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**An Introduction on Non-Commutative Functional Analysis: Quantised Calculus **

**An Introduction on Non-Commutative Functional Analysis: Quantised Calculus**

#### Lecturers

Professor Fedor Sukochev, The University of New South Wales

Dr Galina Levitina, The University of New South Wales

** **Synopsis

Functional analysis a central pillar of modern analysis and its foundations will be covered in this course with an emphasis on the study of compact linear maps on Hilbert spaces. This course provides the basic tools for the development of such areas as quantum mechanics, harmonic analysis and stochastic calculus. Special attention will be given to applications in noncommutative geometry, specifically quantised calculus. The quantised calculus was introduced by Alain Connes in his 1994 book Noncommutative Geometry. This is a means of doing calculus which makes sense in a high degree of generality and has many properties analogous to the classical notion of “infinitesimal calculus”. We will give a brief overview of applications to Julia sets.

#### Course Overview

- Review of general Functional Analysis (measure and integration, Banach spaces and their duals, Hilbert spaces, bounded linear operators on Hilbert spaces).
- Compact operators (spectral theory of compact operators and singular value decomposition)
- Ideals of compact operators: Calkin theorem for two-sided ideals, Schatten and weak Schatten ideals
- Traces on ideals of compact operators: the classical trace and singular traces
- Applications: quantised calculus on the circle and an overview of quantised calculus for fractals, Connes trace theorem.

#### Pre-requisites

- Basics of set theory (countable and uncountable sets, Schroeder-Bernstein Theorem)
- Topological and metric spaces (convergence, open and closed sets, continuity, completeness, contraction mapping theorem)
- Sequences and series of functions on metric spaces (pointwise and uniform convergence, differentiation and integration of limits and sums, Fourier series).

#### Assessment

- Four tests each for 5%
- One assignment for 40%
- Final exam for 40 %

#### Resources/pre-reading (if available)

- J.B. Conway: A Course in Functional Analysis.
- W. Rudin: Functional Analysis.
- M. Reed and B. Simon: Methods of Modern Mathematical Physics. Vol. 1 Functional Analysis.
- K. Yosida: Functional Analysis

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#### Biography

Professor Fedor Sukochev, The University of New South Wales

Professor Fedor Sukochev is Professor of Pure Mathematics within the School of Mathematics and Statistics at The University of New South Wales. He is a world leader in discovering novel analytic approaches to complicated interdisciplinary problems. His research focuses on noncommutative analysis, geometry and probability. In 2012, he was awarded with a three-year ARC Discovery Outstanding Researcher award. In 2016, Professor Sukochev was elected a Fellow of the Australian Academy of Science for his research contributions to noncommutative Analysis and Geometry. In 2017 Professor Sukochev was awarded with Australian Laureate Fellowship Award.

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#### Biography

Dr Galina Levitina, The University of New South Wales

Galina is a Postdoctoral Research Fellow in the School of Mathematics and Statistics at UNSW. Her research is focuses on noncommutative analysis and its applications in mathematical physics.