Dixmier traces

Serge RICHARD Professor

Department: Graduate School of Mathematics

Class Time: 2017 Spring Wednesday
Recommended for: Graduate school of Mathematics

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Course Overview

This course will provide an overview of some classical tools of functional analysis as well as some more advanced material developed over the last 10 to 15 years. After quickly recalling some basic definitions on Hilbert spaces and operators acting on them, we shall introduce many properties of the set of compact operators and the Schatten ideals. The Dixmier trace will then be introduced and compared to the usual trace. In the second part of the course, a more general framework for singular traces will be introduced, and some connections with other branches of mathematics will be presented. In particular, the link with Wodzicki's residue will be sketched. In order to provide a large panorama on the subject together with applications, some details might be omitted, but references will be provided.

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Reference

The two main references for this course will be

  • S. Lord, F. Sukochev, D. Zanin: Singular traces, theory and applications, 2013
  • B. Simon: Trace ideals and their applications, 2005 (second edition)

Plan of the course

  1. Hilbert space and linear operators,
  2. Normed ideals of K(H),
  3. The Dixmier trace,
  4. Heat kernel and zeta-function,
  5. Traces of pseudo-differential operators.

Required Knowledge

Knowledge on standard undergraduate functional analysis.

Attendance

This course is open for any students at Nagoya University as one of the "open subjects" of general education.

Method of Evaluation

Grades based on attendance, a written report, or an examination.

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Lecture Notes

full document (98pages)
Dixmier traces (PDF, 385KB)

Lecture Notes

Title (PDF, 17KB)
Contents (PDF, 61KB)
Chapter 1
Hilbert space and linear operators (PDF, 165KB)
Chapter 2
Normed ideals of K(H) (PDF, 175KB)
Chapter 3
The Diximier trace (PDF, 137KB)
Chapter 4
Heat kernel and zeta-function (PDF, 101KB)
Chapter 5
Traces of pseudo-differntial operators (PDF, 194KB)
Bibliography
Bibliography (PDF, 55KB)

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Page last updated August 9, 2017

The class contents were most recently updated on the date indicated. Please be aware that there may be some changes between the most recent year and the current page.

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