MATH 215B: Differential Topology
Topics: Basics of differentiable manifolds (tangent spaces, vector fields, tensor fields, differential forms), embeddings, tubular neighborhoods, integration and Stokes' Theorem, deRham cohomology, intersection theory via Poincare duality, Morse theory. NOTE: Undergraduates and Masters students who wish to enroll must fill out a Request for Review form:
https://forms.gle/v5RojToYzmYxGvKc7 ; Your request will be reviewed by faculty and you will be notified if you are granted permission to enroll.
Terms: Win
| Units: 3
Instructors:
Abouzaid, M. (PI)
;
Bonciocat, C. (TA)
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 147 or 215B. NOTE: Undergraduates and Masters students who wish to enroll must fill out a Request for Review form:
https://forms.gle/v5RojToYzmYxGvKc7 ; Your request will be reviewed by faculty and you'll be notified if you are granted permission to enroll.
Terms: Spr
| Units: 3
Instructors:
Chodosh, O. (PI)
MATH 216A: Introduction to Algebraic Geometry
Algebraic varieties, and introduction to schemes, morphisms, sheaves, and the functorial viewpoint. May be repeated for credit. NOTE: Undergrad, Coterm, and Master's students require instructor permission to enroll. Those interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Aut
| Units: 3
| Repeatable
for credit
Instructors:
Zhu, X. (PI)
MATH 216B: Introduction to Algebraic Geometry
Continuation of 216A. May be repeated for credit. NOTE: Undergraduates and Masters students who wish to enroll must fill out a Request for Review form:
https://forms.gle/v5RojToYzmYxGvKc7 ; Your request will be reviewed by faculty and you'll be notified if you are granted permission to enroll.
Terms: Win
| Units: 3
| Repeatable
for credit
Instructors:
Zhu, X. (PI)
MATH 216C: Introduction to Algebraic Geometry
Continuation of 216B. May be repeated for credit. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Last offered: Spring 2024
| Units: 3
| Repeatable
for credit
MATH 220A: Partial Differential Equations of Applied Mathematics (CME 303)
Introduction to partial differential equations: basic properties of elliptic, parabolic, and hyperbolic equations; Hamilton-Jacobi equations and applications to optimal control; stochastic modeling, forward and backward Kolmogorov equations; Fourier transform and Fourier series. Prerequisite: multivariable calculus, rigorous courses on basic real analysis and ordinary differential equations. NOTE: Undergrad, Coterm, and Master's students require instructor permission to enroll. Those interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Aut
| Units: 3
Instructors:
Ryzhik, L. (PI)
;
Borges Prado, L. (TA)
MATH 220B: Computational Methods of Applied Mathematics (CME 306)
Numerical methods for solving elliptic, parabolic, and hyperbolic partial differential equations. Algorithms for gradient and Hamiltonian systems. Algorithms for stochastic differential equations and Monte Carlo methods. Algorithms for computational harmonic analysis. Prerequisites: advanced undergraduate level PDE and advanced undergraduate level numerical analysis. NOTE: Undergraduates and Masters students who wish to enroll must fill out a Request for Review form:
https://forms.gle/v5RojToYzmYxGvKc7 ; Your request will be reviewed by faculty and you will be notified if you are granted permission to enroll.
Terms: Win
| Units: 3
Instructors:
Ying, L. (PI)
MATH 221B: Mathematical Methods of Imaging
This is a project based course where the first half is an introduction to imaging: coherent and incoherent, passive and active, migration, time-reversal and optimization-based imaging, with applications to ultrasound, radar, sonar and seismic imaging. The projects come from a close study of recent papers that use neural networks for imaging when large data sets are available and some special applications such as satellite imaging.
Last offered: Spring 2024
| Units: 3
MATH 228: Stochastic Methods in Engineering (CME 308, MS&E 324)
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
Instructors:
Glynn, P. (PI)
MATH 230A: Theory of Probability I (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.
Terms: Aut
| Units: 3