
Math 237B (Topics in Differential Equations)
Course Topics: Mathematical and Computational Geometric PDE
Instructor: Prof. Michael Holst
(5739 AP&M, mholst@math.ucsd.edu)
Term: Spring 2007
Lecture: 9:30a10:50a TuTh, APM 5402 (MOVED FROM 5829)
Textbook(s):

N. Straumann,
General Relativity: With Applications to Astrophysics,
Springer, 2004.
 
R. Wald,
General Relativity,
Univ. of Chicago Press, 1984.
 
A. Cohen,
Numerical Analysis of Wavelet Methods,
JAI Press, 2003.

Class webpage:
http://ccom.ucsd.edu/~mholst/teaching/ucsd/237b_s07
CATALOG DESCRIPTION:
237ABC. Topics in Differential Equations (444)
May be repeated for credit with consent of adviser.
Prerequisite: consent of instructor.
GRADES, EXAMS, DATES:
Your grade in the course is based on attending the lectures and
participating in the class.
 Final: Thursday 13 June 2007 (9am11am, 5402 APM)
 No Class: April 10, April 12, May 15
 First lecture: Tuesday 3 April 2007.
 Last lecture: Thursday 7 June 2007.
 Finals week: 1115 June 2007.
ANNOUNCEMENTS, NOTES, ETC:
 IPAM Workshop Notes
on Elliptic PDE and Approximation theory
 Notes
on Calculus in Banach Spaces, Variational Methods, and Mechanics
SCHEDULE FOR THE LECTURE TOPICS:
The overall plan is to first develop the basic mathematical framework for
understanding a class of PDE arising in geometric settings, and based on
this develop a general approximation theory framework for designing
provably good computational methods for this class of PDE.
Relevant applications include
models of biomembranes,
models of fluid interactions,
geometric flows in material science,
flows of metrics in geometry,
models of black hole collisions in astrophysics,
and physicsbased reality models in computer animation.
Since we only have ten weeks, we will build the lectures primarily around
one particular geometric PDE model (general relativity) describing the
gravitational interaction of massive objects such as black holes and
neutron stars.
However, the notation, theory, and techniques we develop will be
useful for other applications.
The outline below will be updated as the quarter proceeds,
based on how the interests of the participants and the lecturer evolve.
We will leverage much of the material developed last quarter
in 237A, but the course will again be basically selfcontained.
Lecture 
Topics Covered 
1 (4/3/07, week 1) 
Differentiable manifolds, connections, covariant
differentiation, tangent vectors, differential forms,
tangent space, cotangent space, Hausdorff property and
partition of unity.

2 (4/5/07, week 1) 
Fiber bundles, tangent and cotangent bundles,
tangent and cotangent spaces as fibers of a bundle,
tensor fields as sections of a bundle, jet bundles.

3 (4/17/07, week 3) 
Lie derivative, exterior derivative, integration,
manifolds with boundary, Stokes theorem, divergence
theorem.

4 (4/19/07, week 3) 
Metrics, Riemannian and pseudoRiemanian manifolds,
LeviCevita connection, Christoffel symbols.

5 (4/24/07, week 4) 
LaplaceBeltrami operator, properties
of the LaplaceBeltrami operator, weak maximum principle,
a priori Linfinity bounds, sub and supersolutions.

6 (4/26/07, week 4) 
General nonlinear geometric elliptic operators and PDE,
properties of a semilinear LaplaceBeltramilike operator,
weak maximum principle,
a priori Linfinity bounds, sub and supersolutions.

7 (5/1/07, week 5) 
Lecture 1 on G. Dziuk paper
(Lectures Notes in Math 1357, 1988)
on a priori error estimates for the
LaplaceBeltrami operator equation.

8 (5/3/07, week 5) 
Lecture 2 on G. Dziuk paper
(Lectures Notes in Math 1357, 1988)
on a priori error estimates for the
LaplaceBeltrami operator equation.

9 (5/8/07, week 6) 
Lecture 3 on G. Dziuk paper
(Lectures Notes in Math 1357, 1988)
on a priori error estimates for the
LaplaceBeltrami operator equation.

10 (5/10/07, week 6) 
Lecture 1 on
M. Holst paper
(AiCM, 2001)
on a priori and a posteriori estimates for a
general class of semilinear geometric PDE, with application
to the constraints in GR.

11 (5/15/07, week 7) 
Lecture 2 on
M. Holst paper
(AiCM, 2001)
on a priori and a posteriori estimates for a
general class of semilinear geometric PDE, with application
to the constraints in GR.

12 (5/17/07, week 7) 
Lecture 3 on
M. Holst paper
(AiCM, 2001)
on a priori and a posteriori estimates for a
general class of semilinear geometric PDE, with application
to the constraints in GR.

13 (5/22/07, week 8) 
Lecture 1 on
A. Demlow and G. Dziuk paper
(SINUM, 2006)
on a posteriori estimates for the LaplaceBeltrami
operator equation.

14 (5/24/07, week 8) 
Lecture 2 on
A. Demlow and G. Dziuk paper
(SINUM, 2006)
on a posteriori estimates for the LaplaceBeltrami
operator equation.

15 (5/29/07, week 9) 
Lecture 3 on
A. Demlow and G. Dziuk paper
(SINUM, 2006)
on a posteriori estimates for the LaplaceBeltrami
operator equation.

16 (5/31/07, week 9) 
Lecture 1 on
G. Dziuk and C. Elliot paper
(IMA J. Numer. Anal., 2006)
on a priori estimates for finite elements on
evolving surfaces.

17 (6/5/07, week 10) 
Lecture 2 on
G. Dziuk and C. Elliot paper
(IMA J. Numer. Anal., 2006)
on a priori estimates for finite elements on
evolving surfaces.

18 (6/7/07, week 10) 
Lecture 3 on
G. Dziuk and C. Elliot paper
(IMA J. Numer. Anal., 2006)
on a priori estimates for finite elements on
evolving surfaces.

19 (6/13/07, Final Lecture) 
Overview of HuiskenIlmanen paper on inverse
mean curvature flow, Hamilton paper on Ricci flow,
and prospects for explicit (rather than implicit) and
intrinsic (rather than extrinsic) numerical treatment.

