Mail Code: 943054065
Email: mcsinquiries@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:
 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 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 Probability, STATS 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 higherlevel 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 onetime 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 writingintensive 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  
Singlevariable calculus or AP credit. ^{1}  
MATH 19  Calculus  3 
MATH 20  Calculus  3 
MATH 21  Calculus  4 
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 prooforiented sequence)  
Modern Mathematics: Continuous Methods  
Modern Mathematics: Continuous Methods  
Modern Mathematics: Continuous Methods  
Modern Mathematics: Discrete Methods (a prooforiented 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)  2225  
CS 103  Mathematical Foundations of Computing  5 
CS 106A  Programming Methodology  5 
and either  
CS 106B  Programming Abstractions  5 
or CS 106X  Programming Abstractions  
Select two of the following:  710  
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)  711  
MS&E 211X  Introduction to Optimization (Accelerated)  34 
MS&E 221  Stochastic Modeling  3 
Or select three of the following:  911  
Introduction to Optimization  
Introduction to Stochastic Modeling  
Introduction to Optimization  
Introduction to Optimization Theory  
Stochastic Modeling  
Introduction to Stochastic Control with Applications  
Statistics (STATS)  1011  
STATS 116  Theory of Probability  34 
or MATH 151  Introduction to Probability Theory  
STATS 200  Introduction to Statistical Inference  4 
Select one of the following:  3  
Introduction to Applied Statistics  
Introduction to Regression Models and Analysis of Variance  
Writing in the Major (WIM)  35  
Choose one from the MCSdesignated 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  
Modern Statistics for Modern Biology  
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 Electives  9  
Choose three courses in Mathematical and Computational Science 100level 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, CrossValidation, and Sample Reuse  
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  
ObjectOriented 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 Units  7689 
^{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 19 Calculus, MATH 20 Calculus, MATH 21 Calculus; See also the Registrar's Advanced Placement web site (AP or IB exams). Students who place out of MATH 19, 20, and 21 are required to take additional Math classes as discussed with MCS student services and the student's faculty advisor. 
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 141  Biostatistics ^{1}  5 
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:  34  
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:  14  
Modern Statistics for Modern Biology  
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 
^{1}  STATS 141: Biostatistics (BIO 141) 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. 
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 ^{1}  15  
Vector Calculus for Engineers  
Ordinary Differential Equations for Engineers  
Linear Algebra and Partial Differential Equations for Engineers  
STATS 116 may be replaced by:  35  
Statistical Methods in Engineering and the Physical Sciences  
STATS 191/STATS 203 may be replaced by:  34  
Data Mining and Analysis  
Allowable Elective Course Substitutions:  9  
Select one of the following:  34  
Functions of a Complex Variable  
Introduction to Combinatorics and Its Applications  
Complex Analysis  
Metalogic  
Select two of the following:  35  
Dynamics  
Introduction to Chemical Engineering  
ENGR 25B  
ENGR 40  (no longer offered)  
Introduction to Materials Science, Nanotechnology Emphasis  
Feedback Control Design 
^{1}  Only MCS majors pursuing the engineering track may petition their advisor to substitute the required Math series for CME courses listed above. 
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, CrossValidation, and Sample Reuse  
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:

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

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

Related research from a 199 course

Participation for credit in a small group seminar

Directed reading


Complete a final report which should:

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

Be typed with 12pt font, singlespaced, minimum 1 page (no longer than 2 pages) with a oneinch margin at the top and bottom of each page.

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

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

The student's work must demonstrate indepth 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 206  Introduction to Numerical Methods for Engineering  3 
CS/STATS 229  Machine Learning  34 
CS 248  Interactive Computer Graphics  34 
EE 364A  Convex Optimization I  3 
MATH 171  Fundamental Concepts of Analysis  3 
MATH 172  Lebesgue Integration and Fourier Analysis  3 
MATH 205A  Real Analysis  3 
STATS 202  Data Mining and Analysis  3 
STATS 216  Introduction to Statistical Learning  3 
STATS 217  Introduction to Stochastic Processes I  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: 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)  35  
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 106A  Programming Methodology  5 
and either  
CS 106B  Programming Abstractions  5 
or CS 106X  Programming Abstractions  
Management Science and Engineering (MS&E)  34  
Select one of the following:  
Introduction to Optimization  
Stochastic Modeling  
Statistics (STATS)  7  
Select two of the following:  
STATS 116  Theory of Probability  4 
and either  
STATS 191  Introduction to Applied Statistics  34 
or STATS 200  Introduction to Statistical Inference  
Electives  9  
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 upperdivision 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 Units  3234 
COVID19 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 "COVID19 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 degreeprogram requirements and/or alter program requirements as appropriate.
Undergraduate Degree Requirements
Grading
The MCS program counts all courses taken in academic year 202021 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 onepage written final report summarizing the knowledge/experience gained upon completion of the internship in order to receive credit.