## 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 and 63CM.

Terms: Spr
| 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:
Fox, J. (PI)

## MATH 63CM: Modern Mathematics: Continuous Methods

A proof-based course on ordinary differential equations and other applications of derivatives. 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: Win
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Ryzhik, L. (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: Spr
| 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

## MATH 79SI: Proof Positive: Principles of Mathematics

What is a mathematical proof, and where do proofs come from? Students will become comfortable with fundamental techniques of mathematical proof through practice with interesting and accessible examples from many areas of math. Students will additionally hone their communication skills and develop their ability to formulate and answer precise mathematical questions. Topics include direct proof, proof by contrapositive, proof by contradiction, many applications of mathematical induction, constructing good definitions, and useful writing habits. The course is designed to prepare students who have completed or are concurrently enrolled in
MATH 51 to succeed in introductory proof-based math classes at the level of
MATH 115 or
MATH 120, or to simply appreciate the nature of proof at a deeper level than is seen in high school geometry. To be considered for enrollment, please email masonr@stanford.edu and attend the first class meeting on Tuesday, April 3 at 3PM in 300-303.

Last offered: Spring 2018

## MATH 80Q: Capillary Surfaces: Explored and Unexplored Territory

Preference to sophomores. Capillary surfaces: the interfaces between fluids that are adjacent to each other and do not mix. Recently discovered phenomena, predicted mathematically and subsequently confirmed by experiments, some done in space shuttles. Interested students may participate in ongoing investigations with affinity between mathematics and physics.

Last offered: Winter 2018
| UG Reqs: WAY-FR, WAY-SMA

## MATH 83N: 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 seminar develops 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 seminar may be paired with
Math 51; though that course is not a pre- or co-requisite.

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

Instructors:
Sauermann, L. (PI)

## MATH 87Q: Mathematics of Knots, Braids, Links, and Tangles

Preference to sophomores. Types of knots and how knots can be distinguished from one another by means of numerical or polynomial invariants. The geometry and algebra of braids, including their relationships to knots. Topology of surfaces. Brief summary of applications to biology, chemistry, and physics.

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

Instructors:
Wieczorek, W. (PI)

## MATH 101: Math Discovery Lab

MDL is a discovery-based project course in mathematics. Students work independently in small groups to explore open-ended mathematical problems and discover original mathematics. Students formulate conjectures and hypotheses; test predictions by computation, simulation, or pure thought; and present their results to classmates. No lecture component; in-class meetings reserved for student presentations, attendance mandatory. Admission is by application:
http://math101.stanford.edu. Motivated students with any level of mathematical background are encouraged to apply. WIM

Last offered: Winter 2018
| UG Reqs: WAY-FR

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