Catalog Navigation
Contacts
Office: Sequoia Hall, 390 Jane Stanford Way
Mail Code: 94305-4065
Email: mcs-inquiries@stanford.edu
Web Site: https://mcs.stanford.edu/

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 sponsored by Stanford's departments of Statistics, Mathematics, Computer Science, and Management Science & Engineering, providing 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.

Utilizing 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

The Program in Mathematical and Computational Science (MCS) offers a Bachelor of Science in Mathematical and Computational Science. Eligible students may also pursue a Bachelor of Science with Honors. The department also offers a minor in Mathematical and Computational Science.

Suggested Preparation for the Major

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

How to Declare the Major

To declare the major, a student should first meet with an MCS peer advisor to create a proposed study plan and then with the MCS student services officer to discuss the major. Students ordinarily have taken two of the required MATH 50 series courses and a core Statistics course prior to declaration. Once the student has created a proposed study plan, they should connect with the MCS student services officer and declare the major through Axess. Students should have an overall grade point average (GPA) of 3.0 to declare.

Degree Requirements

  • The student must have a grade point average (GPA) of 3.0 or better in all course work used to fulfill the major requirement.
  • At least three quarters before graduation, majors must file with their advisor 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.

  • Students who earn less than a 'C+' 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 advisor (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 advisor. Submit the MCS Program Transfer Credit Form to the student services office.

  • Students may take their three electives courses for credit (CR).

  • Students may be granted a one-time exception to take a core course for credit (CR) with the exception of STATS 116 and STATS 200.

  • 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.

Course Requirements

Units
Mathematics (MATH) 28
Single-variable calculus or AP credit. 1
MATH 19Calculus3
MATH 20Calculus3
MATH 21Calculus4
Students may choose one of the following sequences:15
Multivariable Calculus and Linear Algebra
Linear Algebra, Multivariable Calculus, and Modern Applications
Integral Calculus of Several Variables
Ordinary Differential Equations with Linear Algebra
Modern Mathematics: Continuous Methods (a proof-oriented sequence)
Modern Mathematics: Continuous Methods
Modern Mathematics: Continuous Methods
Modern Mathematics: Continuous Methods
Modern Mathematics: Discrete Methods (a proof-oriented sequence)
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)22-25
CS 103Mathematical Foundations of Computing5
CS 106AProgramming Methodology5
and either
CS 106BProgramming Abstractions5
or CS 106X Programming Abstractions
Select two of the following:7-10
Introduction to Scientific Computing
Computer Organization and Systems
Introduction to the Theory of Computation
Design and Analysis of Algorithms
Computers, Ethics, and Public Policy
Ethics, Public Policy, and Technological Change
Management Science and Engineering (MS&E) 7-11
MS&E 211XIntroduction to Optimization (Accelerated)3-4
MS&E 221Stochastic Modeling3
Or select three of the following:9-11
Introduction to Optimization
Introduction to Stochastic Modeling
Introduction to Optimization
Introduction to Optimization Theory
Stochastic Modeling
Introduction to Stochastic Control with Applications
Statistics (STATS)10-11
STATS 116Theory of Probability3-4
or MATH 151 Introduction to Probability Theory
STATS 200Introduction to Statistical Inference4
Select one of the following:3
Introduction to Applied Statistics
Introduction to Regression Models and Analysis of Variance
Writing in the Major (WIM)3-5
Choose one from the MCS-designated WIM courses to fulfill the Writing in the Major requirement:
Applied Group Theory
Applied Number Theory and Field Theory
Groups and Rings
Fundamental Concepts of Analysis
Computers, Ethics, and Public Policy
Ethics, Public Policy, and Technological Change
Statistical Methods in Computational Genetics
WIM courses offered by other majors may be used in cases of specific concentrations (e.g. biology, decision theory). Advisor approval required.
Mathematical and Computational Science Approved Electives9
Choose three courses in Mathematical and Computational Science 100-level or above, at least 3 units each from two different departments.
Choose three electives:
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
Fundamentals of Data Science: Prediction, Inference, Causality
Introduction to Stochastic Control with Applications
Topics in Social Data
Applied Matrix Theory
Functions of a Complex Variable
Graph Theory
Introduction to Combinatorics and Its Applications
Linear Algebra and Matrix Theory
Introduction to Scientific Computing
Functions of a Real Variable
Complex Analysis
Partial Differential Equations
Stochastic Processes
Basic Probability and Stochastic Processes with Engineering Applications
Discrete Probabilistic Methods
Fundamental Concepts of Analysis
Lebesgue Integration and Fourier Analysis
Metalogic
Mathematics of Sports
Data Science 101
Data Mining and Analysis
Applied Multivariate Analysis
Introduction to Time Series Analysis
Bootstrap, Cross-Validation, and Sample Re-use
Statistical Models in Biology
Introduction to Statistical Learning
Introduction to Stochastic Processes I
Introduction to Stochastic Processes II
Stochastic Processes
Statistical Methods in Finance
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
Software Development for Scientists and Engineers
Numerical Linear Algebra
Object-Oriented Systems Design
Principles of Computer Systems
Operating Systems and Systems Programming
Compilers
Computational Logic
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 advisor's approval, courses other than those listed or offered by the sponsoring departments may be used to fulfill part of the elective requirement. Courses must provide skills relevant to the MCS degree and do not overlap courses in the student's program. Depending on student’s interests, these may be in fields such as, biology, economics, electrical engineering, industrial engineering, and medicine, are otherwise relevant to a mathematical sciences major.
Total Units76-89

Mathematical and Computational Science Tracks

MCS program has designed three tracks to allow majors to pursue their interests in fields where applied mathematics and statistical analysis is utilized. Declared MCS majors are not required to choose a track. These tracks are not declared in Axess and are not printed on the transcript or diploma.    

Biology Track

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 15
Allowable Elective Course Substitutions:
Take three courses from Foundational Biology Core:10
Genetics
Biochemistry & Molecular Biology
Physiology
Evolution
Cell Biology
Or take two courses from the Biology core and one of the following:3-4
Advance Molecular Biology: Epigenetics and Proteostasis
BIO 133
(no longer offered)
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
Genes and Disease (no longer offered)
The following courses are no longer offered, but may be used by students who completed them in fulfillment of this requirement: BIO102, 160A & 160B

Engineering Track

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 advisor, 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 may 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
Metalogic
Select two of the following:3-5
Dynamics
Introduction to Chemical Engineering
ENGR 25B
ENGR 40
(no longer offered)
Introduction to Materials Science, Nanotechnology Emphasis
Feedback Control Design

Statistics Track 

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
Bootstrap, Cross-Validation, and Sample Re-use
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. Students interested in honors should consult with their faculty advisor as soon as possible to allow more opportunities in course planning and concentration area. The honors program allows for a capstone experience, building upon the student’s current academic knowledge and strengthening their understanding in a specific field of study/concentration. Honors work may be concentrated in fields such as biological sciences and medicine, environment, physics, sports analytics, investment science, AI/machine learning, etc. 

Students are required to submit an MCS Honors Proposal Form describing the concentration for honors work, including the courses they intend to use, by the final study list deadline two quarters prior to the expected degree conferral quarter. The honors final report is due no later than the last day of classes of the quarter the student expects to graduate. More information can be found on the MCS Honors Website.

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

  1. Maintain a GPA of at least 3.5 in all major coursework. 

  2. Students should complete 15 units of graduate level coursework. Included in these 15 units can be any of the following:

    1. Related research from a 199 course

    2. ​Participation for credit in a small group seminar

    3. Directed reading

  3. Complete a final report which should:

    1. Include their name, degree and the title of their work.

    2. Be typed with 12pt font, single-spaced, minimum 1 page (no longer than 2 pages) with a one-inch margin at the top and bottom of each page.

    3. Explain a theme between the student’s coursework, their interests, and how they relate to MCS.

    4. Describe how each course selected added to the student's knowledge and understanding in the chosen area of concentration.

    5. The student's work must demonstrate in-depth learning of a topic or shared idea in the breadth of the MCS major (examples are on MCS webpage), and all students are held to Stanford’s Honor Code.

Units
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
MATH 205AReal Analysis3
STATS 202Data Mining and Analysis3
STATS 216Introduction to Statistical Learning3
STATS 217Introduction to Stochastic Processes I3

return to top of page

Minor in Mathematical and Computational Science

The minor in Mathematical and Computational Science is intended to provide an experience of the four constituent areas: Mathematics, Computer Science, Management Science and Engineering, and Statistics. The minor consists of nine courses for a minimum of 32 units. A grade point average (GPA) of 2.75 is required for courses fulfilling the minor. All courses for the minor must be taken for a letter grade, if offered. 

Degree Requirements

Units
Mathematics (MATH) 3-5
Select one of the following:
Linear Algebra, Multivariable Calculus, and Modern Applications
Applied Matrix Theory
Computer Science (CS) 10
Select two of the followning:
CS 106AProgramming Methodology5
and either
CS 106BProgramming Abstractions5
or CS 106X Programming Abstractions
Management Science and Engineering (MS&E) 3-4
Select one of the following:
Introduction to Optimization
Stochastic Modeling
Statistics (STATS) 7
Select two of the following:
STATS 116Theory of Probability4
and either
STATS 191Introduction to Applied Statistics3-4
or STATS 200 Introduction to Statistical Inference
Electives9
The minor requires three courses, two of which must be in different departments.
Select three of the following:
Introduction to Scientific Computing
Mathematical Foundations of Computing
Computer Organization and Systems
Introduction to the Theory of Computation
Design and Analysis of Algorithms
Game Theory and Economic Applications
The Fourier Transform and Its Applications
Introduction to Optimization
Mathematical Programming and Combinatorial Optimization
Stochastic Modeling
Introduction to Stochastic Control with Applications
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
Fundamental Concepts of Analysis
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 MCS program director. Undergraduate majors in the constituent programs may not count courses in their own departments.
Total Units32-34
 

COVID-19 Policies

On July 30, the Academic Senate adopted grading policies effective for all undergraduate and graduate programs, excepting the professional Graduate School of Business, School of Law, and the School of Medicine M.D. Program. For a complete list of those and other academic policies relating to the pandemic, see the "COVID-19 and Academic Continuity" section of this bulletin.

The Senate decided that all undergraduate and graduate courses offered for a letter grade must also offer students the option of taking the course for a “credit” or “no credit” grade and recommended that deans, departments, and programs consider adopting local policies to count courses taken for a “credit” or “satisfactory” grade toward the fulfillment of degree-program requirements and/or alter program requirements as appropriate.


Undergraduate Degree Requirements

Grading

The MCS program counts all courses taken in academic year 2020-21 with a grade of 'CR' (credit) or 'S' (satisfactory) towards satisfaction of undergraduate degree requirements and minor that otherwise require a letter grade.

Faculty

Director: Professor Guenther Walther

Associate Director: Professor Chiara Sabatti

Faculty Advisers: Assistant Professor John Duchi, Professor Bradley Efron,  Associate Professor David Rogosa, Assistant Professor Johan Ugander, Assistant Professor Scott Linderman

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

Courses

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.