Geometric Analysis (course 742)

University of Maryland

Department of Mathematics

Autumn 2014






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 1-4 (Ryan Hunter)
Lectures 5-6 (Jacky Chong)
Lectures 9-10 (Jason Suagee)
Lectures 11-12 (Siming He)
Lectures 13-15 (Zhenfu Wang)
Lectures 16 & 19 (Siming He)
Lectures 17-18 (Bo Tian)
Lectures 20-21 (Bo Tian)


Schedule:

  • Lecture 1
    Overview. Basic definitions of Riemannian geometry. Langrangians and Euler-Lagrange 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 E-L 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: p-Laplacian, 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 1-forms.

  • 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 semi-linear equations (L. Simon's lecture 6).

  • Lecture 19
    Existence of weak solutions of semi-linear equations, Lax-Milgram Lemma, and establishing coercivity under ellipticity assumption. Interpolation inqualities (L. Simon's lecture 7).
    HW 3

  • Lecture 20
    Spectrum of self-adjoint operators (L. Simon's lecture 10).

  • Lecture 21
    Spectrum of self-adjoint 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.