Functional Analysis
This area of analysis concerns itself with the interplay between algebraic and topological structures, and provides essential tools for treating such topics as harmonic analysis, ergodic theory, differential equations and integral equations. Central themes are Banach Spaces, Banach Algebras, Operator Theory, Operator Algebras and Spectral Theory. Individual members of the Department are active in the areas of: Numerical Range and Hermitian Elements; Group and Semigroup Algebras; Entire Functions; Operator Algebras and their connections with Topology, K-Theory, Differential Geometry and Physics.
Nonlinear Analysis
This provides the theoretical setting for applications to nonlinear problems, especially those related to differential equations. The tools are a mixture of analytical and topological such as topological degree, fixed point index theory, finite dimensional approximations, variational methods and critical point theory. The current interest in this Department is in the study of positive solutions for nonlinear differential equations involving nonlocal boundary conditions.
Operator Algebras
Research in Operator Algebras in Glasgow is concerned primarily with C*-algebras and von Neumann algebras. The area has advanced greatly in scope since its beginnings in the 1930's and now embraces many challenging areas of research. The principal topics pursued in this department in recent years have included the structure and properties of nuclear and exact C*-algebras, C*-algebras associated with groups and semigroups, K-theory and exact groups.
Banach Algebras
Banach Algebras are objects with rich algebraic and analytical structures; their study seeks to understand the connection between the two. In particular, Banach Algebras generated by semigroup algebras are being investigated: we have considered algebraic properties, including primitivity, of these.
Operator Theory, Linear Systems and vector-valued Harmonic Analysis
Operator theory derives its motivation from many applications. In the group, mainly Toeplitz, Hankel and related operators are being studied. These operators appear naturally in the study of linear systems, as properties like time-invariance of a linear system translate into an intertwining property of associated operators, giving rise to Toeplitz and Hankel operators. Consideration of systems with multidimensional output spaces then leads into questions of vector-valued Harmonic Analysis. Another related area of research in the group is that of such operators in a multivariable setting, which leads into currently much-investigated questions of Harmonic Analysis in a product setting.
Euclidean Harmonic Analysis and Combinatorics
Euclidean harmonic analysis is the study of the Fourier transform and related operators on n-dimensional Euclidean space. In this context, the rich geometry of Rn (translations, dilations, curvature) plays a crucial role. Research in this area often leads to questions of a fractal-geometric, combinatorial, or even number-theoretic nature. For example, two central problems in harmonic analysis are the restriction problem, which arises in the study of PDEs, and Bochner-Riesz, which arises in Fourier inversion. Both of these imply the Kakeya conjecture on the Hausdorff dimension of sets containing many lines, recent progress on which has used the combinatorial theory of sum-sets.
Teaching Staff
Dr M J Crabb
Dr S Pott
Dr A S Wassermann
Prof J R L Webb
Dr S White
Retired Academics
Dr H R DowsonDr C M McGregor
Dr P G Spain
Postdocs
Research Students
Visitors
- Tosin Mewomo, Abeokuta, Nigeria, 19th–24th February 2007
- Antony Wassermann, Marseille/Newton Institute, December 2006
- Marianna Csörnyei, UCL, September 18th–21st 2006
- Eric Guethner, June 28–30, 2006
- Alexander Helemskii, Moscow, June 26–29, 2006
- Rachid El Harti, Hassan I University, Settat, Morocco, June 3rd–June 24th 2006
- Prof C. Sadosky, Howard University, Washington D.C., May 16–June 1, 2006
- Matt Daws, Oxford, May 15–17, 2006
- Dr B. Haak (LMS Scheme 2 grant visit), May 9
- Gennaro Infante, Universita della Calabria, May 2006
- Prof John Duncan, Fayetteville, Arkansas, April 10–14, 2006
- A. Wiggins, Texas A&M
Seminars and Networks
The Analysis Seminar normally meets on Wednesdays in term time at 2 p.m. There is also a joint Applied Analysis Seminar with the Mathematics department at the University of Strathclyde. In addition, our research students participate in the Postgraduate Seminar. The Analysis group is part of two networks in Analysis which are supported by the London Mathematical Society: the North British Functional Analysis Seminar, which is currently being organised from Glasgow, and the UK Harmonic Analysis and PDEs Research Network.
Members of the group are currently being supported by the Engineering and Physical Sciences Research Council and the Royal Society.
Links
Other analysis-related activities nearby include:
Belfast Functional Analysis Day
Edinburgh Mathematical Society
International Centre for Mathematical Sciences
Maxwell Institute
North British Probability Seminar
See the current Analysis group diary.
Members of the group have recently joined from the following Universities:
University of Edinburgh
University College London
University of Newcastle
Texas A&M University
University of Yaounde I
University of York