Time: Tuesdays, Thursdays at 9:30am.
Room: Mathematics Building 0104.
Teacher: Y.A.
Rubinstein. Office hours: 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.
L. Simon, Lectures on PDEs, 2013.
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 from 2013:
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.
HW 1

Lecture 3
Jacobi theory I.

Lecture 4
Jacobi theory II.

Lecture 5
Jacobi theory III.

Lecture 6
The direct method in the calculus of variations.
Compactness of sublevel sets as motivation for requiring weak sequential
lower semicontinuity and coercivity.

Lecture 7
Situations where the direct method can be applied: pLaplacian, harmonic
maps into Euclidean space (generalizing geodesics).

Lecture 8
Bochner technique, I.

Lecture 9
Bochner technique, II, Killing fields.

Lecture 10
Bochner technique, III, Stokes' theorem and integration on manifolds.

Lecture 11
Bochner technique, IV, application to 1forms.

Lecture 12
The direct method and
optimal transportation.

Lecture 13
Constrained minimization and the direct method.
HW 2

Lecture 14
Cyclical monotonicity. The fundamental theorem of optimal transporatation.

Lecture 15
Cyclical monotonicity and Rockafellar's theorem. The fundamental theorem
of optimal transporatation, continued.

Lecture 16
Brenier's theorem, I.

Lecture 17
Brenier's theorem, II.

Lecture 18
Regularity weak solutions of semilinear equations (L. Simon's lecture 6).

Lecture 19
Existence of weak solutions of semilinear equations, LaxMilgram Lemma,
and establishing coercivity under ellipticity assumption. Interpolation
inqualities
(L. Simon's lecture 7).
HW 3

Lecture 20
Spectrum of selfadjoint operators
(L. Simon's lecture 10).

Lecture 21
Spectrum of selfadjoint operators  continued.
Parabolic equations, heat kernel
(L. Simon's lecture 11).

Lecture 22
Weyl's asymptotic formula (L. Simon's lecture 11).

Lecture 23
Main lemma of interior Schauder theory (Schauder estimates via scaling)
(L. Simon's lecture 12).

Lecture 24
Schauder estimates: the method of freezing coefficients (L. Simon's
lecture 12).

Lecture 25
The (weak and strong) Maximum Principle for second order elliptic
equations.
The Hopf boundary point lemma (L. Simon's
lecture 13).

Lecture 26
Outlook and further reading.
