## Researchers:

- Dr Nurulla Azamov, Emeritus Prof Peter Dodds, Dr Theresa Dodds , Associate Prof Vladimir Ejov (leader), Prof Jerzy Filar, Prof Vladimir Gaitsgory, Emeritus Prof Gopal Gopalsamy, Mr Yohei Tanaka

Plot of the function f(x)=(x^2 - 1)(x - 2 - i)2/(x^2 + 2 + 2i) The hue represents the function argument, while the brightness represents the magnitude. |

It is sometimes said that mathematics has dual nature due to the fact that it is the science of relations as well as the science of quantity and measurement. Arguably, Mathematical Analysis (usually abbreviated as, simply, Analysis) is the one branch of mathematics that is devoted to both of these aspects of mathematics.

Relations are represented as functions and the study of their properties forms the core of mathematics curricula across the world. The discovery of infinitesimal calculus in the 17th century – independently by Newton and Liebnitz - has led to dynamic growth of this field which now includes a wide range of important specialisations such as real, complex and functional analysis. The quantity and measurement aspect is often represented either with the help of extrema of the functions studied, or with the notion of a measure that was pioneered by Lebesgue in the early 20th century. The scientific impact of this branch of mathematics has been profound. Much of modern understanding of science and technology is recorded in the language of functions and their properties.

The work of Flinders' Analysis research group focuses on a selection of problems from this vast field, and challenging, field of mathematics. Specific projects, currently being investigated include: mathematical scattering theory, theory of spectral shift function, perturbation theory of operators in Hilbert space, von Neumann algebras. The three projects listed below are currently receiving priority.

**Professional and community engagement**

- Peter Dodds serves as Associate Editor of the Journal Positivity, published by Springer.
- Jerzy Filar serves as Associate Editor of Elsevier's Journal of Mathematical Analysis and Applications (JMAA) and has been a Fellow of the Australian Mathematical Society since 2003.
- Gopal Gopalsamy serves as co-Editor of Elsevier's Nonlinear Analysis: Real World Applications.

## Absolutely continuous and singular spectral shift functions

Nurulla Azamov has been investigating a new approach to scattering theory of arbitrary self-adjoint operators by arbitrary self-adjoint trace-class perturbations. Scattering theory of self-adjoint operators by self-adjoint trace-class perturbations is a classical branch of analysis, originated in 1950's in the works of T. Kato, M. Rosenblum, M.Sh. Birman, etc. This branch of mathematics is sometimes called abstract or mathematical scattering theory. There is an enormous literature and there are two notable monographs devoted to this subject: H. Baumgartel, M. Wollenberg, Mathematical scattering theory, Birkhauser, 1983 and D.R.Yafaev, Mathematical scattering theory, Providence, R.I., AMS 1992.

The main feature of the new approach, given in the above paper, is its constructiveness. There are two approaches in the literature to this theory, which are called time-dependent scattering theory and stationary scattering theory. The time-dependent approach focuses mainly on wave operators and the scattering operator, which is also called the off-shell scattering operator in physical literature. In stationary approach the main focus is on the so-called wave matrices and the scattering matrix which is also called the on-shell scattering operator. The on-shell scattering operator is the main object of interest for physicists. The drawback of both approaches is that they do not allow to define explicitly the set of values of energy for which the scattering matrix exists. This is a big hindrance; it implies in particular that the scattering matrix cannot be considered as a function of a physical parameter for a given value of the parameter.

In Azamov's forthcoming manuscript - *Absolutely continuous and singular spectral shift functions* the set of values of energy (called regular values) for which the scattering matrix exists is defined upfront as a set of numbers, which satisfy a well-known theorem called limiting absorption principle, before the development of the proper scattering theory. For each regular value of energy all the main objects of scattering theory, such as scattering matrix, wave matrices, the Hilbert space of asymptotic states etc, are constructed and their properties are proved. It is also shown that these objects coincide with similar objects of time-dependent theory. Many classical results of mathematical scattering theory are given new proofs, including Kato-Rosenblum theorem, Birman-Krein formula, multiplicative property of the wave matrices etc.

However, this paper does more than supply a new presentation of a well-known theory. The approach of this paper allows us to consider the scattering matrix (that is, the on-shell scattering operator) as a function of physical parameters, such as coupling constant, in the abstract setting. In particular, it is shown that scattering theory possesses a certain coupling-constant regularity property. This coupling-constant regularity property allows us to introduce a natural splitting of the so-called spectral shift function into absolutely continuous and singular parts. One of the main results of this paper asserts that the singular part of the spectral shift function is an almost everywhere integer-valued function of energy for arbitrary trace-class perturbations of arbitrary self-adjoint operators. This paper makes it possible to pose many new questions in abstract scattering theory, which did not make sense before. The manuscript which is 119 pages long will appear in Dissertationes Mathematicae.

*Integral Property Azamov’s singular spectral shift function*

## Non-commutative integration

Peter Dodds - in collaboration with Ben dePagter (Delft University of Technology, Netherlands) and Fedor Sukochev ( UNSW) is working on a research monograph that will serve as an introductory graduate level text as well as a basic reference for more established mathematicians with interests in non-commutative analysis and probabliity. Its origins lie in two apparently distinct areas of mathematical analysis: the theory of operator ideals going back to von Neumann and Schatten and the theory of rearrangement invariant Banach lattices of measurable functions, which has its roots in many areas of classical analysis related to the development of the well-known Lp-spaces. The principal aim of the monograph is to create a single theory which contains each of these motivating areas as special cases of the more general theory. The heart of this theory is the development of non-commutative integration created by Dixmier and Segal in the 1950's, which in turn is based on the fundamental work von Neumann and Murray on algebras of operators on some Hilbert space. In this theory, the notion of a classical measure is replaced by that of a normal trace on a semifinite von Neumann algebra, and the notion of complex measurable function is replaced by that of a measurable operator affiliated with the corresponding von Neumann algebra.

The introduction of non-commutative measure theory permits the detailed study of Banach spaces whose elements are measurable operators. This study parallels the classical theory of Banach function spaces. From this general study, the theory of Schatten ideals as well as much of the classical theory of rearrangement-invariant Banach function spaces now emerge as special cases. This unification, in turn, yields considerable new insight.

The monograph will contain chapters on topics such as the construction of non-commutative functions spaces from rearrangement-invariant spaces on the positive semi-axis,; the development of non-commutative rearrangement inequalities which find their roots in the study of singular value inequalities familiar from matrix theory; duality theory and non-commutative interpolation; Banach space geometry of non-commutative symmetric spaces, including convexity and concavity properties; Kadec-Klee properties; UMD and non-commutative Khintchine and martingale inequalities.

## Analytic perturbation theory and applications

Jerzy Filar – in collaboration with K. Avrachenkov (INRIA, France) and P. Howlett (UniSA) – is nearing the completion of a graduate level text book that was under contract to be published in SIAM’s prestigious Studies in Applied and Numerical Mathematics series.

This book was motivated by the observation that in a vast majority of applications of mathematics the systems of governing equations include parameters that are assumed to have known values. Of course, in practice, these values may only be known up to a certain level of accuracy. Hence, it is essential to understand how deviations from their nominal values may affect solutions of these governing equations. Naturally, there is a desire to study the effect of all possible deviations. However, in its most general setting, this is a formidable challenge and hence structural assumptions are usually required if strong, constructive, results are to be explicitly derived. Frequently, parameters of interest will be coefficients of a matrix.

Therefore, it is natural to begin investigations by analyzing matrices with perturbed elements. Historically, there was a lot of interest in understanding how such perturbations influence key properties of the matrix. For instance, how will the eigenvalues and eigenvectors of this matrix be affected?

In this book we study a range of problems that are more general than spectral analysis. In particular, we analyse the behavior of solutions to perturbed linear and polynomial systems of equations, perturbed mathematical programming problems, perturbed Markov chains and Markov Decision Processes and some corresponding extensions to operators in Hilbert and Banach spaces.

## Complex Analysis and Cauchy-Riemann Differential Geometry

Vladimir Ejov collaborates with many researchers in this area. In particular, his notable results are listed below.

In collaboration with M. Eastwood and A. Isaev in the paper *Towards a classification of homogeneous tube domains in C^{4}* (2004), the authors give a classification of all tube domains in

**C**

^{4}, with a base on which an affine group acts transitively and which contains in its boundary a homogeneous hypersurface. The Lie group of automorphisms of such domains is also described. There are three classes of such domains: generalization of balls, nil-balls in the sense of Penney and a class of four new domains.

In collaboration with M. Kolar and G. Schmalz in the paper

*Degenerate hypersurfaces with a two-parametric family of automorphisms*(2007), the authors give a complete classification of Levi-degenerate hypersurfaces if finite type in

**C**

^{2}with 2-dimensional symmetry groups. The analysis is based on classification of 2-dimensional Lie algebras and explicit description of isotropy groups for such hypersurfaces that follows from construction of Chern-Moser type normal forms at points of finite type.

In collaboration with G. Schmalz in the paper

*Non-linearizable CR-automorphisms, torsion-free elliptic CR-manifolds and second order ODE*(2005), the authors show the existence of an elliptic CR manifold of CR dimension and CR codimension 2 not related to quadrics whose isotropy group is nonlinearizable. In addition, a complete description is given of such manifolds by using Cartan's method of equivalence to obtain a complete classification of second-order ordinary differential equations admitting Lie groups of fibre-preserving point symmetries which correspond to nonlinearizable automorphisms.

Currently Ejov is working with G. Schmalz on the classification of rigid spheres in

**C**

^{2}, the problem formulated by N. Stanton in 1991, which, in particular, would provide the solution for the zero curvature PDE:

for the surface defined as Im *w = h(z) *and *f(z) = *log Δ*h(z)*.

## Construction of Near Optimal Oscillatory Regimes in Singularly Perturbed Control Systems via Solutions of Hamilton-Jacobi-Bellman Inequalities

Vladimir Gaitsgory is an applied mathematician with broad areas of interest in control, optimisation (operations research), dynamical systems and games theories, and their applications. His most important contributions are in the areas of singularly perturbed (SP) Markov Decision processes and mathematical programming problems and in the area of SP dynamical control systems. He has authored and co-authored three research monographs and numerous papers. He has attracted a significant ARC funding to support his research. His is currently an ARC DORA Fellow, working on the following project.

Problems of optimal control of systems evolving in multiple time scales arise in a great variety of applications (from diet to environmental modelling). We address the challenge of analytically and numerically constructing rapidly oscillating controls that “near optimally coordinate" the slow and fast dynamics. The ability to optimally control dynamical systems evolving in multiple time scales bene?ts a great variety of applications. We develop a set of new algorithms for construction of such controls based on solutions of Hamilton-Jacobi-Bellman inequalities. These are extensively tested on numerical examples and models of real-life systems arising in biomedical and environmental research.