Math 2303: Graduate Analysis 3
MWF 2:00 - 2:50pm -- Thackeray 524
Homework 1 , Homework 2
, Midterm and solutions.
- Instructor. Dr. Marta Lewicka (office hours in Thack
408, Monday, Wednesday 3:00-4:00pm)
- Prerequisites. This is the 3rd course in the Graduate Analysis sequel, directed at
students who have taken 2301 Analysis 1; so that knowledge of Lebesgue
measure and integration, Lebesgue spaces L^p and basic knowledge of
Sobolev spaces W^{1,p} is assumed.
The analysis and linear algebra material in the Math Preliminary Exams syllabus is also assumed.
- Textbooks. The course will be self-contained.
If in doubt, consult with the books:
- Brezis: Functional analysis, Sobolev spaces and Partial
Differential Equations (also, you may read the shorter French
edition: Analyse fonctionelle - Theorie et applications)
- Lieb and Loss: Analysis
- Evans and Gariepy: Measure Theory and Fine Properties of Functions
- Ambrosio, Fusco and Pallara: Functions of Bounded Variation
and Free Discontinuity Problems
- Grades. Grades will be based on one midterm (25%),
homeworks (25%) and a presentation (50%).
Incompletes and make-up exams will almost never be given,
and only for cases of extreme personal tragedy.
- Topics. In addition to the standard theorems below the course will touch upon some
more recent subjects in the Mathematical Analysis.
I. Measure theory and BV functions:
I.1. Nonnegative Radon measures, Vitali-Besicovich theorem, differentiation of Radon measures,
I.2. Radon-Nikodym theorem, Lebesgue deconposition theorem,
I.3. Riesz representation theorem,
I.4. Vector measures, variation of a measure, Jordan, Hahn, polar decompositions,
I.5. Functions of bounded variation, perimeter of a set,
I.6. Helly's compactness theorem,
I.7. The 1-dimensional case.
II. Spectral theory of compact operators:
II.1. Compact and finite rank operators,
II.2. Schauder fixed point theorem,
II.3. Adjoint operator and Schauder's theorem,
II.4. Orthogonality relations,
II.5. Complementarity and projections in Banach spaces.
II.6. Noncompactness of a ball in infinitely dimensional spaces,
II.7. Fredholm alternative,
II.8. spectrum of a compact operator,
II.9. Lax-Milgram theorem,
II.10. Spectral theory of compact self-adjoint operators,
II.11. Courant-Fischer formulas,
II.12. Spectrum of a compact positive oparator,
II.13. Noether index theory.
- Calendar of presentations:
17 Sept: "The Vitali and the Besikovich covering theorems"
(from Evans and Gariepy).
Presentation by Irina Navrotskaya.
15 Oct: "Hausdorf measure and dimension. Isodiametric
inequality, Steiner symmetrization" (from Evans and Gariepy).
Presentation by Xiaodan Zhou.
29 Oct: "Convex integration with constraints and applications
to phase transitions and partial differential equations" by Muller and
Sverak. Link to paper.
- Partial differential relations, convex integration, Tartar's square.
Presentation by Pablo Ochoa.
- Relation between divergence free fields and incompressible
maps. Presentation by Maria Medina.
- The main approximation lemma and proof of the main
result. Presentation by Jilong Hu and Guoqing Liu.
3 Dec: "A new proof of the uniqueness of the flow
for ordinary differential equations with BV vector fields" by
Hauray and Le Bris. Link to
paper. Presentation by Matthew Wheeler and Aliaksandra Yarosh.