Introductory Courses

Dusty Grundmeier

2021 Fall (2 Credits)

Schedule: TR 9:00 AM - 10:15 AM

Instructor Permissions: Instructor

Enrollment Cap: n/a

In his seminal work from 1945, How to Solve It, George Polya introduced principles of mathematical problem solving that are widely applicable to problems in science and engineering. This year-long class focuses on building a powerful and portable problem-solving and modeling tool kit while bridging the divide between mathematics and science courses. The second semester will be organized around projects in areas of student interest. The projects are designed to launch students into undergraduate research opportunities. Math ESPb topics often include differential equations, mathematical modeling, data analysis and visualization, and mathematical and scientific programming tools. Both Math ESPA and Math ESPB must be taken in the same academic year to receive credit.

Additional Course Attributes::

Attribute Value(s)
All: Cross Reg Availability Available for Harvard Cross Registration
Full Year Course Indivisible Course
FAS: Course Level Primarily for Undergraduate Students
FAS Divisional Distribution Science & Engineering & Applied Science

Reshma Menon

2022 Spring (2 Credits)

Schedule: TBD

Instructor Permissions: Instructor

Enrollment Cap: n/a

In his seminal work from 1945, How to Solve It, George Polya introduced principles of mathematical problem solving that are widely applicable to problems in science and engineering. This year-long class focuses on building a powerful and portable problem-solving and modeling tool kit while bridging the divide between mathematics and science courses. The second semester will be organized around projects in areas of student interest. The projects are designed to launch students into undergraduate research opportunities. Math ESPb topics often include differential equations, mathematical modeling, data analysis and visualization, and mathematical and scientific programming tools. Both Math ESPA and Math ESPB must be taken in the same academic year to receive credit.

Additional Course Attributes::

Attribute Value(s)
FAS: Course Level Primarily for Undergraduate Students
Course Search Attributes Display Only in Course Search
All: Cross Reg Availability Available for Harvard Cross Registration
Full Year Course Indivisible Course
FAS Divisional Distribution Science & Engineering & Applied Science

Reshma Menon

2021 Fall (4 Credits)

Schedule: TBD

Instructor Permissions: None

Enrollment Cap: n/a

The study of functions and their rates of change. Fundamental ideas of calculus are introduced early and used to provide a framework for the study of mathematical modeling involving algebraic, exponential, and logarithmic functions. Thorough understanding of differential calculus promoted by year-long reinforcement. Applications to biology and economics emphasized according to the interests of our students.

Course Notes:
This is a lecture course taught in small sections. This course, when taken together with Mathematics Mb, can be followed by Mathematics 1b. Mathematics Ma and Mb together cover all the material in Mathematics 1a (and more).

Additional Course Attributes:

Attribute Value(s)
All: Cross Reg Availability Available for Harvard Cross Registration
Quantitative Reasoning with Data Yes
FAS: Course Level Primarily for Undergraduate Students
FAS Divisional Distribution Science & Engineering & Applied Science

Voula Collins

2022 Spring (4 Credits)

Schedule: TBD

Instructor Permissions: None

Enrollment Cap: n/a

Continued investigation of functions and differential calculus through modeling; an introduction to integration with applications; an introduction to differential equations. Solid preparation for Mathematics 1b.

Course Notes:
This is a lecture course taught in small sections. This course, when taken together with Mathematics Ma, can be followed by Mathematics 1b. Mathematics Ma and Mathematics Mb together cover all the material in Mathematics 1a (and more).
Requirements:
Prerequisite: Mathematics MA

Additional Course Attributes:

Attribute Value(s)
FAS: Course Level Primarily for Undergraduate Students
All: Cross Reg Availability Available for Harvard Cross Registration
FAS Divisional Distribution Science & Engineering & Applied Science

Brendan Kelly

2021 Fall (4 Credits)

Schedule: MWF 12:00 PM - 1:15 PM

Instructor Permissions: None

Enrollment Cap: n/a

This course develops a portable toolkit of quantitative skills that supports students’ strategic thinking. At the center of the course is a set of case studies from a broad range of organizations that require comprehensive quantitative analysis to properly diagnose and address the problems presented. After taking this course, students’ strategic thinking will be bolstered by the ability to develop mathematical models, apply core ideas from differential and integral calculus to problems in economics and social science, and make use of the R statistical package to carry out data analysis. Each analytical tool comes to life in an authentic application. The course focuses not just on how to carry out the analysis, but how to communicate results to a non-mathematical audience in simple functional language. This course should be seen as an applied alternative for Math Ma/Math Mb or Math 1a for students interested in Economics and the Social Sciences. Students completing Math Qa and Math Qb will satisfy the Math 1a requirement for the Economics Concentration. Completing both Math Qa and Math Qb can count as a course toward the Mathematics Concentration. After completing Math Qa and Math Qb, students’ next math course could be Math 1b or Math 18a, depending on students’ interests.

Additional Course Attributes:

Attribute Value(s)
All: Cross Reg Availability Available for Harvard Cross Registration
FAS: Course Level Primarily for Undergraduate Students
FAS Divisional Distribution Science & Engineering & Applied Science

Brendan Kelly

2022 Spring (4 Credits)

Schedule: MWF 12:00 PM - 1:15 PM

Instructor Permissions: None

Enrollment Cap: n/a

This course develops a portable toolkit of quantitative skills that supports students’ strategic thinking. At the center of the course is a set of case studies from a broad range of organizations that require comprehensive quantitative analysis to properly diagnose and address the problems presented. After taking this course, students’ strategic thinking will be bolstered by the ability to develop mathematical models, apply core ideas from differential and integral calculus to problems in economics and social science, and make use of the R statistical package to carry out data analysis. Each analytical tool comes to life in an authentic application. The course focuses not just on how to carry out the analysis, but how to communicate results to a non-mathematical audience in simple functional language. This course should be seen as an applied alternative for Math Ma/Math Mb or Math 1a for students interested in Economics and the Social Sciences. Students completing Math Qa and Math Qb will satisfy the Math 1a requirement for the Economics Concentration. Completing both Math Qa and Math Qb can count as a course toward the Mathematics Concentration. After completing Math Qa and Math Qb, students’ next math course could be Math 1b or Math 18a, depending on students’ interests.

Additional Course Attributes:

Attribute Value(s)
All: Cross Reg Availability Available for Harvard Cross Registration
FAS: Course Level Primarily for Undergraduate Students
FAS Divisional Distribution Science & Engineering & Applied Science

Tyler Bongers

2021 Fall (4 Credits)

Schedule: TBD

Instructor Permissions: None

Enrollment Cap: n/a

The development of calculus by Newton and Leibniz ranks among the greatest achievements of the past millennium. This course will help you see why by introducing: how differential calculus treats rates of change; how integral calculus treats accumulation; and how the fundamental theorem of calculus links the two. These ideas will be applied to problems from many other disciplines.

Course Notes:
In the fall, Math 1a is taught in small sections. In the spring, Math 1a is taught in a larger lecture format. Mathematics Ma and Mb together cover all of the material in Mathematics 1a (and more).
Recommended Prep:
A solid background in precalculus.
Requirements:
Anti-requisite: cannot be taken for credit if MATH S-1AB already complete

Additional Course Attributes:

Attribute Value(s)
All: Cross Reg Availability Available for Harvard Cross Registration
Quantitative Reasoning with Data Yes
FAS Divisional Distribution Science & Engineering & Applied Science
FAS: Course Level Primarily for Undergraduate Students
Oliver Knill

2022 Spring (4 Credits)

Schedule: TBD

Instructor Permissions: None

Enrollment Cap: n/a

The development of calculus by Newton and Leibniz ranks among the greatest achievements of the past millennium. This course will help you see why by introducing: how differential calculus treats rates of change; how integral calculus treats accumulation; and how the fundamental theorem of calculus links the two. These ideas will be applied to problems from many other disciplines.

Course Notes:
In the fall, Math 1a is taught in small sections. In the spring, Math 1a is taught in a larger lecture format. Mathematics Ma and Mb together cover all of the material in Mathematics 1a (and more).
Recommended Prep:
A solid background in precalculus.
Requirements:
Anti-requisite: cannot be taken for credit if MATH S-1AB already complete

Additional Course Attributes:

Attribute Value(s)
All: Cross Reg Availability Available for Harvard Cross Registration
Quantitative Reasoning with Data Yes
FAS Divisional Distribution Science & Engineering & Applied Science
FAS: Course Level Primarily for Undergraduate Students
Cliff Taubes

2021 Fall (4 Credits)

Schedule: TBD

Instructor Permissions: None

Enrollment Cap: n/a

Speaking the language of modern mathematics requires fluency with the topics of this course: infinite series, integration, and differential equations. Model practical situations using integrals and differential equations. Learn how to represent interesting functions using series and find qualitative, numerical, and analytic ways of studying differential equations. Develop both conceptual understanding and the ability to apply it.

Course Notes:
This is a lecture taught in small sections.
Recommended Prep:
Mathematics 1a or Ma and Mb; or 5 on the AB advanced placement test; or an equivalent background in mathematics.
Requirements:
Anti-requisite: cannot be taken for credit if MATH S-1AB already complete

Additional Course Attributes:

Attribute Value(s)
All: Cross Reg Availability Available for Harvard Cross Registration
Quantitative Reasoning with Data Yes
FAS Divisional Distribution Science & Engineering & Applied Science
FAS: Course Level Primarily for Undergraduate Students
Hakim Walker

2022 Spring (4 Credits)

Schedule: TBD

Instructor Permissions: None

Enrollment Cap: n/a

Speaking the language of modern mathematics requires fluency with the topics of this course: infinite series, integration, and differential equations. Model practical situations using integrals and differential equations. Learn how to represent interesting functions using series and find qualitative, numerical, and analytic ways of studying differential equations. Develop both conceptual understanding and the ability to apply it.

Course Notes:
This is a lecture taught in small sections.
Recommended Prep:
Mathematics 1a or Ma and Mb; or 5 on the AB advanced placement test; or an equivalent background in mathematics.
Requirements:
Anti-requisite: cannot be taken for credit if MATH S-1AB already complete

Additional Course Attributes:

Attribute Value(s)
All: Cross Reg Availability Available for Harvard Cross Registration
Quantitative Reasoning with Data Yes
FAS Divisional Distribution Science & Engineering & Applied Science
FAS: Course Level Primarily for Undergraduate Students

2021 Fall (4 Credits)

Schedule: TBD

Instructor Permissions: None

Enrollment Cap: n/a

Focus on concepts and techniques of multivariable calculus most useful to those studying the social sciences, particularly economics: functions of several variables; partial derivatives; directional derivatives and the gradient; constrained and unconstrained optimization, including the method of Lagrange multipliers. Covers linear and polynomial approximation and integrals for a single variable and multivariable functions; modeling with derivatives. Covers topics from Math 21a most useful to social sciences.

Course Notes:
Mathematics 21b can be taken before or after Mathematics 18. Examples draw primarily from economics and the social sciences, though Mathematics 18 may be useful to students in certain natural sciences. Students whose main interests lie in the physical sciences, mathematics, or engineering should consider Math 21a or Applied Math 22a.
Recommended Prep:
Mathematics 1b or equivalent, or a 5 on the BC Advanced Placement Examination in Mathematics.
Requirements:
Anti-requisite: Not to be taken in addition to Mathematics 21a or Applied Mathematics 22a

Additional Course Attributes:

Attribute Value(s)
All: Cross Reg Availability Available for Harvard Cross Registration
Quantitative Reasoning with Data Yes
FAS Divisional Distribution Science & Engineering & Applied Science
FAS: Course Level Primarily for Undergraduate Students
John Cain

2021 Fall (4 Credits)

Schedule: MWF 10:30 AM - 11:45 AM

Instructor Permissions: None

Enrollment Cap: n/a

Considers the construction and analysis of mathematical models that arise in the life sciences, ecology and environmental life science. Introduces mathematics that includes multivariable calculus, differential equations in one or more variables, vectors, matrices, and linear and non-linear dynamical systems. Taught via examples from current literature (both good and bad).

Course Notes:
This course is recommended over Math 21a for those planning to concentrate on the life sciences and ESPP. Can be taken with or without Mathematics 21a,b. Students with interests in the social sciences and economics might consider Mathematics 18. This course can be taken before or after Mathematics 18.
Recommended Prep:
A course in one variable calculus preferably at the level of Mathematics 1b.

Additional Course Attributes:

Attribute Value(s)
All: Cross Reg Availability Available for Harvard Cross Registration
Quantitative Reasoning with Data Yes
FAS Divisional Distribution Science & Engineering & Applied Science
FAS: Course Level Primarily for Undergraduate Students
Brendan Kelly

2022 Spring (4 Credits)

Schedule: MWF 01:30 PM - 02:45 PM

Instructor Permissions: None

Enrollment Cap: n/a

Probability, statistics and linear algebra with applications to life sciences, chemistry, environmental sciences, economics, and social sciences. Students will learn to use computing software to perform relevant calculations on data sets coming from these areas of study. Linear algebra includes matrices, eigenvalues, eigenvectors, determinants, and applications to probability, statistics, dynamical systems. Basic probability and statistics are introduced, as are standard models, techniques, and their uses including the central limit theorem, Markov chains, curve fitting, regression, and pattern analysis.

Additional Course Attributes:

Attribute Value(s)
FAS Divisional Distribution Science & Engineering & Applied Science
All: Cross Reg Availability Available for Harvard Cross Registration
Quantitative Reasoning with Data Yes
FAS: Course Level Primarily for Undergraduate Students
Janet Chen

2021 Fall (4 Credits)

Schedule: TBD

Instructor Permissions: None

Enrollment Cap: n/a

To see how calculus applies in practical situations described by more than one variable, we study: Vectors, lines, planes, parameterization of curves and surfaces, partial derivatives, directional derivatives and the gradient, optimization and critical point analysis, including constrained optimization and the Method of Lagrange Multipliers, integration over curves, surfaces and solid regions using Cartesian, polar, cylindrical, and spherical coordinates, divergence and curl of vector fields, and the Green’s, Stokes’s, and Divergence Theorems.

Course Notes:
This course is taught in small sections. May not be taken for credit by students who have passed Applied Mathematics 22a.
Recommended Prep:
Mathematics 1b or an equivalent background in mathematics.
Requirements:
Anti-requisite: Not to be taken in addition to AM 22b.

Additional Course Attributes:

Attribute Value(s)
All: Cross Reg Availability Available for Harvard Cross Registration
Quantitative Reasoning with Data Yes
FAS Divisional Distribution Science & Engineering & Applied Science
FAS: Course Level Primarily for Undergraduate Students
David Freund

2022 Spring (4 Credits)

Schedule: TBD

Instructor Permissions: None

Enrollment Cap: n/a

To see how calculus applies in practical situations described by more than one variable, we study: Vectors, lines, planes, parameterization of curves and surfaces, partial derivatives, directional derivatives and the gradient, optimization and critical point analysis, including constrained optimization and the Method of Lagrange Multipliers, integration over curves, surfaces and solid regions using Cartesian, polar, cylindrical, and spherical coordinates, divergence and curl of vector fields, and the Green’s, Stokes’s, and Divergence Theorems.

Course Notes:
This course is taught in small sections. May not be taken for credit by students who have passed Applied Mathematics 22a.
Recommended Prep:
Mathematics 1b or an equivalent background in mathematics.
Requirements:
Anti-requisite: Not to be taken in addition to AM 22b

Additional Course Attributes:

Attribute Value(s)
FAS Divisional Distribution Science & Engineering & Applied Science
All: Cross Reg Availability Available for Harvard Cross Registration
FAS: Course Level Primarily for Undergraduate Students
Quantitative Reasoning with Data Yes
Jill Guerra

2021 Fall (4 Credits)

Schedule: TBD

Instructor Permissions: None

Enrollment Cap: n/a

Matrices provide the algebraic structure for solving myriad problems across the sciences. We study matrices and related topics such as linear transformations and linear spaces, determinants, eigenvalues, and eigenvectors. Applications include dynamical systems, ordinary and partial differential equations, and an introduction to Fourier series.

Course Notes:
This is a lecture taught in small sections. May not be taken by students who have passed Applied Mathematics 21b.
Recommended Prep:
Mathematics 1b or an equivalent background in mathematics. Mathematics 21a is commonly taken before Mathematics 21b, but is not a prerequisite, although familiarity with partial derivatives is useful.
Requirements:
Anti-requisite: Not to be taken in addition to AM 22a.

Additional Course Attributes:

Attribute Value(s)
FAS Divisional Distribution Science & Engineering & Applied Science
All: Cross Reg Availability Available for Harvard Cross Registration
FAS: Course Level Primarily for Undergraduate Students
Quantitative Reasoning with Data Yes
Janet Chen, Tyler Bongers

2022 Spring (4 Credits)

Schedule: TBD

Instructor Permissions: None

Enrollment Cap: n/a

Matrices provide the algebraic structure for solving myriad problems across the sciences. We study matrices and related topics such as linear transformations and linear spaces, determinants, eigenvalues, and eigenvectors. Applications include dynamical systems, ordinary and partial differential equations, and an introduction to Fourier series.

Course Notes:
This is a lecture taught in small sections. May not be taken by students who have passed Applied Mathematics 21b.
Recommended Prep:
Mathematics 1b or an equivalent background in mathematics. Mathematics 21a is commonly taken before Mathematics 21b, but is not a prerequisite, although familiarity with partial derivatives is useful.
Requirements:
Anti-requisite: Not to be taken in addition to AM 22a.

Additional Course Attributes:

Attribute Value(s)
FAS Divisional Distribution Science & Engineering & Applied Science
All: Cross Reg Availability Available for Harvard Cross Registration
FAS: Course Level Primarily for Undergraduate Students
Quantitative Reasoning with Data Yes
Dusty Grundmeier

2021 Fall (4 Credits)

Schedule: MWF 12:00 PM - 01:15 PM

Instructor Permissions: None

Enrollment Cap: n/a

Mathematics 22 covers multivariable calculus and linear algebra for students interested in mathematical sciences. It covers the same topics as Mathematics 21, but does so with more rigor. Students are taught techniques of proof and mathematical reasoning. The workload and content is comparable with the Mathematics 21 sequence. But unlike the latter, the linear algebra and calculus are more interlinked. The content of Math 22a is mostly aligned with Math 21b (linear algebra), and the content of Math 22b is mostly aligned with Math 21a (multivariable calculus).

Additional Course Attributes:

Attribute Value(s)
FAS Divisional Distribution Science & Engineering & Applied Science
All: Cross Reg Availability Available for Harvard Cross Registration
FAS: Course Level Primarily for Undergraduate Students
Quantitative Reasoning with Data Yes
Dusty Grundmeier

2022 Spring (4 Credits)

Schedule: MWF 12:00 PM - 01:15 PM

Instructor Permissions: Instructor

Enrollment Cap: n/a

A continuation of Mathematics 22a

Additional Course Attributes:

Attribute Value(s)
FAS Divisional Distribution Science & Engineering & Applied Science
All: Cross Reg Availability Available for Harvard Cross Registration
FAS: Course Level Primarily for Undergraduate Students
Paul Bamberg

2022 Spring (4 Credits)

Schedule: F 03:00 PM - 05:45 PM

Instructor Permissions: None

Enrollment Cap: n/a

Proof strategies and logic. Sets, countability, sigma fields, and axiomatic foundations of probability. Summation of series and evaluation of multiple integrals, with emphasis on calculation of expectation and variance. Abstract vector spaces and inner product spaces, with applications to the analysis of large datasets. Key functions and theorems of mathematical statistics. A brief introduction to classical vector calculus as used in electromagnetic theory. Students will learn to use some of the statistical and graphical display tools in the R scripting language.

Course Notes:
This course is a sequel to Mathematics 23a, recommended for students who are not sure of their concentration or who are thinking about a concentration in the social sciences, economics, computer science, life sciences or data science. Graduate students wishing to take this course for credit should speak with Dr. Bamberg to arrange enrollment in Mathematics 370 instead.
Recommended Prep:
Mathematics 23a or Mathematics 21a and 21b. The latter option is for seniors who are preparing for graduate programs in statistics, computer science, or data science.
Requirements:
Prerequisite: MATH 23A OR (MATH 21A AND MATH 21B) AND (Not to be taken in addition to MATH 23B)

Additional Course Attributes:

Attribute Value(s)
FAS: Course Level Primarily for Undergraduate Students
Quantitative Reasoning with Data Yes
All: Cross Reg Availability Available for Harvard Cross Registration
FAS Divisional Distribution Science & Engineering & Applied Science
John Cain

2021 Fall (4 Credits)

Schedule: MWF 09:00 AM - 10:15 AM

Instructor Permissions: None

Enrollment Cap: n/a

A rigorous treatment of linear algebra. Topics include Construction of number systems; fields, vector spaces, and linear transformations; eigenvalues and eigenvectors, determinants and inner products. Metric spaces, compactness, and connectedness.

Course Notes:
Expect to spend a lot of time doing mathematics.
Recommended Prep:
5 on the Calculus BC Advanced Placement Examination and some familiarity with writing proofs, or the equivalent as determined by the instructor.

Additional Course Attributes:

Attribute Value(s)
FAS Divisional Distribution Science & Engineering & Applied Science
FAS: Course Level Primarily for Undergraduate Students
All: Cross Reg Availability Available for Harvard Cross Registration
John Cain

2022 Spring (4 Credits)

Schedule: MWF 09:00 AM - 10:15 AM

Instructor Permissions: None

Enrollment Cap: n/a

A rigorous treatment of basic analysis. Topics include convergence, continuity, differentiation, the Riemann integral, uniform convergence, the Stone-Weierstrass theorem, Fourier series, differentiation in several variables. Additional topics, including the classical results of vector calculus in two and three dimensions, as time allows.

Course Notes:
Expect to spend a lot time doing mathematics.
Requirements:
Prerequisite: Mathematics 25A OR Mathematics 55A

Additional Course Attributes:

Attribute Value(s)
All: Cross Reg Availability Available for Harvard Cross Registration
FAS Divisional Distribution Science & Engineering & Applied Science
FAS: Course Level Primarily for Undergraduate Students
Denis Auroux

2021 Fall (4 Credits)

Schedule: MWF 10:30 AM - 11:45 AM

Instructor Permission: Instructor

Enrollment Cap: n/a

A rigorous introduction to abstract algebra, including group theory and linear algebra. This course covers the equivalent of Mathematics 25a and Mathematics 122 and prepares students for Mathematics 123 and other advanced courses in number theory and algebra. (A course in analysis such as Mathematics 25b or 55b is recommended for Spring semester.)

Course Notes:
Mathematics 55a is an intensive course for students who are comfortable with abstract mathematics. (Students without this background will gain it and learn the material from Math 55a,b in other courses by continuing into the Mathematics Concentration as sophomores.) Students can switch between Mathematics 55a and either Mathematics 25a, 23a, 22a, 21a during the first three weeks without penalty. Live lectures will take place Mondays, Wednesdays, and Fridays, 10: 30-11:45am, US Eastern time. Attendance is recommended, but not mandatory. The instructor will offer other opportunities for interaction with students who cannot attend the live lecture.
Recommended Prep:
Familiarity with proofs and abstract reasoning; and commitment to a fast-moving course.

Additional Course Attributes:

Attribute Value(s)
FAS: Course Level Primarily for Undergraduate Students
All: Cross Reg Availability Available for Harvard Cross Registration
FAS Divisional Distribution Science & Engineering & Applied Science
Denis Auroux

2022 Spring (4 Credits)

Schedule: MWF 10:30 AM - 11:45 AM

Instructor Permissions: Instructor

Enrollment Cap: n/a

A rigorous introduction to real and complex analysis. This course covers the equivalent of Mathematics 25b and Mathematics 113, and prepares students for Mathematics 114 and other advanced courses in analysis.

Course Notes:
Mathematics 55b is an intensive course for students having significant experience with abstract mathematics.
Requirements:
Prerequisite: Mathematics 55A

Additional Course Attributes:

Attribute Value(s)
FAS: Course Level Primarily for Undergraduate Students
All: Cross Reg Availability Available for Harvard Cross Registration
FAS Divisional Distribution Science & Engineering & Applied Science