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Mathematics
School of Humanities & Sciences
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  1. Home
  2. Programs
  3. Undergraduate
  4. Mathematics
  5. Major: Bachelor of Science

Major: Bachelor of Science

Major
  • Mathematics opens in a new tab
Program Minimum Units
61 Units
University Minimum Units
180 Units
Interests
Experimentation & Discovery, Technology, Data and Information Sciences
Academic Areas
Natural Sciences, Mathematics & Data Sciences, Engineering & Computer Science

Overview

The undergraduate program in Mathematics aims to provide students with a broad understanding of mathematics encompassing logical reasoning, generalization, abstraction, and formal proof. Courses in the program teach students to create, analyze, and interpret mathematical models and to communicate sound arguments based on mathematical reasoning and careful data analysis. The mathematics degree prepares students for careers in the tech and finance industries, graduate programs in quantitative fields, computational sciences, and the corporate sector.

Requirements

Core Program Requirements

Math Course Requirement

8 Math courses at the 100-level and above (including Math 56) plus 25 more units of Math classes (usually fulfilled using single and multivariable calculus).
  • At least eight Math courses above MATH 63CM/MATH 63DM.

  • MATH 56 may count toward the eight courses.

  • STATS 116 (No longer offered), STATS118 opens in a new tab or STATS200 opens in a new tab(at most one among STATS 118 and STATS 200, but not both, and neither if STATS 116 is being used), PHIL 151, & PHIL 152 count as 100-level Math courses. No other classes outside the Math department can count in this way. There are no exceptions to this for any reason.

  • The following course do not count toward the Math Course Requirement:

    MATH 190, MATH 193, MATH193X opens in a new tab, MATH197 opens in a new tab, MATH198 opens in a new tab, and MATH199 opens in a new tab (MATH 190, MATH193X and MATH199 can count towards the Elective Course Requirement, see below.)

  • Up to 10 units of Single Variable Calculus can be used as part of the Math Course Requirement.

    • If you have not earned 10 units of single-variable calculus credit but have Math course credit that affirms your mastery of single-variable calculus, please email mathstudentservices@stanford.edu and specify what that course credit is (it must entail substantial use of both derivatives and integrals). Examples include Math 51 + either Math 52 or Math 53, Math 61CM, or Math 61DM + 63DM. To fulfill the Math Major, you will need to make up whatever portion of such 10 units you don’t have on your transcript, but can do so with approved electives.

  • Only Math 50/60CM/60DM series and first-year calculus can be double-counted toward any other major or minor (Note: The completion of 15 units from the multivariable math sequences is recommended but not a requirement, and at most 15 of such units can be used towards the major).

Electives

Required Electives

Four elective courses are required for the Math major.

  • Math Department courses numbered Math 101 and above can be used as electives except for MATH193 opens in a new tab, MATH197 opens in a new tab, and MATH198 opens in a new tab

  • MATH190 opens in a new tab, MATH193X opens in a new tab and MATH199 opens in a new tab may only be counted toward electives, not toward the 57 Core units above.

  • In addition, there are approved out-of-department electives that can be used as electives (See list below).

Non-MATH Elective Options

Complete any of the following course(s):
AA 216

Projection-Based Model Order Reduction

3 Units
APPPHYS 229

Statistical Mechanics of Learning and Computation

3 Units
APPPHYS 237

Quantitative Evolutionary Dynamics and Genomics

3 Units
BIO 141

Introduction to Statistics for Biology

5 Units
BIO 165

Quantitative Approaches in Modern Biology

3 Units
BIO 188

Systems Biology: Principles of Cell Signaling

3 Units
BIOE 101

Systems Biology

3 Units
CHEM 151

Inorganic Chemistry I

4 Units
CHEM 153

The Chemistry of Extended Solids

3 Units
CHEM 171

Linking the Molecular to the Macroscopic: Foundations of Physical Chemistry

4 Units
CHEM 173

Physical Chemistry II

3 Units
CHEM 175

Physical Chemistry III

3 Units
CLASSICS 136

The Greek Invention of Mathematics

3-5 Units
CME 108

Introduction to Scientific Computing with Machine Learning Applications

3 Units
CS 103

Mathematical Foundations of Computing

3-5 Units
CS 123

A Hands-On Introduction to Building AI-Enabled Robots

3 Units
CS 109

Introduction to Probability for Computer Scientists

3-5 Units
CS 129

Applied Machine Learning

3-4 Units
CS 148

Introduction to Computer Graphics and Imaging

3-4 Units
CS 154

Introduction to the Theory of Computation

3-4 Units
CS 157

Computational Logic

3 Units
CS 161

Design and Analysis of Algorithms

3-5 Units
CS 166

Advanced Data Structures

3-4 Units
CS 168

The Modern Algorithmic Toolbox

3-4 Units
CS 205L

Continuous Mathematical Methods with an Emphasis on Machine Learning

3 Units
CS 221

Artificial Intelligence: Principles and Techniques

3-4 Units
CS 224N

Natural Language Processing with Deep Learning

3-4 Units
CS 224R

Deep Reinforcement Learning

3 Units
CS 228

Probabilistic Graphical Models: Principles and Techniques

3-4 Units
CS 229

Machine Learning

3-4 Units
CS 229M

Machine Learning Theory

3 Units
CS 230

Deep Learning

3-4 Units
CS 233

Geometric and Topological Data Analysis

3 Units
CS 234

Reinforcement Learning

3 Units
CS 235

Computational Methods for Biomedical Image Analysis and Interpretation

3 Units
CS 250

Algebraic Error Correcting Codes

3 Units
CS 251

Cryptocurrencies and blockchain technologies

3 Units
CS 254

Computational Complexity

3 Units
CS 255

Introduction to Cryptography

3 Units
CS 256

Algorithmic Fairness

3 Units
CS 259Q

Quantum Computing

3 Units
CS 261

Combinatorial Optimization

3 Units
CS 265

Randomized Algorithms and Probabilistic Analysis

3 Units
CS 354

Topics in Intractability: Unfulfilled Algorithmic Fantasies

3 Units
CS 355

Advanced Topics in Cryptography

3 Units
ECON 50

Economic Analysis I

5 Units
ECON 51

Economic Analysis II

5 Units
ECON 52

Economic Analysis III

5 Units
ECON 102A

Introduction to Statistical Methods (Postcalculus) for Social Scientists

5 Units
ECON 102B

Applied Econometrics

5 Units
ECON 102C

Advanced Topics in Econometrics

5 Units
ECON 111

Money and Banking

5 Units
ECON 136

Market Design

5 Units
ECON 137

Decision Modeling and Information

5 Units
ECON 140

(FIN I) Introduction to Corporate Finance

5 Units
ECON 141

Investments in Financial Markets

5 Units
ECON 160

Game Theory and Economic Applications

5 Units
ECON 167G

Game Theory and Social Behavior

5 Units
ECON 180

Accelerated Game Theory

5 Units
ECON 202

Microeconomics I

2-5 Units
ECON 203

Microeconomics II

5 Units
ECON 204

Microeconomics III

5 Units
ECON 284

Simplicity and Complexity in Economic Theory

3 Units
EE 363

Linear Dynamical Systems

3 Units
EE 387

Algebraic Error Correcting Codes

3 Units
ENGR 14

Intro to Solid Mechanics

4 Units
ENGR 15

Dynamics

3 Units
ENGR 60

Engineering Economics and Sustainability

3 Units
ENGR 62

Introduction to Optimization

3-4 Units
ENGR 76

Information Science and Engineering

5 Units
ESS 246A

Atmosphere, Ocean, and Climate Dynamics: The Atmospheric Circulation

3 Units
MS&E 111

Introduction to Optimization

3-4 Units
MS&E 111DS

Introduction to Optimization: Data Science

3-4 Units
MS&E 111X

Introduction to Optimization (Accelerated)

3-4 Units
MS&E 112

Graph and Combinatorial Optimization

3 Units
MS&E 120

Introduction to Probability

4 Units
MS&E 121

Introduction to Stochastic Modeling

4 Units
MS&E 135

Networks

3 Units
MS&E 145

Introduction to Finance and Investment

4 Units
MS&E 211

Introduction to Optimization

3-4 Units
MS&E 211X

Introduction to Optimization (Accelerated)

3-4 Units
MS&E 220

Probabilistic Analysis

3 Units
MS&E 221

Stochastic Modeling

3 Units
MS&E 223

Stochastic Simulation and Monte Carlo Methods

3 Units
MS&E 226

Fundamentals of Data Science: Prediction, Inference, Causality

3 Units
MS&E 232

Introduction to Game Theory

3 Units
MS&E 232H

Introduction to Game Theory (Accelerated)

3 Units
MS&E 244

Statistical Arbitrage

3 Units
MS&E 245A

Investment Science

4 Units
MS&E 245B

Advanced Investment Science

3 Units
MS&E 310

Linear Programming

3 Units
MS&E 338

Aligning Superintelligence

3 Units
MUSIC 320

Introduction to Audio Signal Processing

2-4 Units
MUSIC 423

Graduate Research in Music Technology

1-10 Units
PHIL 150

Mathematical Logic

4 Units
or
PHIL 250

Mathematical Logic

4 Units
PHIL 151

Metalogic

4 Units
PHIL 152

Computability and Logic

4 Units
PHIL 154

Modal Logic

4 Units
or
PHIL 254

Modal Logic

4 Units
PHIL 162

Philosophy of Mathematics

4 Units
PHYSICS 14N

Quantum Information: Visions and Emerging Technologies

3 Units
PHYSICS 41

Mechanics

4 Units
PHYSICS 43

Electricity and Magnetism

4 Units
PHYSICS 45

Light and Heat

4 Units
PHYSICS 61

Mechanics and Special Relativity

4 Units
PHYSICS 71

Quantum and Thermal Physics

4 Units
PHYSICS 81

Electricity and Magnetism Using Special Relativity and Vector Calculus

4 Units
PHYSICS 100

Introduction to Observational Astrophysics

4 Units
PHYSICS 104

Introduction to Electronic Techniques for Physics Research Instrumentation 

4 Units
PHYSICS 108

Advanced Physics Laboratory: Project

5 Units
PHYSICS 110

Advanced Mechanics

3-4 Units
PHYSICS 111

Partial Differential Equations of Mathematical Physics

4 Units
PHYSICS 112

Mathematical Methods for Physics

4 Units
PHYSICS 113

Computational Physics

4 Units
PHYSICS 120

Electromagnetism

4 Units
PHYSICS 121

Electrodynamics

4 Units
PHYSICS 130

Quantum Mechanics I

4 Units
PHYSICS 131

Quantum Mechanics II

4 Units
PHYSICS 134

Advanced Topics in Quantum Mechanics

3-4 Units
PHYSICS 160

Introduction to Stellar and Galactic Astrophysics

3 Units
PHYSICS 161

Introduction to Cosmology and Extragalactic Astrophysics

3 Units
PHYSICS 170

Thermodynamics, Kinetic Theory, and Statistical Mechanics I

4 Units
PHYSICS 171

Thermodynamics, Kinetic Theory, and Statistical Mechanics II

4 Units
PHYSICS 172

Solid State Physics

3-4 Units
PHYSICS 210

Advanced Mechanics

3-4 Units
PHYSICS 212

Statistical Mechanics

3 Units
PHYSICS 216

Back of the Envelope Physics

2 Units
PHYSICS 220

Classical Electrodynamics

3 Units
PHYSICS 230

Graduate Quantum Mechanics I

3 Units
PHYSICS 231

Graduate Quantum Mechanics II

3 Units
PHYSICS 234

Advanced Topics in Quantum Mechanics

3-4 Units
PHYSICS 240

Introduction to the Physics of Energy

3 Units
PHYSICS 241

Introduction to Nuclear Energy

3 Units
PHYSICS 252

Introduction to Particle Physics I

3 Units
PHYSICS 260

Introduction to Stellar and Galactic Astrophysics

3 Units
PHYSICS 261

Introduction to Cosmology and Extragalactic Astrophysics

3 Units
PHYSICS 262

General Relativity

3 Units
PHYSICS 266

Statistical Methods in Experimental Physics

4 Units
PHYSICS 269

Neutrinos in Astrophysics and Cosmology

3 Units
PHYSICS 331

Quantum Field Theory II

3 Units
POLISCI 356A

Formal Theory I

3-5 Units
POLISCI 356B

Formal Theory II

3-5 Units
STATS 110

Introduction to Statistics for Engineering and the Sciences

5 Units
STATS 117

Introduction to Probability Theory

3-4 Units
STATS 118

Probability Theory for Statistical Inference

3-4 Units
STATS 141

Introduction to Statistics for Biology

5 Units
STATS 191

Introduction to Applied Statistics

3 Units
STATS 200

Introduction to Theoretical Statistics

3-4 Units
STATS 202

Statistical Learning and Data Science

3 Units
STATS 203

Regression Models and Analysis of Variance

3 Units
STATS 205

Introduction to Nonparametric Statistics

3 Units
STATS 206

Applied Multivariate Analysis

3 Units
STATS 207

Time Series Analysis

3 Units
STATS 217

Introduction to Stochastic Processes I

3 Units
STATS 218

Introduction to Stochastic Processes II

3 Units
STATS 229

Machine Learning

3-4 Units
STATS 270

Bayesian Statistics

3 Units
STATS 271

Applied Bayesian Statistics

3 Units
STATS 305A

Applied Statistics I

3 Units
STATS 315B

Modern Applied Statistics: Learning II

3 Units
STATS 318

Modern Markov Chains

3 Units
  • MUSIC 423: Limit of three units.

  • PHIL 151 & PHIL 152: This may count toward either the Math Course Requirement or Elective Course Requirement, but not both.

  • PHYSICS 41 and 61 are mandatory S/NC grading, so at most, one may be counted as a Math elective.

  • STATS 116 (No longer offered) and STATS 118 or STATS 200 (at most one, but not both, and neither if STATS 116 is being used) may count toward the Math Course Requirement or Elective Course Requirement, but not both.

Writing in the Major (WIM)

Writing in the Major (WIM)

Complete at least 1 of the following courses:
MATH 190

Communicating Mathematics

3 Units
MATH 195

Senior Exploration and Synthesis

4 Units

All majors, including transfer students, need to complete the WIM requirement.

The two course options for fulfilling WIM, Math 190 and Math 195, are each offered twice per year in Fall and in Winter.  Enrollment in each of these courses is limited to officially approved Math majors, and is done via an application process that begins near the end of the preceding Spring (all Math majors who have not fulfilled WIM will be contacted about applying in late Spring).  

Math 190 is a 3-unit class with background prerequisites in proof-based math (such as Math 113 or 115 with a grade of at least B)  and does not entail learning new mathematics, so it does not count as one of the eight 100-level classes toward the Math major but does count as an elective. It can be taken in the second year or later. 

Math 195 is a 4-unit class limited to senior Math majors only, and also fulfills the capstone requirement. This course entails learning new mathematics, and as such it does count as one of the eight 100-level classes toward the Math major.

Capstone

Mathematics Capstone Requirements

Mathematics majors have interests in many possible directions (e.g., applied math, algebra, geometry, probability, CS, finance, etc.). The capstone in Mathematics aims to ensure that each student takes part in some independent work or course experience in their area(s) of mathematical interest that exposes them to a culmination of the subject in these areas.  

Such experiences can expose a student to an underlying and more abstract unity among disparate methods, a body of ideas displaying the intellectual beauty of the subject, or a compelling application of abstract mathematical concepts and techniques to a real-world context. Using these goals as a guide, the capstone in Mathematics can be fulfilled by carrying out (at least) one of the following three options: 

Complete any of the following requirements:
Honors Thesis Capstone Option

Completion of the honors program.

Earn at least 6 units in the following course(s):
MATH 197

Senior Honors Thesis

1-6 Units
or
Undergraduate Course Option
Complete at least 1 of the following courses:
MATH 117

Advanced Complex Analysis

4 Units
MATH 122

Modules and Group Representations

4 Units
MATH 147

Differential Topology

4 Units
MATH 154

Algebraic Number Theory

4 Units
MATH 155

Analytic Number Theory

4 Units
MATH 158

Probability and Stochastic Differential Equations for Applications

4 Units
MATH 173

Theory of Partial Differential Equations

4 Units
PHIL 152

Computability and Logic

4 Units
or
Graduate Course Option
Complete at least 1 of the following courses:
MATH 205A

Real Analysis

3 Units
MATH 205B

Real Analysis

3 Units
MATH 210A

Modern Algebra I

3 Units
MATH 210B

Modern Algebra II

3 Units
MATH 210C

Lie Theory

3 Units
MATH 215A

Algebraic Topology

3 Units
MATH 215B

Differential Topology

3 Units
MATH 215C

Differential Geometry

3 Units
MATH 216A

Introduction to Algebraic Geometry

3 Units
MATH 230A

Theory of Probability I

3 Units
or
Capstone Project Option
Complete all of the following course(s):
MATH 195

Senior Exploration and Synthesis

4 Units

Subplans

Optional
Students have the option to complete a subplan as part of this program. A subplan is an officially declared and defined set of requirements that allows students to go deeper into a discipline or focus within a program.

Computer Science Theory/Discrete Mathematics Subplan

Required Courses

Complete all of the following requirements:
Required Courses
Complete all of the following course(s):
MATH 116

Complex Analysis

4 Units
or
MATH 143

Differential Geometry

4 Units
or
MATH 144

Introduction to Topology and Geometry

4 Units
or
MATH 147

Differential Topology

4 Units
MATH 120

Groups and Rings

4 Units
MATH 171

Fundamental Concepts of Analysis

4 Units
CS 154

Introduction to the Theory of Computation

3-4 Units
CS 161

Design and Analysis of Algorithms

3-5 Units

Subplan students must include the following courses in their program of study (which can count toward the both the Math Course Requirement and the Elective Course Requirement).

All subplan courses must be taken for a Letter grade.

Combinatorics Course Requirement

Complete at least 2 of the following courses:
MATH 61DM

Modern Mathematics: Discrete Methods

5 Units
MATH 107

Graph Theory

4 Units
MATH 108

Introduction to Combinatorics and Its Applications

4 Units
MATH 233A

Topics in Combinatorics

3 Units
or
MATH 233C

Topics in Combinatorics

3 Units

Probability Course Requirement

Complete at least 1 of the following courses:
MATH 63DM

Modern Mathematics: Discrete Methods

5 Units
MATH 151

Introduction to Probability Theory

4 Units
MATH 230A

Theory of Probability I

3 Units
STATS 118

Probability Theory for Statistical Inference

3-4 Units

Math Course Requirement

Complete at least 2 of the following courses:
MATH 113

Linear Algebra and Matrix Theory

4 Units
MATH 152

Elementary Theory of Numbers

4 Units
MATH 154

Algebraic Number Theory

4 Units
MATH 155

Analytic Number Theory

4 Units
MATH 159

Discrete Probabilistic Methods

4 Units
MATH 161

Set Theory

4 Units

Computer Science Course Requirement

Complete at least 4 of the following courses:
CS 168

The Modern Algorithmic Toolbox

3-4 Units
CS 250

Algebraic Error Correcting Codes

3 Units
CS 254

Computational Complexity

3 Units
CS 254B

Computational Complexity II

3 Units
CS 255

Introduction to Cryptography

3 Units
CS 256

Algorithmic Fairness

3 Units
CS 259Q

Quantum Computing

3 Units
CS 261

Combinatorial Optimization

3 Units
CS 263

Counting and Sampling

3 Units
CS 265

Randomized Algorithms and Probabilistic Analysis

3 Units
CS 269I

Incentives in Computer Science

3 Units

Honors

Optional
Students may choose to pursue department honors by completing a program of special advanced work, such as an honors thesis, in addition to their major requirements.

Honors

Required Courses

Honors students must complete the core program requirements for all Math majors; see Core Program Requirements (above).

Honors students must have a minimum GPA of 3.3 across all Math courses to apply to the program.

It is strongly recommended that either the Math 60CM or Math 60DM sequences be taken during the student's freshman year. This is a recommendation, not a requirement.

Complete all of the following requirements:
I. Required Courses
Complete all of the following course(s):
MATH 120

Groups and Rings

4 Units
MATH 171

Fundamental Concepts of Analysis

4 Units

and
II. Required Courses
Complete at least 1 of the following courses:
MATH 116

Complex Analysis

4 Units
MATH 143

Differential Geometry

4 Units
MATH 144

Introduction to Topology and Geometry

4 Units
MATH 147

Differential Topology

4 Units

Honors Course Requirements

III. 7 courses numbered Math 121 and above that are not being used for requirements I and II. This must include at least one course from the three areas below: Algebra, Analysis, and Geometry/Topology.
  • MATH 193 does not count toward this requirement.

  • MATH 197 does not count toward this requirement.

  • MATH 198 does not count toward this requirement.

  • PHIL 151 and PHIL 152 may count toward this requirement.

  • One 3-unit MATH 199 course may count toward this requirement with approval from the Director of Undergraduate Studies before the start of the course. Honors students must submit the request to the Student Services Officer two weeks before the beginning of the quarter for review. The criteria for approval include weekly homework graded by the instructor and topic coverage appropriate for Math honors.

Algebra Course Requirement

Complete at least 1 of the following courses:
MATH 121

Galois Theory

4 Units
MATH 122

Modules and Group Representations

4 Units
MATH 145

Algebraic Geometry

4 Units
MATH 152

Elementary Theory of Numbers

4 Units
MATH 154

Algebraic Number Theory

4 Units

Analysis Course Requirement

Complete at least 1 of the following courses:
MATH 116

Complex Analysis

4 Units
MATH 117

Advanced Complex Analysis

4 Units
MATH 136

Stochastic Processes

4 Units
MATH 151

Introduction to Probability Theory

4 Units
MATH 155

Analytic Number Theory

4 Units
MATH 172

Fourier Analysis

4 Units
MATH 173

Theory of Partial Differential Equations

4 Units
MATH 175

Elementary Functional Analysis

4 Units

Geometry/Topology Course Requirement

Complete at least 1 of the following courses:
MATH 143

Differential Geometry

4 Units
MATH 144

Introduction to Topology and Geometry

4 Units
MATH 147

Differential Topology

4 Units
MATH 148

Algebraic Topology

4 Units

Graduate Level Course Recommended

Recommended to complete at least 1 of the following Courses:

MATH 205A, MATH 210A, MATH 215A, OR MATH230A opens in a new tab

Note: This is a recommendation. This is not required to complete Math honors.

Thesis Requirement. Completion of an honors thesis is required.

Earn at least 6 units in the following course(s):
MATH 197

Senior Honors Thesis

1-6 Units
  • MATH 197 must be taken over all three quarters for 2-3 units per quarter.

  • The 6 units of MATH 197 is in addition to the Math Course Requirement and Elective Course Requirement for the Math major.

  • A complete draft of the honors thesis is due by the end of week 7 in Spring quarter. A final draft is due by 5pm on the last day of classes in Spring quarter.