Math 218 (Advanced Parabolic and Hyperbolic Partial Differential Equations), Spring 2022

Lecture time and location: TuTh 11:40am-1:15pm, McHenry Library Classroom 1270.

Announcement: Class on Thursday, April 21 will meet on Zoom. See Canvas for details.

Instructor: Joey Zou

Email: yzou34(at)ucsc(dot)edu

Office Hours: Tu 2:30-3:30pm, in McHenry Library, Room 4167, or by appointment over Zoom.

Canvas site: https://canvas.ucsc.edu/courses/52775. I will use Canvas to send announcements.

Syllabus: here

Lecture notes (periodically updated)

The course will focus on various aspects of linear parabolic and hyperbolic equations, with a particular emphasis on the construction and properties of parametrices and solution operators to such equations. Towards the end of the quarter, I'll plan on giving an introduction to microlocal analysis, with a particular focus on applications to parabolic and hyperbolic problems.

Problem sets (optional):

• Homework 1
• Homework 2
• Homework 3

Tentative Schedule:

• Lectures 1, 2, 3 (03/29, 03/31, 04/05): Distribution Theory
• Lecture 4 (04/07): Fourier Transform
• Lectures 5 and 6 (04/12, 04/14): Introduction to Heat Equation
• Lectures 7, 8, 9 (04/19, 04/21, 04/26): Parabolic regularity
• Lecture 10 (04/28): Heat Kernels on Compact Manifolds
• Lectures 11 and 12 (05/03, 05/05): Introduction to Wave Equation
• Lectures 13, 14, and 15 (05/10, 05/12, 05/17): Geometric optics ansatz for hyperbolic equations
• Lectures 16 and 17 (05/19, 05/24): Introduction to Microlocal Analysis and Parametrices
• Lecture 18 (05/26): Parametrices for Parabolic Equations
• Lectures 19 and 20 (05/31, 06/02): Parametrices for Hyperbolic Equations