2019-2020 Intensive Courses
NEW: "Spectral theory, singular integral operators and harmonic analysis" by Professor Sergei Treil, Brown University
Dates: Wednesday 5th and 12th February 2020
Venue: Room S4.23, Floor 4, KCL, Strand Building, London, WC2R 2LS
Abstract:
Harmonic analysis and spectral theory are closely interconnected: harmonic analysis provides an important tool for the spectral theory, and spectral theory serves as a motivation for many interesting problems in harmonic analysis.
In the mini-course I'll explain the connections between spectral theory and harmonic analysis, and present some recent result in spectral theory obtained using modern developments in harmonic analysis.
Topics to be covered include:
- review of the spectral theorem for self-adjoint and unitary operators; von Neumann direct integrals;
- basic theory of rank one perturbations, Kato-Rosenblum and Aronszajn-Donoghue theorems;
- harmonic analysis problems related to rank one perturbations: two-weight estimates for the Hilbert transform, regularization of singular integral operators;
- non-selfadjoint perturbations, model spaces and Clark measures; classical results and new developments;
- matrix measures and higher rank perturbations.
No special knowledge except basic operator theory and measure is needed.
Background reading:
M.Sh.Birman, M.Z.Solomjak, "Spectral theory of self-adjoint operators in Hilbert space", Chapters 5 and 6.
B.Simon, "Trace Ideals and Their Applications", Chapter 12.
A.Poltoratskii, "Boundary behaviour of pseudocontinuable functions", St.Petersburg Math. J. vol. 5 no.2 (1994).
C.Liaw, S.Treil, "Clark model in the general situation", Journal d'Analyse Mathematique vol. 130 no. 1 (2016).
C.Liaw, S.Treil, "Rank one perturbations and singular integral operators", Journal of Functional Analysis, vol. 257 no. 6 (2009).
=====================
This is a free course, open to all. To register, please email office@ltcc.ac.uk.
Intensive Course 2019
Methods of Noncommutative Analysis
by Dr. Rauan Akylzhanov, Research Associate, School of Mathematical Sciences, Queen Mary University of London
Starts 15 May (1pm to 5pm), ends: 16 May (9am to 1pm) / Venue: Room 500, 5th Floor, Department of Mathematics, 25 Gordon Street, UCL, London, WC1H 0AY
Noncommutative analysis is a young, newly emerging research field at the intersection of noncommutative geometry and classical (mainly harmonic) analysis. This course will provide an introduction up to research level. After some background from classical analysis, we introduce semi-finite von Neumann algebras and tools to 'measure the size' of 'noncommutative measurable functions'. Motivated by work of Alain Connes, we see how a Dirac-like operator can encode aspects of geometry and classical analysis. In particular, we deduce Lp - Lq bounds for linear operators affiliated with quantum group von Neumann algebras. The course concludes with a discussion of open problems. Some exposure to basic functional analysis would be helpful.
To register for this course, please complete the registration form above or contact office@ltcc.ac.uk.
