TOPICS IN ADVANCED ANALYSIS B

Degree course: 
Corso di Second cycle degree in MATHEMATICS
Academyc year when starting the degree: 
2018/2019
Year: 
1
Academyc year when helding the course: 
2018/2019
Course type: 
Compulsory subjects, characteristic of the class
Credits: 
8
Period: 
First Semester
Standard lectures hours: 
64
Detail of lecture’s hours: 
Lesson (64 hours)
Requirements: 

Linear Algebra, a knowledge of the basic tools of Functional Analysis, Lebesgue integration theory and the fundamentals of Spectral Theory.

Final Examination: 
Orale

Verification of learning will consist of two parts:

1) A short written thesis about a chosen topic to be exposed to the class as a lecture.
2) A traditional oral exam, during which the student will have to show that she/he has acquired the basic notions and the proofs of the main theorems.

Assessment: 
Voto Finale

The purpose of the teaching is to let the student acquire, via both abstract results and concrete examples, the basic concepts from the theory of locally compact groups and their representations. The course aims also to introduce applications of Operator Algebra Theory, Harmonic Analysis and Representation Theory. At the end of the course we expect that:

1) The student has acquired the fundamental notions pertaining to the above mentioned fields
2) On the base of the proofs illustrated during the lectures, the student is able to make by him/herself reasonings of medium complexity that lead him/her to deduce abstract properties of the above mentioned objects;
3) The student is able to investigate the main properties of the objects alluded to above in concrete situations.

Banach Algebras: Basic Concepts
Gelfand Theory
Nonunital Banach Algebras
The Spectral Theorem
Spectral Theory of ∗-Representations
Von Neumann Algebras

Topological Groups
Haar Measure
Convolutions
Homogeneous Spaces
Unitary Representations

Representations of a Group and Its Group Algebra

Functions of Positive Type


Analysis on Locally Compact Abelian Groups

The Dual Group

The Fourier Transform

The Pontrjagin Duality Theorem

Representations of Locally Compact Abelian Groups

Closed Ideals in L1(G)

Spectral Synthesis


Analysis on Compact Groups

Representations of Compact Groups

The Peter-Weyl Theorem

Fourier Analysis on Compact Groups


Induced Representations

The Inducing Construction
The Frobenius Reciprocity Theorem
Pseudomeasures and Induction in Stages
Systems of Imprimitivity
The Imprimitivity Theorem
Introduction to the Mackey Machine

Gerald B. Folland A Course in Abstract Harmonic Analysis, 2nd Edition, CRC Press, 2015

Lectures

Professors

Borrowers