## MATH 53: Ordinary Differential Equations with Linear Algebra

Ordinary differential equations and initial value problems, systems of linear differential equations with constant coefficients, applications of second-order equations to oscillations, matrix exponentials, Laplace transforms, stability of non-linear systems and phase plane analysis, numerical methods. Prerequisite: 51 or equivalents.

Terms: Aut, Win, Spr, Sum
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Ionel, E. (PI)
;
La, J. (PI)
;
Lee, S. (PI)
;
Varolgunes, U. (PI)
;
Ottolini, A. (TA)
;
Simper, M. (TA)

## MATH 53A: Ordinary Differential Equations with Linear Algebra, ACE

Additional problem solving session for
Math 53 guided by a course assistant. Concurrent enrollment in
Math 53 required. Application required:
https://engineering.stanford.edu/students-academics/equity-and-inclusion-initiatives/undergraduate-programs/additional-calculus

Terms: Aut, Win, Spr, Sum
| Units: 1

## MATH 56: Proofs and Modern Mathematics

How do mathematicians think? Why are the mathematical facts learned in school true? In this course students will explore higher-level mathematical thinking and will gain familiarity with a crucial aspect of mathematics: achieving certainty via mathematical proofs, a creative activity of figuring out what should be true and why. This course is ideal for students who would like to learn about the reasoning underlying mathematical results, but at a pace and level of abstraction not as intense as
Math 61CM/DM, as a consequence benefiting from additional opportunity to explore the reasoning. Familiarity with one-variable calculus is strongly recommended at least at the AB level of AP Calculus since a significant part of the course develops some of the main results in that material systematically from a small list of axioms. We also address linear algebra from the viewpoint of a mathematician, illuminating notions such as fields and abstract vector spaces. This course may be paired with
Math 51; though that course is not a pre- or co-requisite.

Terms: Aut, Win
| Units: 3
| UG Reqs: WAY-FR

## MATH 61CM: Modern Mathematics: Continuous Methods

This is the first part of a theoretical (i.e., proof-based) sequence in multivariable calculus and linear algebra, providing a unified treatment of these topics. Covers general vector spaces, linear maps and duality, eigenvalues, inner product spaces, spectral theorem, metric spaces, differentiation in Euclidean space, submanifolds of Euclidean space as local graphs, integration on Euclidean space, and many examples. The linear algebra content is covered jointly with
Math 61DM. Students should know 1-variable calculus and have an interest in a theoretical approach to the subject. Prerequisite: score of 5 on the BC-level Advanced Placement calculus exam, or consent of the instructor.

Terms: Aut
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Ryzhik, L. (PI)
;
Hernandez, F. (TA)

## MATH 61DM: Modern Mathematics: Discrete Methods

This is the first part of a theoretical (i.e., proof-based) sequence in discrete mathematics and linear algebra. Covers general vector spaces, linear maps and duality, eigenvalues, inner product spaces, spectral theorem, counting techniques, and linear algebra methods in discrete mathematics including spectral graph theory and dimension arguments. The linear algebra content is covered jointly with
Math 61CM. Students should have an interest in a theoretical approach to the subject. Prerequisite: score of 5 on the BC-level Advanced Placement calculus exam, or consent of the instructor.nnThis sequence is not appropriate for students planning to major in natural sciences, economics, or engineering, but is suitable for majors in any other field (such as MCS ("data science"), computer science, and mathematics).

Terms: Aut
| Units: 5
| UG Reqs: WAY-FR

Instructors:
Vondrak, J. (PI)
;
Pham, H. (TA)

## MATH 62CM: Modern Mathematics: Continuous Methods

A proof-based introduction to manifolds and the general Stokes' theorem. This includes a treatment of multilinear algebra, further study of submanifolds of Euclidean space (with many examples), differential forms and their geometric interpretations, integration of differential forms, Stokes' theorem, and some applications to topology. Prerequisites:
Math 61CM.

Terms: Win
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Chodosh, O. (PI)

## MATH 62DM: Modern Mathematics: Discrete Methods

This is the second part of a proof-based sequence in discrete mathematics. This course covers topics in elementary number theory, group theory, and discrete Fourier analysis. For example, we'll discuss the basic examples of abelian groups arising from congruences in elementary number theory, as well as the non-abelian symmetric group of permutations. Prerequisites: 61DM or 61CM.

Terms: Win
| Units: 5
| UG Reqs: WAY-FR

Instructors:
Soundararajan, K. (PI)

## MATH 63CM: Modern Mathematics: Continuous Methods

A proof-based course on ordinary differential equations. Topics include the inverse and implicit function theorems, implicitly-defined submanifolds of Euclidean space, linear systems of differential equations and necessary tools from linear algebra, stability and asymptotic properties of solutions to linear systems, existence and uniqueness theorems for nonlinear differential equations, behavior of solutions near an equilibrium point, and Sturm-Liouville theory. Prerequisite:
Math 61CM.

Terms: Sum
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Luk, J. (PI)

## MATH 63DM: Modern Mathematics: Discrete Methods

Third part of a proof-based sequence in discrete mathematics. The first half of the quarter gives a fast-paced coverage of probability and random processes with an intensive use of generating functions. The second half treats entropy, Shannon¿s coding theorem, game theory, probabilistic methods in solving non-probabilistic problems; some of these topics may vary from year to year. nnPrerequisite:
Math 61DM or 61CM

Terms: Sum
| Units: 5
| UG Reqs: WAY-FR

Instructors:
Tokieda, T. (PI)

## MATH 70SI: The Game of Go: Strategy, Theory, and History

Strategy and mathematical theories of the game of Go, with guest appearance by a professional Go player.

Last offered: Autumn 2014

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