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

This interdisciplinary undergraduate degree program in MCS is administrated by the departments of Mathematics, Computer Science, and Statistics. 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.

Using the faculty and courses of the departments listed above, this major 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 and statistics 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:

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

Bachelor of Science in Mathematical and Computational Science

Suggested Preparation for the Major

Students ordinarily would have taken 2 of the required Math courses (MATH 51 Linear Algebra and Differential Calculus of Several Variables/MATH 52 Integral Calculus of Several Variables/MATH 53 Ordinary Differential Equations with Linear Algebra) and one of the required Statistics courses (STATS 116 Theory of Probability, STATS 191 Introduction to Applied Statistics) before declaring MCS during their freshman or sophomore year.

How to Declare

To declare the major, a student must submit the Declaration of Major in Axess. Following a review of the student's transcript, a department adviser is assigned to the student upon declaration approval. It is recommended that students meet with their adviser at least once per quarter to discuss progress towards degree completion.

Course Requirements for the MCS Bachelor's Degree (78-84 units)

Units
Mathematics (MATH) 28
MATH 41Calculus 15
MATH 42Calculus 15
Students may choose to follow the Math 50 or Math 60 series (3 courses); Math 60 series replaces Math 50 Honors:15
Linear Algebra and Differential Calculus of Several Variables
Integral Calculus of Several Variables
Ordinary Differential Equations with Linear Algebra
Math 60 series: Continuous Methods
Modern Mathematics: Continuous Methods
Modern Mathematics: Continuous Methods
Modern Mathematics: Continuous Methods
Math 60 series: Discrete Methods
Modern Mathematics: Discrete Methods
Modern Mathematics: Discrete Methods
Modern Mathematics: Discrete Methods
Select one of the following:3
Applied Matrix Theory
Linear Algebra and Matrix Theory
Computer Science (CS)22-24
CS 103Mathematical Foundations of Computing5
CS 106AProgramming Methodology5
and either
CS 106BProgramming Abstractions5
or CS 106X Programming Abstractions (Accelerated)
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
MS&E 211Linear and Nonlinear Optimization4
MS&E 221Stochastic Modeling3
Or select three of the following:
Introduction to Optimization
Introduction to Stochastic Modeling
Linear and Nonlinear Optimization
Stochastic Modeling
Stochastic Control
Statistics (STATS)11-12
STATS 116Theory of Probability5
STATS 200Introduction to Statistical Inference3
Select one of the following:3
Introduction to Applied Statistics
Introduction to Regression Models and Analysis of Variance
1

Students who scored a 5 on both the Calculus AB and BC advanced placement exams (total of 10 units) can be waived out of MATH 41 and MATH 42. See also the Registrar's Advanced Placement web site (AP or IB exams).

Writing in the Major Requirement

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

Units
Choose one from the MCS-designated WIM courses to fulfill the Writing in the Major 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
WIM courses offered by other majors may be used in cases of specific concentrations (e.g. biology, decision theory). Adviser approval required.

Mathematical and Computational Science Electives

Choose three courses in Mathematical and Computational Science 100-level or above, at least 3 units each from two different departments. At least one must be from following list:

Units
Choose three courses from the following:9 units
Advanced Topics in Econometrics
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 I
Convex Optimization II
Probabilistic Analysis
Simulation
"Small" Data
Stochastic Control
Mathematics of Sports
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
Stochastic Processes
Fundamental Concepts of Analysis
Lebesgue Integration and Fourier Analysis
Calculus of Variations
Metalogic
Data Mining and Analysis
Applied Multivariate Analysis
Introduction to Time Series Analysis
Introduction to the Bootstrap
Statistical Models in Biology
Introduction to Statistical Learning
Introduction to Stochastic Processes I
Introduction to Stochastic Processes II
Stochastic Processes
Statistical Methods in Finance
Bayesian Statistics I
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
Software Development for Scientists and Engineers
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
Software Project
Artificial Intelligence: Principles and Techniques
Introduction to Robotics
Experimental Robotics
Probabilistic Graphical Models: Principles and Techniques
Machine Learning
Program Analysis and Optimizations
Mining Massive Data Sets
Interactive Computer Graphics
Electives that are not offered this year, but may be offered in subsequent years, are eligible for credit toward the major.
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, etc., that might be relevant to a mathematical sciences major, depending on a student's interests.

Grade and Course 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.
  • The student must have a grade point average (GPA) of 3.0 or better in all course work used to fulfill the major requirement.
  • Students who earn less than a 'B-' in STATS 116 Theory of Probability or STATS 200 Introduction to Statistical Inference must repeat the course.
  • Only one MCS core course can be substituted by filing a petition with their adviser (with the exception of STATS 200 Introduction to Statistical Inference which cannot be substituted). The Course Substitution Form must be submitted the quarter prior to enrolling in the course.
  • Course transfer credit is subject to department evaluation and to the Office of the Registrar's external credit evaluation. These courses may result in a replacement course for MCS required course or may establish placement in a higher-level course. Transfer requests must first be submitted to Student Services Center prior to being evaluated by your adviser. Submit the MCS Program Transfer Credit Form to the student services office.
  • At least three quarters before graduation, majors must file with their adviser a plan for completing degree requirements.

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 141Biostatistics 13-5
Allowable Elective Course Substitutions:
Take three courses from the Biology Core:10
Genetics, Biochemistry, and Molecular Biology
Cell Biology and Animal Physiology
Plant Biology, Evolution, and Ecology
Or take two courses from the Biology core and one of the following:3-4
Evolution
Conservation Biology: A Latin American Perspective
Theoretical Population Genetics (offered alternate years)
Molecular and Cellular Immunology
Honors students select the following three courses:1-4
Statistical Methods in Computational Genetics
Fundamentals of Molecular Evolution
Population Studies
The following courses are no longer offered, but may be used by students who completed them in fulfillment of this requirement: BIO102, 160A & 160B
1

Can replace STATS 191 Introduction to Applied Statistics or STATS 203 Introduction to Regression Models and Analysis of Variance from the major's Statistics core requirement.

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
With consent of an MCS adviser, MATH 51, MATH 52, MATH 53 series may be substituted for CME 100, CME 102, CME 104. Depending on the exact material taught in relevant years, an additional math course might be necessary 115
Vector Calculus for Engineers
Ordinary Differential Equations for Engineers
Linear Algebra and Partial Differential Equations for Engineers
STATS 116 may be replaced by:3-5
Statistical Methods in Engineering and the Physical Sciences
STATS 191/STATS 203 may be replaced by:3-4
Data Mining and Analysis
Allowable Elective Course Substitutions:9
Select one of the following:3-4
Functions of a Complex Variable
Introduction to Combinatorics and Its Applications
Complex Analysis
Mathematics of Computation
MATH 132
Calculus of Variations
Metalogic
Select two of the following:3-5
Dynamics
Introduction to Chemical Engineering
Biotechnology
Engineering Thermodynamics
Introductory Electronics
Introduction to Materials Science, Nanotechnology Emphasis
Feedback Control Design
1

Only M&CS majors pursuing the engineering track may petition their adviser to substitute the required Math series for CME courses listed above.

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 - (87 units total)

Required:
Units
Additional Courses for the Statistics Track:9
Introduction to Stochastic Processes I
Advanced CS, such as:3
Mining Massive Data Sets
Advanced MS&E, such as:3
Probabilistic Analysis
or
Simulation
Allowable Elective Course Substitutions:9
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
Bayesian Statistics I

Honors Program

The honors program is designed to encourage a more intensive study of mathematical sciences than the B.S. program. 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. Students are required to submit an outline describing the concentration for honors work and the courses they intend to use two quarters prior to expected degree conferral.

In addition to meeting all requirements for the B.S., the student must:

  1. Maintain an average letter grade equivalent to at least a 3.5 in all academic work.
  2. 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:
    1. An approved upper-level or graduate course
    2. Participation in a small group seminar
    3. At least 3 units of directed reading
  3. Prepare a statement describing major area of concentration for honors work.
  4. Describe how each course selected added to the student's knowledge and understanding in area chosen for concentration.
  5. Honors statement should be submitted to the adviser by the last day of classes of the student's graduation quarter using the honors approval form.
Suggested electives for students pursuing Honors:
CME 206Introduction to Numerical Methods for Engineering3
CS/STATS 229Machine Learning3-4
CS 248Interactive Computer Graphics3-4
EE 364AConvex Optimization I3
MATH 171Fundamental Concepts of Analysis3
MATH 172Lebesgue Integration and Fourier Analysis3
STATS 202Data Mining and Analysis3
STATS 216Introduction to Statistical Learning3
STATS 217Introduction to Stochastic Processes I2-3

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:3-5
Linear Algebra and Differential Calculus of Several Variables
or
Applied Matrix Theory
Select two of the followning:10
Programming Methodology
and either
Programming Abstractions
Programming Abstractions (Accelerated)
Select one of the following:3-4
Linear and Nonlinear Optimization
or
Stochastic Modeling
Select two of the following:8
STATS 116Theory of Probability5
and either
Introduction to Applied Statistics
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
Game Theory and Economic Applications
The Fourier Transform and Its Applications
Linear and Nonlinear Optimization
Mathematical Programming and Combinatorial Optimization
Stochastic Modeling
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
MATH 132
Fundamental Concepts of Analysis
Calculus of Variations
Metalogic
Introduction to Applied Statistics
Introduction to Statistical Inference
Data Mining and Analysis
Introduction to Regression Models and Analysis of Variance
Introduction to Stochastic Processes I

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

Faculty Advisers: Assistant Professor John Duchi, Professor Bradley Efron, Professor Susan Holmes, Associate Professor Chiara Sabatti

Steering Committee: Takeshi Amemiya (Economics, emeritus), Emmanuel Candès (Mathematics, Statistics), Darrel Duffie (Economics & GSB), Gunnar Carlsson (Mathematics), Richard Cottle (Management Science and Engineering, emeritus), John Duchi (Electrical Engineering & Statistics),  Bradley Efron (Statistics), Peter Glynn (Management Science and Engineering), Susan Holmes (Statistics), Ramesh Johari (Management Science and Engineering), Percy Liang (Computer Science & Statistics), Parviz Moin (Engineering), George Papanicolaou (Mathematics), Eric Roberts (Computer Science, emeritus), David Rogosa (Education & Statistics), Tim Roughgarden (Computer Science), Chiara Sabatti (Biomedical Data Science & Statistics), David Siegmund (Statistics), Jonathan Taylor (Statistics), Brian White (Mathematics).

Courses

MCS 100. Mathematics of Sports. 3 Units.

The use of mathematics, statistics, and probability in the analysis of sports performance, sports records, and strategy. Topics include mathematical analysis of the physics of sports and the determinations of optimal strategies. New diagnostic statistics and strategies for each sport. Corequisite: STATS 60, 110 or 116.
Same as: STATS 50

MCS 198. Practical Training. 1 Unit.

For students majoring in Mathematical and Computational Science only. Students obtain employment in a relevant industrial or research activity to enhance their professional experience. Students may enroll in summer quarters only and for a total of three times. Students must first notify their MCS adviser before enrolling in their course section, and must submit a one-page written final report summarizing the knowledge/experience gained upon completion of the internship in order to receive credit.