## 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.