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Functional Analysis

Functional analysis has developed from its origins in differential equations and Fourier analysis into a major area of modern mathematics, with applications in representation theory, dynamical systems, quantum physics and control theory, among others. It is an area of enormous current vitality: at least four practitioners have won Fields medals (the mathematical equivalent of the Nobel Prize) over the past 25 years. The group at Newcastle has expertise across a broad range of subdisciplines and interests which spread deeply into neighbouring areas.

Researchers:

PhD Students:

Noncommutative Geometry and Quantum Groups
(Wojciech Szymanski, Roberto Conti)

Noncommutative geometry is a relatively new area of pure mathematics studying geometric objects with help of functional analysis and algebra, in which noncommutative algebras play the role of function algebras on classical topological spaces and differentiable manifolds. Examples of such understood 'noncommutative spaces' include the phase space of quantum mechanics, noncommutative torus and duals of nonabelian groups. Instead of group actions, their symmetries are captured by actions of Hopf algebras and quantum groups, which themselves provide interesting examples in this theory. In this project we focus on: constructions and analysis of new examples, developing topological theory of quantum fibre bundles (especially principal and associated bundles), noncommutative index computations, the C*-algebraic structure of quantum groups and homogeneous spaces (and especially their relation with graph algebras and other generalized Cuntz-Krieger algebras), and on spectral aspects of noncommutative geometry a là Connes involving construction and analysis of abstract Dirac operators. Key external collaborators on this project include: J. H. Hong (Korea), P. M. Hajac, R. Matthes, A. Sitarz (all Poland), P. F. Baum (USA), P. Bertozzini, W. Lewkeeratiyutkul (both Thailand). PhD students: David Robertson.

Totally Disconnected Locally Compact Groups
(George Willis, Udo Baumgartner)

Discrete mathematics, number theory and algebra are among the branches of mathematics where totally disconnected groups are found. They occur as, for example, automorphism groups of relational structures (such as binary trees), matrix groups over the p-adic numbers or as pro-finite groups. We are investigating these diverse groups from a unified point of view that is based on concepts known as 'the scale function' and 'tidy subgroup'. In collaboration with researchers from Australia, India, Israel, Europe and North America, we are also applying these new techniques to totally disconnected groups occurring in other branches of mathematics. The conference 'Totally disconnected groups, graphs and geometry', held at Blaubeuren (Germany) in 2007 and organized by members of our research group, brought together 60 mathematicians from different research areas to explore these applications. The conference website is http://science-it.newcastle.edu.au/~ub563/BLAUBEUREN/Blaubeuren-Home_Page.htm. PhD students: Daniel Horadam.

Non-Linear Analysis and Fixed Point Theory
(Brailey Sims)

Nonlinear optimization and control problems lead to variational inequalities and hence ultimately to fixed points of a related nonlinear operator. Equilibria of dynamical systems correspond to fixed points of nonlinear maps on infinite dimensional spaces. When the system is dissipative, the mapping is often nonexpansive with respect to an appropriate metric. The convergence and ergodic structure of orbits and various iterative schemes, such as those of Pickard and Krasnoselskii, relate to the stability and long-term average behaviour of the system. Our work furthers the study of nonexpansive and related types of mappings and multifunctions, with a focus on establishing readily verified, yet widely applicable, criteria ensuring the existence of fixed points, together with effective algorithms by which they can be approximated. Special emphasis is given to the more difficult cases, where the underlying space lacks nice geometric structure such as that exhibited by a Hilbert space, or where there is no intrinsic linear structure present. For this we exploit ultraproduct methods and hyperbolicity in geodesic metric spaces. This work involves collaboration with mathematicians from Asia, Europe and North America, including Tomas Benevides (Seville), Sompong Dhompongsa (Chiang Mai,Thailand), Enrique Llorens Fuster (Valencia), Art Kirk (Iowa). PhD students: Francisco Eduardo Castillo Santos, Ian Searston, Mark Smith.

Banach Spaces and Banach Algebras
(George Willis, Brailey Sims, Venta Terauds, John Giles)

Spaces of functions or sequences are the context for many mathematical models. The geometry of such spaces is often important when solving the models and this geometry forms part of the subject matter of Banach space theory. In many models it is also important that the functions or sequences in the space can be multiplied, in which case the theoretical setting is that of Banach algebras. We are investigating various aspects including Banach space geometry (convexity and smoothness properties of the norm), theory and applications of ultraproduct constructions, approximation properties, non-commutative 2 harmonic analysis, radical algebras, and spectral theory and functional calculus of Banach space operators. Our work involves collaboration with mathematicians from Asia, Europe and North America.

C*-Algebras
(Wojciech Szymanski, Roberto Conti)

This project is focused on structural properties and applications of C*-algebras, especially generalized Cuntz-Krieger algebras and C*-algebras related to locally compact groups. The applications we are primarily interested in include noncommutative geometry, quantum groups, and Fourier analysis. Key external collaborators on this project include: J. H. Hong (Korea), E. Bedos (Norway).