## MATH 221B: Mathematical Methods of Imaging (CME 321B)

Array imaging using Kirchhoff migration and beamforming, resolution theory for broad and narrow band array imaging in homogeneous media, topics in high-frequency, variable background imaging with velocity estimation, interferometric imaging methods, the role of noise and inhomogeneities, and variational problems that arise in optimizing the performance of array imaging algorithms.

Terms: Spr
| Units: 3

Instructors:
Papanicolaou, G. (PI)

## MATH 226: Numerical Solution of Partial Differential Equations (CME 306)

Hyperbolic partial differential equations: stability, convergence and qualitative properties; nonlinear hyperbolic equations and systems; combined solution methods from elliptic, parabolic, and hyperbolic problems. Examples include: Burger's equation, Euler equations for compressible flow, Navier-Stokes equations for incompressible flow. Prerequisites:
MATH 220A or
CME 302.

Terms: Spr
| Units: 3

Instructors:
Ying, L. (PI)
;
Liu, F. (TA)

## MATH 227: Partial Differential Equations and Diffusion Processes

Parabolic and elliptic partial differential equations and their relation to diffusion processes. First order equations and optimal control. Emphasis is on applications to mathematical finance. Prerequisites:
MATH 131 and
MATH 136/
STATS 219, or equivalents.

Last offered: Winter 2015

## MATH 228: Stochastic Methods in Engineering (CME 308)

The basic limit theorems of probability theory and their application to maximum likelihood estimation. Basic Monte Carlo methods and importance sampling. Markov chains and processes, random walks, basic ergodic theory and its application to parameter estimation. Discrete time stochastic control and Bayesian filtering. Diffusion approximations, Brownian motion and an introduction to stochastic differential equations. Examples and problems from various applied areas. Prerequisites: exposure to probability and background in analysis.

Terms: Spr
| Units: 3

## MATH 228A: Probability, Stochastic Analysis and Applications

The basic limit theorems of probability theory and their application to maximum likelihood estimation. Basic Monte Carlo methods and importance sampling. Markov chains and processes, random walks, basic ergodic theory and its application to parameter estimation. Discrete time stochastic control and Bayesian filtering. Diffusion approximations, Brownian motion and basic stochastic differential equations. Examples and problems from various applied areas. Prerequisites: exposure to probability and background in analysis.

Terms: Spr
| Units: 3

Instructors:
Papanicolaou, G. (PI)

## MATH 230A: Theory of Probability (STATS 310A)

Mathematical tools: sigma algebras, measure theory, connections between coin tossing and Lebesgue measure, basic convergence theorems. Probability: independence, Borel-Cantelli lemmas, almost sure and Lp convergence, weak and strong laws of large numbers. Large deviations. Weak convergence; central limit theorems; Poisson convergence; Stein's method. Prerequisites: 116,
MATH 171.

Terms: Aut
| Units: 2-4

## MATH 230B: Theory of Probability (STATS 310B)

Conditional expectations, discrete time martingales, stopping times, uniform integrability, applications to 0-1 laws, Radon-Nikodym Theorem, ruin problems, etc. Other topics as time allows selected from (i) local limit theorems, (ii) renewal theory, (iii) discrete time Markov chains, (iv) random walk theory,nn(v) ergodic theory. Prerequisite: 310A or
MATH 230A.

Terms: Win
| Units: 2-3

## MATH 230C: Theory of Probability (STATS 310C)

Continuous time stochastic processes: martingales, Brownian motion, stationary independent increments, Markov jump processes and Gaussian processes. Invariance principle, random walks, LIL and functional CLT. Markov and strong Markov property. Infinitely divisible laws. Some ergodic theory. Prerequisite: 310B or
MATH 230B.

Terms: Spr
| Units: 2-4

## MATH 231A: An Introduction to Random Matrix Theory (STATS 351A)

Patterns in the eigenvalue distribution of typical large matrices, which also show up in physics (energy distribution in scattering experiments), combinatorics (length of longest increasing subsequence), first passage percolation and number theory (zeros of the zeta function). Classical compact ensembles (random orthogonal matrices). The tools of determinental point processes.

Last offered: Autumn 2008

## MATH 231C: Free Probability

Background from operator theory, addition and multiplication theorems for operators, spectral properties of infinite-dimensional operators, the free additive and multiplicative convolutions of probability measures and their classical counterparts, asymptotic freeness of large random matrices, and free entropy and free dimension. Prerequisite:
STATS 310B or equivalent.

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