | ||

Game Theory and Economic Applications | ||

Experimental Economics | ||

The Fourier Transform and Its Applications | ||

Introduction to Linear Dynamical Systems | ||

Introduction to Statistical Signal Processing | ||

Computer Systems Architecture | ||

Convex Optimization II | ||

Stochastic Control | ||

Applied Matrix Theory | ||

Functions of a Complex Variable | ||

Introduction to Combinatorics and Its Applications | ||

Linear Algebra and Matrix Theory | ||

Functions of a Real Variable | ||

Complex Analysis | ||

Partial Differential Equations I | ||

Stochastic Processes | ||

Fundamental Concepts of Analysis | ||

Lebesgue Integration and Fourier Analysis | ||

Calculus of Variations | ||

First-Order Logic (Winte) | ||

Data Mining and Analysis | ||

Introduction to the Bootstrap | ||

Introduction to Stochastic Processes | ||

Introduction to Stochastic Processes | ||

Stochastic Processes | ||

A Course in Bayesian Statistics | ||

For Computer Science (CS), electives can include courses not taken as units under the CS list above and the following: | ||

Introduction to Numerical Methods for Engineering | ||

Numerical Linear Algebra | ||

Object-Oriented Systems Design | ||

Principles of Computer Systems | ||

Operating Systems and Systems Programming | ||

Compilers | ||

Logic and Automated Reasoning | ||

Design and Analysis of Algorithms | ||

Computing with Physical Objects: Algorithms for Shape and Motion | ||

Software Project | ||

Artificial Intelligence: Principles and Techniques | ||

Introduction to Robotics | ||

Experimental Robotics | ||

Probabilistic Graphical Models: Principles and Techniques | ||

Machine Learning | ||

Program Analysis and Optimizations | ||

Interactive Computer Graphics |

With the adviser's approval, courses other than those offered by the sponsoring departments may be used to fulfill part of the elective requirement. These may be in fields such as biology, economics, electrical engineering, industrial engineering, and medicine, that might be relevant to a mathematical sciences major, depending on a student's interests.

- At least three quarters before graduation, majors must file with their advisers a plan for completing degree requirements.
- All courses used to fulfill major requirements must be taken for a letter grade with the exception of courses offered satisfactory/no credit only.
- A course used to fulfill the requirements of one section of the program may not be applied toward the fulfillment of the requirements of another section.
- The student must have a grade point average (GPA) of 2.0 or better in all course work used to fulfill the major requirement.
- Electives that are not offered this year, but may be offered in subsequent years, are eligible for credit toward the major: CME 311, ECON 102C Advanced Topics in Econometrics, MCS 100/STATS 50.

### Mathematical and Computational Science Biology Track (Option)

Students in the Biology track take the introductory courses for the Mathematics and Computational Science major with the following allowable substitutions as electives.

Units | ||
---|---|---|

STATS/BIO 141 | Biostatistics ^{1} | 3-5 |

Select two of the following from the Biology Core: | ||

Genetics, Biochemistry, and Molecular Biology | ||

Cell Biology and Animal Physiology | ||

Plant Biology, Evolution, and Ecology | ||

Select one of the following: | ||

Genetics, Biochemistry, and Molecular Biology | ||

Cell Biology and Animal Physiology | ||

Plant Biology, Evolution, and Ecology | ||

Demography: Health, Development, Environment | ||

Evolutionary Paleobiology | ||

Evolution | ||

Conservation Biology: A Latin American Perspective | ||

Developmental Biology I | ||

Developmental Biology II | ||

Theoretical Population Genetics | ||

Molecular and Cellular Immunology | ||

Honors students select the following three courses: | ||

Fundamentals of Molecular Evolution | ||

Population Studies | ||

Statistical Methods in Computational Genetics |

^{1} | Replaces STATS 191 Introduction to Applied Statistics/STATS 203 Introduction to Regression Models and Analysis of Variance with STATS 141 Biostatistics/BIO 141 Biostatistics |

### Mathematical and Computational Science Engineering Track (Option)

Students in the Engineering track take the introductory courses for the Mathematics and Computational Sciences major with the following allowable substitutions.

Units | ||
---|---|---|

The MATH 51-53 series may be replaced by: | ||

Vector Calculus for Engineers | ||

Ordinary Differential Equations for Engineers | ||

Linear Algebra and Partial Differential Equations for Engineers | ||

Linear Algebra with Application to Engineering Computations | ||

STATS 116 may be replaced by: | ||

Statistical Methods in Engineering and the Physical Sciences | ||

STATS 191/STATS 203 may be replaced by: | ||

Data Mining and Analysis | ||

Engineering Track Electives: | ||

Select one of the following: | ||

Functions of a Complex Variable | ||

Introduction to Combinatorics and Its Applications | ||

Complex Analysis | ||

Calculus of Variations | ||

First-Order Logic | ||

Select two of the following: | ||

Dynamics | ||

Introduction to Chemical Engineering | ||

Biotechnology | ||

Engineering Thermodynamics | ||

Introductory Electronics | ||

Introduction to Materials Science, Nanotechnology Emphasis | ||

Feedback Control Design |

### Mathematical and Computational Science Statistics Track (Option)

Students in the Statistics track take the introductory courses for the Mathematics and Computational Sciences major with the following additional courses - (85 units total)

##### Required:

Units | ||
---|---|---|

STATS 217 | Introduction to Stochastic Processes | 3 |

Advanced CS, such as: | ||

CS 246 | Mining Massive Data Sets | 3-4 |

Advanced MS&E, such as: | ||

MS&E 220 | Probabilistic Analysis | 3-4 |

or | ||

Simulation | ||

Statistics Track Electives: | ||

Select three of the following: | ||

Data Mining and Analysis | ||

Applied Multivariate Analysis | ||

Introduction to Time Series Analysis | ||

Introduction to the Bootstrap | ||

Introduction to Statistical Learning | ||

Stochastic Processes | ||

A Course in Bayesian Statistics |

### Honors Program

The honors program is designed to encourage a more intensive study of mathematical sciences than the B.S. program. In addition to meeting all requirements for the B.S., the student must:

- Maintain an average letter grade equivalent to at least a
**3.6**in all academic work. - Complete at least 15 units in mathematical sciences in addition to the requirements for the major listed above. Include in these 15 units at least one of the following:
- An approved higher-level graduate course
- Participation in a small group seminar
- At least 3 units of directed reading

- Prepare a statement describing major area of concentration for honors work.
- Describe how each course selected added to the student's knowledge and understanding in area chosen for concentration.
- Students interested in honors should consult with their adviser by last quarter of their junior year to prepare their program of study. Honors work may be concentrated in fields such as biological sciences, environment, physics, etc.
- Suggested electives for students pursuing Honors: EE 364, CME 206, CS 229, CS 248, MATH 171, MATH 172, STATS 202, STATS 216, STATS 217.

## Minor in Mathematical and Computational Science

The minor in Mathematical and Computational Science is intended to provide an experience of the four constituent areas: Computer Science, Mathematics, Management Science and Engineering, and Statistics. Five basic courses are required:

Units | ||
---|---|---|

Select one of the following: | 3 | |

CS 106X | Programming Abstractions (Accelerated) | 3-5 |

or | ||

Programming Methodology and Programming Abstractions | ||

MATH 51 | Linear Algebra and Differential Calculus of Several Variables | 5 |

or | ||

Applied Matrix Theory | ||

MS&E 211 | Linear and Nonlinear Optimization | 3-4 |

and | ||

Stochastic Modeling | ||

STATS 116 | Theory of Probability | 3-5 |

and | ||

STATS 191 | Introduction to Applied Statistics | 3-4 |

or | ||

Introduction to Statistical Inference |

In addition to the above, the minor requires three courses from the following, two of which must be in different departments:

Units | ||
---|---|---|

Select three of the following: | 9 | |

Introduction to Scientific Computing | ||

Mathematical Foundations of Computing | ||

Computer Organization and Systems | ||

Introduction to Automata and Complexity Theory | ||

Design and Analysis of Algorithms | ||

The Fourier Transform and Its Applications | ||

Game Theory and Economic Applications | ||

Stochastic Control | ||

Applied Matrix Theory | ||

Functions of a Complex Variable | ||

Introduction to Combinatorics and Its Applications | ||

Applied Group Theory | ||

Applied Number Theory and Field Theory | ||

Functions of a Real Variable | ||

Partial Differential Equations I | ||

Fundamental Concepts of Analysis | ||

Calculus of Variations | ||

First-Order Logic | ||

Introduction to Applied Statistics | ||

Introduction to Statistical Inference | ||

Data Mining and Analysis | ||

Introduction to Regression Models and Analysis of Variance | ||

Introduction to Stochastic Processes |

Other upper-division courses appropriate to the program major may be substituted with consent of the program director. Undergraduate majors in the constituent programs may not count courses in their own departments.

*Co-Directors:* Bradley Efron, Susan Holmes

*Committee in Charge:* Takeshi Amemiya (Economics, emeritus), Emmanuel Candes (Mathematics, Statistics), Gunnar Carlsson (Mathematics), Richard Cottle (Management Science and Engineering, emeritus), Bradley Efron (Statistics), Margot Gerritsen (ICME), Susan Holmes (Statistics), Parviz Moin (Engineering), George Papanicolaou (Mathematics), Eric Roberts (Computer Science), David Rogosa (Education), Tim Roughgarden (Computer Science), Chiara Sabatti (Statistics), Amin Saberi (Management Science and Engineering), David Siegmund (Statistics), Jonathan Taylor (Statistics), Brian White (Mathematics).