Time: Tuesdays, Thursdays at 11am.
Room: 1308 Mathematics Building.
Teacher: Y.A.
Rubinstein. Office hours: Tuesdays, Thursdays 1:102pm and
by appointment.
Course plan:
The goal will be to give an introduction to Geometric Analysis
that is accessible to beginning students interested in
PDE/Analysis or Geometry but not necessarily in both nor
necessarily with background in both. Topics will range, e.g.,
from Calculus of Variations, Bochner technique, Morse theory, weak
solutions and elliptic regularity, maximum principle for elliptic and
parabolic equations, Green's function of the Laplacian, isoperimetric and
Sobolev inequalities, continuity method, curvature and comparison results,
harmonic maps, curvature prescription problems.
Requirement: each student taking the course for a grade will be asked
to prepare and typeset notes for a block of lectures as well as the
solutions of the homework
exercises assigned during those lectures.
Main references:
T. Aubin,
Some nonlinear problems in Riemannian geometry, Springer, 1998.
D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of
Second Order, Springer, 2001.
P. Li, Geometric Analysis, Cambridge University Press, 2012.
P. Petersen, Riemannian Geometry, Springer, 2006.
M.M. Postnikov, Geometry VI: Riemannian Geometry, Springer, 2001.
R. Schoen, S.T. Yau, Lectures on Differential Geometry, Int. Press, 1994.
M. Struwe, Variational Methods, 4th Ed., Springer, 2008.
Additional references:
Ambrosio, Gigli, A user's guide to optimal transportation (available
online).
Frederic Robert,
Notes on the construction of Green's function on a Riemannian
manifold.
Lecture notes:
Lectures 14 (Ryan Hunter)
Lectures 56 (Jacky Chong)
Lectures 910 (Jason Suagee)
Lectures 1112 (Siming He)
Lectures 1315 (Zhenfu Wang)
Lectures 16 & 19 (Siming He)
Lectures 1718 (Bo Tian)
Lectures 2021 (Bo Tian)
Schedule:

Lecture 1
Overview. Basic definitions of Riemannian geometry. Langrangians
and EulerLagrange equations. The length Lagrangian and its
EL equation.

Lecture 2
More basic definitions of Riemannian geometry.
Parallel translation and the geodesic equation.
Comparison with the EL equation from last time.

Lecture 3
Jacobi theory I.

Lecture 4
Jacobi theory II.

Lecture 5
The direct method in the calculus of variations.
Compactness of sublevel sets as motivation for requiring weak sequential
lower semicontinuity and coercivity.

Lecture 6
Situations where the direct method can be applied: pLaplacian, harmonic
maps into Euclidean space (generalizing geodesics), optimal transportation
plans.

Lecture 7
Rockafellar's theorem. Basic convex analysis. The fundamental theorem of optimal transportation.

Lecture 8
The fundamental theorem of optimal transportation  continued.
Dual formulation. Brenier's theorem.

Lecture 9
The Weitzenbock formula. Bochner's method for Killing fields and harmonic 1forms.

Lecture 10
Weak solutions and local elliptic regularity (L. Simon's lecture 6)

Lecture 11
Interior Schauder estimates (L. Simon's lecture 12).

Lecture 12
Interior Schauder estimates  continued.

Lecture 13
Interior Schauder estimates  continued.

Lecture 14
Maximum principle.

Lecture 15
Maximum principle and Green's function
in Euclidean space (and domains).

Lecture 16
Green's function on a Riemannian manifold (F. Robert's notes).

Lecture 17
Heat equation and kernel (L. Simon's lecture 11).

Lecture 18
Heat equation and kernel  continued. Weyl's law.

Lecture 19
Green's function on a Riemannian manifold  parametrix
construction continued.

Lecture 20
Nash, Moser, De Giorgi theory (L. Simon's lecture 17

Lecture 21
Nash, Moser, De Giorgi theory  continued.

Lecture 22
Applications of Nash, Moser, De Giorgi theory.

Lecture 23
Applications of Nash, Moser, De Giorgi theory to nonlinear equations.
