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Pure and Applicable Analysis – University of Reading

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Pure and Applicable Analysis

Pure and Applicable Analysis is one of the core mathematical disciplines offering a wide range of research topics ranging from investigations in Number Theory to applications in important questions triggered by applications in areas such as fluid dynamics.

The research group in Mathematical Analysis in Reading is one of the biggest in the department, and covers many of these different research strands.

Mission

The mission of the network is to foster a collaborative environment for intradisciplinary research challenges involving any and all aspects of applied complex analysis, including numerical methods and computational schemes based on complex analysis and conformal geometry. The range of interests of the members spans the physical sciences (fluid mechanics, solid mechanics) and mathematical physics, through to more mathematical topics such as function and operator theory, random matrix theory, spectral analysis and integrable systems.

Events

The network holds regular dedicated workshops on special topics, organizes mini-symposia at larger general meetings/conferences, as well as "Student Chapter Meetings" where PhD students and Post-Docs convene to discuss current projects and share ideas. The focus of the research is this area is the application of complex analytical techniques to the study of integrable systems, boundary value problems, water waves, orthogonal polynomials and random matrix theory.

Research

Nonlinear Partial Differential Equations (Nikos Katzourakis)

At the centre of our research in this area lies the rigorous mathematical analysis of non-linear partial differential equations, including free-boundary problems for non-linear partial differential equations, analysis of non-linear PDE models of large scale atmospheric flow, fully non-linear elliptic PDE's and connections to variational problems, boundary value problems for integrable nonlinear equations.

Applied and Computational Complex Analysis (Tristan Pryer, Jani Virtanen)

The focus of the research is this area is the application of complex analytical techniques to the study of integrable systems, boundary value problems, water waves, orthogonal polynomials and random matrix theory.

Spectral Theory (Michael Levitin)

Our research interests are mainly in foundations, applications and numerical methods of spectral theory. Most of our research deals with the spectral analysis of differential operators, including spectral problems in waveguides, spectral geometry, spectral properties of non-selfadjoint operators and operator pencils, and the Fokas transform approach to linear PDEs.

Number Theory and Complex Analysis (Titus Hilberdink, Rachel Newton)

Number theory contains a wealth of fascinating and easily stated problems, both solved and unsolved, which can be tackled using many different mathematical techniques - analytic, algebraic, geometric, probabilistic etc.: to mention just a few, the Prime Number Theorem, Fermat's Last Theorem, the twin prime conjecture, the ABC conjecture, and perhaps the greatest open problem of all - the Riemann Hypothesis.

Our main research themes are:

  • analytic number theory, using both complex analysis and functional analysis
  • rational points on varieties, local-global principles and obstructions
  • additive combinatorics

Operator Theory (Jani Virtanen)

The central theme of operator theory is the investigation of linear mappings between (infinite dimensional) Banach spaces. The linear mappings under consideration can be very concrete operators, such as Toeplitz operators, or can be studied in an abstract framework.

Of particular interest are Toeplitz, Hankel and singular integral operators on various (analytic) function spaces such as Hardy, Bergman and Fock spaces, and their most important properties such as boundedness, compactness, and other spectral properties, as well as operator algebras and Jordan triple systems of operators acting upon Hilbert spaces, mathematical foundations of quantum mechanics and, more recently, the theory of operator spaces (alias, quantised Banach spaces).

Linear Wave Equations (Nick Biggs, Peter Chamberlain)

Of interest are all aspects of propagation and scattering of linear waves, often investigated now with the aid of perturbation methods, in particular the propagation and trapping of waves in waveguides which are slowly-varying in some sense (e.g. containing a bulge whose width varies slowly along the waveguide, or a bend whose curvature varies slowly along the waveguide).

A more recent topic of interest is that of edge waves trapped in the vicinity of a periodic but slowly-varying boundary.

A final area of interest is the derivation of so-called embedding formulae, which allow solutions of certain scattering problems to be expressed in terms of solutions to other related scattering problems. These formulae are interesting in their own right, but also allow the computation of a wide range of solutions to be carried out extremely efficiently.

People

Name Position Telephone
+44 (0) 118 378
Email
@reading.ac.uk
Dr Nick R T Biggs Associate Professor 8998 n.r.t.biggs
Dr Peter Chamberlain Director of Undergraduate Studies, Associate Professor 5005 p.g.chamberlain
Dr Titus Hilberdink Lecturer 5020 t.w.hilberdink
Dr Nikos Katzourakis Lecturer 7463 n.katzourakis
Professor Michael Levitin Professor of Applied Mathematics 8997 m.levitin
Dr Rachel Newton Lecturer 8914 r.d.newton
Dr Jani Virtanen Lecturer 7930 j.a.virtanen

Former group members:

Professor Beatrice Pelloni

Dr Eugen Varvaruca

Professor Les Bunce

Dr Simon Baker

Dr Sean Prendiville

Dr Oliver Roche-Newton 

Publications

Publications of the members of the research group, available at Reading Publications Database Centaur, can by found by clicking the link below.

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