96756 - Advanced Mathematical Analysus

Learning outcomes

At the end of the course, students will possess the knowledge of the main instruments of advance mathematical analysis: Sobolev spaces, spaces of generalized functions, Fourier transform. These tools will be the main instruments necessary to the quantitative and qualitative study of properties of the solutions to PDEs.

Course contents

Metric function spaces

Measure theory

Hilbert space and applications

Basic Functional Analysis

Tempered distributions and Fourier transforms

Sobolev spaces on the real line

Readings/Bibliography

Lecture notes on Virtuale

Suggested reading:

Richard F. Bass, Real Analysis for Graduate Students Version 4.3, 2022 https://bass.math.uconn.edu/real.html

Gerald B. Folland, Real Analysis: Modern Techniques and Their Applications 2nd Edition, Wiley 1999

Michael Reed, Barry Simon, Functional Analysis, 1981

Walter Rudin, Real and Complex Analysis, McGraw-Hill 1986

Terence Tao, An Introduction to Measure Theory, AMS 2011 https://terrytao.files.wordpress.com/2012/12/gsm-126-tao5-measure-book.pdf

Teaching methods

Lectures and exercise sessions.

Assessment methods

The exam has two parts: a written test with exercise, followed by a written test on theory. The exercise part mantains its value for the duration of the exam session (Winter session/Summer session). Both exams have to be passed with grade not inferior to 18.

Instead of the exercise exam, the student can do the homeworks which will be assigned during the course. The grade, in this case, mantains its value for the whole academic year 2024/2025.

Important. Advanced Analysis is part of the same course together with Riemannian Geometry. Both the exams of Riemannian Geometry and Advanced Analysis have to be passed within the academic year 2024/2025 in order to have the grade registered.

Teaching tools

Lecture notes and exercise sheets will be made available on Virtuale.

Office hours

See the website of Nicola Arcozzi