Advanced Topics in Real Analysis

A.Y. 2020/2021
Overall hours
Learning objectives
Complete a modern and robust foundation of measure theory and integration and differentiability of functions begun in Mathematical Analysis 4 and Real Analysis. In particular, the extension of the fundamental theorems of integral calculus to weakly differentiable vector fields and rough sets. In addition the study of differentiability of convex functions and their approximation by semicontinuous functions, which is the basis of viscosity methods for fully nonlinear partial differential equations.
Expected learning outcomes
Capacity to apply the theorems of Radon-Nikodym and weak compactness for Radon Measures. Capacity to verify the validity and to apply integration by parts formulas for weakly differentiable functions on rough sets. Capacity to reduce questions of differentiability to exceptional sets of zero measure.
Course syllabus and organization

Single session

Lesson period
Second semester
The lectures will be conducted remotely in virtual classrooms (on the zoom platform) via live streaming, allowing for real-time interaction between the students and the instructor.
Course syllabus
-- Measure theory: review of measure theory. Borel and Radon measures in Euclidean spaces. Measurable functions and the the theorems of Lusin and Egoroff. Review integrazion of masurable functions emphasizing Lebesgue and Hausdorff measures. The covering theorems of Vitali and Besicovitch. Differentiation of Radon measures, the Radon-Nikodym theorem, and Lebesgue's differentiation theorem. The Riesz representation theorem for Radon measures, weak convergence and weak compactness for Radon measures.

-- Area and coarea fromulas: Lipschitz functions and Rademacher's theorem. Linear maps and the Jacobian of Lischitz maps. Area and coarea formulas.

-- Functions of bounded variation (BV) in several variables and sets of finite perimeter: Definitions of BV functions and sets of finite perimeter, structure theorem for BV functions. Approximation and compactness for BV functions. Boundary traces and extensions for BV functions. Coarea formula, isoperimetric inequality and relations between capacity and Hausdorff measure. The reduced boundary and structure theorem for sets of finite perimeter. The divergence theorem for BV vector fields on sets of finite perimeter.

-- Differentiation of convex functions in several variables: the subdifferential and the first order theory. Weak derivatives of first and second order. Alexandroff's theorem on the second order differentiability in the sense of Peano. Sub and superdifferentials of second order, semicontinuous functions and semiconvex functions.
Prerequisites for admission
Mathematical Analysis 4 and Real Analysis
Teaching methods
Traditional blackboard lectures. Attendance strongly suggested.
Teaching Resources
-- L. Ambrosio, N. Fusco and D. Pallara - Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.
-- L. C. Evans and R. F. Gariepy - Measure Theory and Fine Properties of Functions, Advanced Studies in Mathematics, CRC Press, Boca Raton, 1992.
-- F. R. Harvey and H. B. Lawson., Jr. - Notes on the differentiation of quasi-convex functions. arXiv:1309.1772v3, 30 July 2016, 1-17, 2016.
Assessment methods and Criteria
The final exam consists of an oral exam in one of two possible forms: a traditional oral examination concerning the entire syllabus of the course or a seminar focused on some advanced topic (agreed upon in advance) which requires some independent study on the part of the student and the ability to integrate the seminar into the context of the syllabus of the course.
MAT/05 - MATHEMATICAL ANALYSIS - University credits: 6
Lessons: 42 hours
Professor: Payne Kevin Ray