Mathematical analysis continues the development of calculus and the theory of real and complex functions. It is an exciting, vibrant field of immense depth and variety with wide-ranging applications in both pure and applied mathematics, as well as in physics, biology, chemistry, and engineering. Mathematical analysis is a major strength of the Mathematics group at Kent with strong connections to geometry, topology, mathematical physics, and nonlinear systems.
The fields of expertise within the group include:
Differential equations and applications
Painlevé equations
In the late nineteenth and early twentieth century, Painlevé and co-workers performed a classification of second-order ordinary differential equations (ODEs) whose solutions are single-valued in the neighbourhood of all movable singularities, i.e. they have no movable critical points.
In the process Painlevé et al discovered six new nonlinear ODEs whose general solution define new transcendental functions since they are not able to be expressible in terms of previously known functions, such as elementary and elliptic functions, and/or in terms of solutions to linear ODEs, and can be thought of as nonlinear analogues of the classical special functions.
In the latter half of the twentieth-century Painlevé equations – although discovered from Mathematical considerations – were found to have applications in a vast range of areas ranging from random matrices and quantum gravity to critical phenomena in wave propagation. Current research is mainly concerned with special solutions of Painlevé equations (rational solutions, and those given in terms of linear special functions), as well as links with orthogonal polynomials and random matrix theory. Other research involves discrete Painlevé equations, which arise from these same contexts, as well as from nonlinear superposition formulae (Bäcklund transformations) for Painlevé ODEs.
Geometric and nonlinear functional analysis
Geometric and Nonlinear Functional Analysis lies at the crossroads of functional analysis, topology and geometry, and is an important and vast area of mathematical analysis. Research at Kent in this field focuses on the fascinating interfaces between algebraic and geometric topology and dynamics and bifurcation theory. In addition, geometrically motivated nonlinear differential equations, and connections between metric geometry and operator algebras are investigated. More details on our research in this area can be found below.
Jordan algebras and metric geometry
The concept of a Jordan algebra has a rich history in mathematics. It was originally introduced by P. Jordan, J. von Neumann and E. Wigner in the 1930s as an algebraic model for quantum mechanics, but soon after unexpected connections with Lie theory, geometry and harmonic analysis were found. A beautiful connection between formally real Jordan algebras and the geometry of cones was discovered by M. Koecher and E. Vinberg. They showed that the cones of squares infinite dimensional formally real Jordan algebras are precisely the symmetric cones, i.e., self-dual cones on which the group of linear automorphisms acts transitively on the interior. The Koecher-Vinberg characterisation provides a striking link with the theory of Riemannian symmetric spaces.
In infinite dimensions, no such characterisation of real Jordan algebras exists, as most real Jordan algebras are realised as Banach spaces rather than Hilbert spaces. Recent findings, however, indicate that there exist alternative characterisations of real Jordan algebras in terms of the geometry of their cones of squares. The research at Kent in this area focusses on further unravelling the connections between real Jordan algebras and geometry, and intertwines ideas from the analysis, Jordan algebras, and metric geometry.
Dynamics of nonexpansive mappings
Nonexpansive mappings are Lipschitz mappings with constant one. Arguably they are, besides isometries, the most fundamental mappings on metric spaces. A central problem is to understand the fixed points and the iterative behaviour of nonexpansive mappings. In case the mapping is a Lipschitz contraction on a complete metric space, Banach ’s contraction mapping theorem provides a solution. If, however, one merely assumes the mapping to be nonexpansive, it is much harder to decide if it has a fixed point, and the iterative behaviour can be complicated.
In the past decades, some surprising discoveries have been made in this area. Among other results it has been shown that every bounded orbit of a nonexpansive mapping on a finite-dimensional normed space with a polyhedral unit ball, converges to a periodic orbit, and, moreover, there exist a priori upper bounds on the possible period lengths. Also, intriguing analogues of the classic Denjoy-Wolff theorem concerning the dynamics of fixed point free holomorphic self-mappings of the open unit disc, have been obtained for fixed-point free nonexpansive mappings on metric spaces that possess features of nonpositive curvature. Research at Kent in this area uses a remarkable mixture of analysis, topology, and metric and discrete geometry.
Topological methods in bifurcation theory
Bifurcation theory has been used for centuries to explain various phenomena in the natural sciences where a physical system depends on a parameter and changes its qualitative behaviour once the parameter crosses a threshold. Some typical examples are the buckling of the Euler rod in statics, the appearance of Taylor vortices in fluid dynamics, the onset of oscillations in an electric circuit in electrical engineering, and the bromination of malonic acid in chemistry.
Topological methods have been employed in bifurcation theory from the very beginning of its systematic study and they have often revealed surprising interfaces between analysis and topology.
At Kent, we apply classical and modern tools from algebraic topology and global analysis to bifurcation problems. This builds a remarkable bridge between entirely different fields of mathematics connecting questions from applied mathematics with deep results of theoretical mathematics like the Atiyah-Singer Index Theorem.
Nonlinear differential equations in geometry
Many problems in modern geometry lead to nonlinear differential equations. Nonlinear equations can usually not be solved explicitly, but the existence of solutions and their qualitative behaviour is an exciting question for which powerful tools have been developed over the centuries. A typical example is the geodesic equation, which is an ordinary differential equation that is obtained when looking for the shortest paths between two points in a curved space.
At Kent, we are interested in the geodesic equations of semi-Riemannian manifolds (including models of spacetime according to Einstein’s General Theory of Relativity). More generally, we study various kinds of Hamiltonian systems that play an important role in symplectic geometry, as well as various types of geometrically motivated partial differential equations.
Operator theory
Operator theory is an important branch of functional analysis which studies linear and nonlinear maps between topological or normed vector spaces. It usually focuses on analysing the spectrum, eigenvalues and eigenfunctions of the operators. At Kent, we study both abstract classes of operators and their properties, as well as concrete differential operators arising in physical or biological applications. More details on the research at Kent in this area can be found below.
Nonlinear Perron-Frobenius theory
There is a vast theory, going back to Perron and Frobenius, describing the spectral properties of linear operators that leave a cone in an ordered vector space invariant. In the past few decades, a number of strikingly detailed nonlinear extensions of the Perron-Frobenius theory have been obtained. These results provide an extensive analysis of the eigenvectors and eigenvalues of various classes of monotone nonlinear operators and give information about their iterative behaviour.
Much of the work in this area is motivated by applications in monotone dynamical systems theory, i.e., dynamical systems that preserve a partial ordering on the state space, which frequently occurs as models in mathematical biology.
Research at Kent aims to further develop this theory and create new tools to analyse nonlinear problems.
Spectral theory of non-selfadjoint operators
Non-selfadjoint operators and associated spectral problems arise naturally in many physical application areas, such as hydrodynamics, magneto-hydrodynamics, lasers, and nuclear scattering. The spectral behaviour of these operators is considerably more complex than that of selfadjoint operators. This leads to various new and unexpected phenomena which have implications for both the analysis and computation of spectral results. In particular, the spectrum is no longer necessarily confined to the real axis and the standard tools available for the selfadjoint case, such as the spectral theorem and variational principles, are no longer applicable.
In recent years there have been many developments in techniques for non-selfadjoint operators, including functional models for various classes of non-selfadjoint operators and the use of so-called boundary triples. Boundary triples allow the introduction of M-functions (Dirichlet-to-Neumann-type maps) for many operators. These are determined completely by boundary data, leading to interesting inverse problems e.g. how much spectral information is contained in the M-function of a non-selfadjoint operator? Related to this are questions from extension theory of operators – how can we characterize all extensions of an operator with certain properties (e.g. self-adjointness, a reality of the spectrum or maximal dissipativity)?
At Kent, we study these questions, both in terms of abstract operator theory and for concrete examples, such as elliptic differential operators or highly singular ordinary differential operators.
Spectral effects of perturbations to linear operators
In many situations, one is interested in making small changes (perturbations) to a physical system to produce a new effect. One example of this is in the propagation of electromagnetic waves in periodically structured media, such as photonic crystals and metamaterials which play a role in many areas, such as lasers and cloaking devices. Typically, the propagation of waves in such media exhibits band-gaps, i.e. intervals on the frequency or energy axis where propagation is forbidden. Mathematically, these correspond to gaps in the spectrum of the operator describing a problem with periodic background medium.
When the periodicity is perturbed by point or line defects, localization may take place in band gaps. The use of line defects in photonic crystals has been proposed in the context of waveguide applications. The gap localization gives rise to guided modes which decay exponentially into the bulk structure and propagate along the direction of the line defect. For lasers, this opens up the possibility of producing light that is highly localised, both in frequency and location. For this reason, it is of great importance to know whether a given line defect produces gap modes.
At Kent, we study these questions making use of methods of the theory of linear operators, including the Floquet-Bloch decomposition of periodic problems and variational characterisations of the spectrum.
Another example is embedded eigenvalues for periodic Schroedinger operators. These can be traced all the way back to von Neumann and Wigner, where perturbations by oscillating potentials were used to produce the desired bound states. It has since been suggested that these bound states might be found in particular molecular and atomic systems.
At Kent, we are looking at the corresponding discrete problem by analysing the asymptotics of the (perturbed) transfer matrices.
Spectral flow and applications
The spectral flow is an integer-valued homotopy invariant for paths of selfadjoint Fredholm operators that was invented by Atiyah, Patodi and Singer in the seventies in order to study the fascinating relationship between spectra of differential operators on manifolds and the underlying geometry. Since then it has been used in various areas such as gauge theory in Mathematical Physics, Floer theory in a symplectic analysis, and bifurcation theory in nonlinear functional analysis.
At Kent, we study applications of the spectral flow in all previously mentioned areas. In particular, we are interested in formulas that equate the spectral flow with analytic and geometric invariants as well as their applications to bifurcation theory.
There is a strong overlap with the research in geometric and nonlinear functional analysis.
Special functions
Special functions are important analytical and computational instruments in mathematics as well as in physics and engineering. They appear and are studied from many different viewpoints using analysis, combinatorics, algebraic techniques, probability, numerical analysis and approximation theory, among other research areas. Examples include Airy and Bessel functions, the Gamma and zeta function, hypergeometric functions, classical orthogonal polynomials, and more recently Painlevé transcendents.
Painlevé equations
In the late nineteenth and early twentieth century, Painlevé and co-workers performed a classification of second-order ordinary differential equations (ODEs) whose solutions are single-valued in the neighbourhood of all movable singularities, i.e. they have no movable critical points.
In the process Painlevé et al discovered six new nonlinear ODEs whose general solution define new transcendental functions since they are not able to be expressible in terms of previously known functions, such as elementary and elliptic functions, and/or in terms of solutions to linear ODEs, and can be thought of as nonlinear analogues of the classical special functions.
In the latter half of the twentieth-century Painlevé equations – although discovered from Mathematical considerations – were found to have applications in a vast range of areas ranging from random matrices and quantum gravity to critical phenomena in wave propagation. Current research is mainly concerned with special solutions of Painlevé equations (rational solutions, and those given in terms of linear special functions), as well as links with orthogonal polynomials and random matrix theory. Other research involves discrete Painlevé equations, which arise from these same contexts, as well as from nonlinear superposition formulae (Bäcklund transformations) for Painlevé ODEs.