## MATH 121: Galois Theory

Field of fractions, splitting fields, separability, finite fields. Galois groups, Galois correspondence, examples and applications. Prerequisite:
Math 120 and (also recommended) 113.

Terms: Win
| Units: 3
| UG Reqs: GER:DB-Math

Instructors:
Bump, D. (PI)
;
Stanton, C. (TA)

## MATH 122: Modules and Group Representations

Modules over PID. Tensor products over fields. Group representations and group rings. Maschke's theorem and character theory. Character tables, construction of representations. Prerequisite:
Math 120. Also recommended: 113.

Terms: Spr
| Units: 3

Instructors:
Wilson, J. (PI)
;
Kuhn, N. (TA)

## MATH 131P: Partial Differential Equations

An introduction to PDE; particularly suitable for non-Math majors. Topics include physical examples of PDE's, method of characteristics, D'Alembert's formula, maximum principles, heat kernel, Duhamel's principle, separation of variables, Fourier series, Harmonic functions, Bessel functions, spherical harmonics. Students who have taken
MATH 171 should consider taking
MATH 173 rather than 131P. Prerequisite: 53.

Terms: Win
| Units: 3
| UG Reqs: GER:DB-Math

Instructors:
Zhu, X. (PI)
;
Ottolini, A. (TA)

## MATH 136: Stochastic Processes (STATS 219)

Introduction to measure theory, Lp spaces and Hilbert spaces. Random variables, expectation, conditional expectation, conditional distribution. Uniform integrability, almost sure and Lp convergence. Stochastic processes: definition, stationarity, sample path continuity. Examples: random walk, Markov chains, Gaussian processes, Poisson processes, Martingales. Construction and basic properties of Brownian motion. Prerequisite:
STATS 116 or
MATH 151 or equivalent. Recommended:
MATH 115 or equivalent.
http://statweb.stanford.edu/~adembo/math-136/

Terms: Aut
| Units: 3
| UG Reqs: GER:DB-Math

Instructors:
Dembo, A. (PI)
;
Hui, Y. (TA)

## MATH 143: Differential Geometry

Geometry of curves and surfaces in three-space and higher dimensional manifolds. Parallel transport, curvature, and geodesics. Surfaces with constant curvature. Minimal surfaces.

Terms: Aut
| Units: 3
| UG Reqs: GER:DB-Math

Instructors:
Fredrickson, L. (PI)
;
Zachos, E. (TA)

## MATH 146: Analysis on Manifolds

Differentiable manifolds, tangent space, submanifolds, implicit function theorem, differential forms, vector and tensor fields. Frobenius' theorem, DeRham theory. Prerequisite: 62CM or 52 and familiarity with linear algebra and analysis arguments at the level of 113 and 115 respectively.

Terms: Win
| Units: 3
| UG Reqs: GER:DB-Math

Instructors:
Starkston, L. (PI)
;
Fauteux-Chapleau, F. (TA)

## MATH 147: Differential Topology

Smooth manifolds, transversality, Sards' theorem, embeddings, degree of a map, Borsuk-Ulam theorem, Hopf degree theorem, Jordan curve theorem. Prerequisite: 115 or 171.

Terms: Spr
| Units: 3
| UG Reqs: GER:DB-Math

Instructors:
Wieczorek, W. (PI)
;
Ungemach, W. (TA)

## MATH 152: Elementary Theory of Numbers

Euclid's algorithm, fundamental theorems on divisibility; prime numbers; congruence of numbers; theorems of Fermat, Euler, Wilson; congruences of first and higher degrees; quadratic residues; introduction to the theory of binary quadratic forms; quadratic reciprocity; partitions.

Terms: Win
| Units: 3
| UG Reqs: GER:DB-Math

Instructors:
Tsai, C. (PI)

## MATH 155: Analytic Number Theory

Topics in analytic number theory such as the distribution of prime numbers, the prime number theorem, twin primes and Goldbach's conjecture, the theory of quadratic forms, Dirichlet's class number formula, Dirichlet's theorem on primes in arithmetic progressions, and the fifteen theorem. Prerequisite: 152, or familiarity with the Euclidean algorithm, congruences, residue classes and reduced residue classes, primitive roots, and quadratic reciprocity.

Terms: Spr
| Units: 3
| UG Reqs: GER:DB-Math

Instructors:
Thorner, J. (PI)
;
Zavyalov, B. (TA)

## MATH 158: Basic Probability and Stochastic Processes with Engineering Applications (CME 298)

Calculus of random variables and their distributions with applications. Review of limit theorems of probability and their application to statistical estimation and basic Monte Carlo methods. Introduction to Markov chains, random walks, Brownian motion and basic stochastic differential equations with emphasis on applications from economics, physics and engineering, such as filtering and control. Prerequisites: exposure to basic probability.

Terms: Spr
| Units: 3

Instructors:
Ying, L. (PI)

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