## MATH 114: Introduction to Scientific Computing (CME 108)

Introduction to Scientific Computing Numerical computation for mathematical, computational, physical sciences and engineering: error analysis, floating-point arithmetic, nonlinear equations, numerical solution of systems of algebraic equations, banded matrices, least squares, unconstrained optimization, polynomial interpolation, numerical differentiation and integration, numerical solution of ordinary differential equations, truncation error, numerical stability for time dependent problems and stiffness. Implementation of numerical methods in MATLAB programming assignments. Prerequisites:
MATH 51, 52, 53; prior programming experience (MATLAB or other language at level of
CS 106A or higher).

Terms: Win, Sum
| Units: 3
| UG Reqs: GER:DB-EngrAppSci, WAY-AQR, WAY-FR

## MATH 115: Functions of a Real Variable

The development of real analysis in Euclidean space: sequences and series, limits, continuous functions, derivatives, integrals. Basic point set topology. Includes introduction to proof-writing. Prerequisite: 21.

Terms: Aut, Spr
| Units: 3
| UG Reqs: GER:DB-Math, WAY-FR

## MATH 116: Complex Analysis

Analytic functions, Cauchy integral formula, power series and Laurent series, calculus of residues and applications, conformal mapping, analytic continuation, introduction to Riemann surfaces, Fourier series and integrals. (
Math 106 offers a less theoretical treatment.) Prerequisites: 52, and 115 or 171.

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

Instructors:
Eliashberg, Y. (PI)
;
Cant, D. (TA)

## MATH 118: Mathematics of Computation

Notions of analysis and algorithms central to modern scientific computing: continuous and discrete Fourier expansions, the fast Fourier transform, orthogonal polynomials, interpolation, quadrature, numerical differentiation, analysis and discretization of initial-value and boundary-value ODE, finite and spectral elements. Prerequisites:
MATH 51 and 53.

Last offered: Winter 2018
| UG Reqs: GER:DB-Math

## MATH 120: Groups and Rings

Recommended for Mathematics majors and required of honors Mathematics majors. Similar to 109 but altered content and more theoretical orientation. Groups acting on sets, examples of finite groups, Sylow theorems, solvable and simple groups. Fields, rings, and ideals; polynomial rings over a field; PID and non-PID. Unique factorization domains. WIM.

Terms: Aut, Spr, Sum
| Units: 3
| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Bump, D. (PI)
;
Grzegrzolka, P. (PI)
;
Ye, L. (PI)
;
Fushida-Hardy, S. (TA)
;
Helfer, J. (TA)

## 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, WAY-FR

Instructors:
Huh, J. (PI)
;
Raksit, A. (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
| UG Reqs: WAY-FR

Instructors:
Diaconis, P. (PI)

## 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, WAY-FR

Instructors:
Kleinhenz, P. (PI)

## 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: Win
| Units: 3
| UG Reqs: GER:DB-Math, WAY-FR

Instructors:
Dembo, A. (PI)

## MATH 137: Mathematical Methods of Classical Mechanics

Newtonian mechanics. Lagrangian formalism. E. Noether's theorem. Oscillations. Rigid bodies. Introduction to symplectic geometry. Hamiltonian formalism. Legendre transform. Variational principles. Geometric optics. Introduction to the theory of integrable systems. Prerequisites: 51, 52, 53, or 61CM, 62CM, 63CM.

Last offered: Spring 2019
| UG Reqs: GER:DB-Math

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