## MATH 215C: Differential Geometry

This course will be an introduction to Riemannian Geometry. Topics will include the Levi-Civita connection, Riemann curvature tensor, Ricci and scalar curvature, geodesics, parallel transport, completeness, geodesics and Jacobi fields, and comparison techniques. Prerequisites 146 or 215B

Terms: Spr
| Units: 3

Instructors:
Hershkovits, O. (PI)

## MATH 216A: Introduction to Algebraic Geometry

Algebraic curves, algebraic varieties, sheaves, cohomology, Riemann-Roch theorem. Classification of algebraic surfaces, moduli spaces, deformation theory and obstruction theory, the notion of schemes. May be repeated for credit. Prerequisites: 210ABC or equivalent.

Terms: Aut
| Units: 3
| Repeatable for credit

Instructors:
Vakil, R. (PI)

## MATH 216B: Introduction to Algebraic Geometry

Continuation of 216A. May be repeated for credit.

Terms: Win
| Units: 3
| Repeatable for credit

Instructors:
Vakil, R. (PI)

## MATH 216C: Introduction to Algebraic Geometry

Continuation of 216B. May be repeated for credit.

Terms: Spr
| Units: 3
| Repeatable for credit

Instructors:
Kemeny, M. (PI)

## MATH 217C: Complex Differential Geometry

Complex structures, almost complex manifolds and integrability, Hermitian and Kahler metrics, connections on complex vector bundles, Chern classes and Chern-Weil theory, Hodge and Dolbeault theory, vanishing theorems, Calabi-Yau manifolds, deformation theory.

Last offered: Winter 2015
| Repeatable for credit

## MATH 220: Partial Differential Equations of Applied Mathematics (CME 303)

First-order partial differential equations; method of characteristics; weak solutions; elliptic, parabolic, and hyperbolic equations; Fourier transform; Fourier series; and eigenvalue problems. Prerequisite: Basic coursework in multivariable calculus and ordinary differential equations, and some prior experience with a proof-based treatment of the material as in
Math 171 or
Math 61CM (formerly
Math 51H).

Terms: Aut
| Units: 3

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

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

Image denoising and deblurring with optimization and partial differential equations methods. Imaging functionals based on total variation and l-1 minimization. Fast algorithms and their implementation.

Last offered: Winter 2014

## 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.

Last offered: Spring 2016

## 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)

## 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 136/
STATS 219 (or equivalents) and
MATH 131P +
MATH 115/171 or
MATH 173 or
MATH 220.

Last offered: Winter 2015

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