Advanced Topics in Analysis

Unit aims

The aim of the unit is to give an introduction to several topics of modern analysis such as Fourier analysis, Harmonic analysis, distributions, Sobolev spaces, and geometric measure theory.

Unit description

The course contains the following parts:

  1. Introduction to Fourier analysis
  2. Introduction to Function spaces
  3. Introduction to Geometric measure theory

Sobolev spaces play a major role in modern analysis, spectral theory and partial Differntial Equations. As of today, Bristol is one of the few places in the UK, offering a course in Sobolev spaces to undergraduates.

In addition, the course covers such fundamentals of modern analysis as the Fourier transform, distributions, Sobolev inequalities, Hausdorff dimension, Hardy-Littlewood maximal operators etc. The main thrust of the course is to prepare students so that the body of modern analysis literature, such as monographs, research papers becomes accessible to them.

Relation to other units

This is the final element of a sequence of Analysis courses at Levels C/4, I/5, and H/6.

Learning objectives

After taking this unit, students should be equipped to read some of the current research in Analysis. In addition, the unit is aimed to give students basic skills of making mathematical presentations. This is a rare opportunity important for their future development.

Transferable skills

Independent study in mathematics, participation in seminars, logical analysis and problem solving.

Syllabus

  1. Lebesgue measure Fubini Theorem
    Convolutions
    Introduction to the Fourier Transform
  2. Introduction to Sobolev spaces and distributions
    Foundations of Harmonic analysis and theory of maximal operators
    Lebesgue points
  3. Hausdorff metric
    Hausdorff measure
    Fractals
    Hausdorff dimensions
    Box dimension

Reading and References

E.B.Davis: Spectral Theory and Differntial Operators, Graduate text, Cambridge University Press , 1995

E.H.Lieb, M.Loss: Analysis, Graduate Studies in Mathematics Volume 14, AMS, 1997

V.Maz'ya, S. Poborchi: Differentiable functions on bad domains, World Scientific, 1997

Unit code: MATHM0020
Level of study: M/7
Credit points: 20
Teaching block (weeks): 2 (13-24)
Lecturer: Dr Yuri Netrusov

Pre-requisites

MATH20901 Multivariable Calculus, MATH20006 Metric Spaces and MATH 30007 Measure Theory and Integration

Co-requisites

None

Methods of teaching

Lectures, guided reading from a textbook for student presentations, discussion of problems and student seminars.

Methods of Assessment 

The pass mark for this unit is 50.

The final mark is calculated as follows:

  • 80% from assessed coursework
  • 20% based on active participation in seminars*

* A student is expected to give two presentations throughout the semester. Two thirds of this component will be awarded for demonstrating thorough understanding of the material through participation/presentations in seminars. One third will be awarded for the clarity, quality, lucidity and style of seminar presentations.

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.

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