The analysis seminar is organised under the joint auspices of the School of Mathematics and Statistics and MASCOS. It deals with a wide variety of modern analysis, at a more specialist and detailed level than the Departmental Seminar or Joint Colloquium. A detailed schedule is below.

The talks are held at 12noon on Wednesdays in the Red-Centre room RC-4082 or RC-3084. All participants are welcome to attend and to present a talk, or a series of talks.

If you are interested in presenting a talk, or you (will) have a visitor who would be able to present a talk, please contact Sergey Ajiev.

For a list of past seminars, see the Analysis Seminar Almanac.

Title: Lowering topological entropy in topological dynamics. Part II.
Speaker: A/Prof. Guo Hua Zhang (Fudan University and UNSW)
Abstract:
By a topological dynamical system we mean a compact metric space equipped with a self-homeomorphism. To understand the complexity of a given topological dynamical system, people are interested in the study of factors and subsets of it. There are many concepts reflecting the complexity of a topological dynamical system, such as transitivity, sensitivity, chaos, complexity function, entropy, and so on.

In this talk, we shall only discuss the concept of entropy in topological dynamics along the lines of both factors and subsets. The first part concerns some results of Lindenstrauss, Shub and Weiss obtained in the period of 1991-2000. The second part, the main part of the talk, concerns some recent results of Huang, Ye and myself.

Title: Lowering topological entropy in topological dynamics.
Speaker: A/Prof. Guo Hua Zhang (Fudan University and UNSW)
Abstract:
By a topological dynamical system we mean a compact metric space equipped with a self-homeomorphism. To understand the complexity of a given topological dynamical system, people are interested in the study of factors and subsets of it. There are many concepts reflecting the complexity of a topological dynamical system, such as transitivity, sensitivity, chaos, complexity function, entropy, and so on.

In this talk, we shall only discuss the concept of entropy in topological dynamics along the lines of both factors and subsets. The first part concerns some results of Lindenstrauss, Shub and Weiss obtained in the period of 1991-2000. The second part, the main part of the talk, concerns some recent results of Huang, Ye and myself.

Title: New approach to abstract scattering theory.
Speaker: Dr. Nurulla Azamov (Flinders)
Abstract:
In this talk I shall discuss a new approach to scattering theory of self-adjoint operators by self-adjoint trace class perturbations. It is based on my paper "Absolutely continuous and singular spectral shift functions".

The main feature and difference of this theory from the classical Birman-Entina theory, an exposition of which can be found in D. Yafaev's book "Mathematical scattering theory", is its constructiveness. Recall that in the trace class scattering theory one first defines the wave operators W_± and the scattering operator S, and after that one shows existence a.e. of the wave matrices w_±(λ) and of the scattering matrix A(λ). Further, one proves the stationary formula for the scattering matrix for a.e. λ. The set of full Lebesgue measure, for which the stationary formula holds, is not described, so this approach has a flavor of existence theorems.

In the new theory, we present upfront the set of full Lebesgue measure such that for all points of that set the scattering matrix exists and the stationary formula holds. Moreover, the scattering matrix and the wave matrices (as well as many other necessary ingredients, such as fiber Hilbert spaces) are explicitly constructed. The wave operators and the scattering operator are defined as direct integrals of (respectively) the wave matrices and of the scattering matrix. Their usual time dependent definitions become theorems.

Some applications of the new scattering theory will be given, including the following theorem.

Theorem. The singular part of the spectral shift function is a.e. integer-valued.

Title: Estimates for solutions of a parameter elliptic multi-order system of differential equations.
Speaker: Prof. Melvin Faierman (UNSW)
Abstract:
This talk is concerned with a boundary problem defined over a bounded region in an Euclidean space, and in particular is devoted to the establishment of a priori estimates for solutions of a parameter elliptic multiorder (i.e., of Douglis-Nirenberg type) system of differential equations under limited smoothness assumptions.

We present results which extend those of Agranovich et al. for non-multi-order systems as well as those of Kozhevnikov and Denk-Volevich who deal with multi-order systems, but whose theories do not cover the important case of Dirichlet boundary conditions.

Title: Pseudo-localization of singular integrals.
Speaker: Prof. Tuomas Hytönen (University of Helsinki)
Abstract:
As an intermediate step in the development of a noncommutative Calderón-Zygmund theory, J. Parcet (JFA, 2009) established a new "pseudo-localization principle" of classical singular integral operators in L₂. For a given function f in L₂, this interesting principle provides a set outside of which any normalized singular integral of f has small norm.

I will discuss this result and its extension to the reflexive L_p spaces, which was left open in Parcet's work. The new method of proof, based on martingale techniques, even slightly sharpens the original L₂ result.

Title: On the regularity of the Dirichlet Green potential in convex domains.
Speaker: Prof. Dorina Mitrea (Missouri)
Abstract:
In this talk I will discuss mapping properties of the Dirichlet Green potential on the scale of Besov and Triebel-Lizorkin spaces when the underlying domain satisfies a uniform exterior ball condition or, it is a convex domain. In the process, the Dirichlet and Regularity problems for harmonic functions in convex domains, with optimal nontangential maximal function estimates, will be treated.

Title: Comparison of spaces of Hardy type.
Speaker: Dr. Andrea Carbonaro (Genova)
Abstract:

Title: Comparison of spaces of Hardy type.
Speaker: Dr. Andrea Carbonaro (ANU)
Abstract:
We will define local Hardy spaces of forms h^p_D that are adapted to the Hodge-Dirac operator D = d + d^* on a complete Riemannian manifold M, which can have exponential volume growth. If Δ = D² is the correspinding Laplacian, then we will see that the local geometric Riesz transform D(Δ + a)^-1/2 is bounded on h^p_D, provided that a > 0 is large enough compared to the exponential growth of M.

In particular, the localisation techniques, exponential off-diagonal estimates and atomic characterisation that we have used will be discussed. This is joint work with Andrea Carbonaro and Alan McIntosh.

Title: On concentration, deviation and Dvoretzky's theorem for function and other spaces.
Speaker: Dr. Sergey Ajiev (UNSW)
Abstract:
The anisotropic spaces of functions defined on open subsets of a Euclidean spaces of Besov and Lizorkin-Triebel type endowed with various norms and a wide class of independently generated spaces introduced earlier are considered along with the subspaces of all these spaces and their duals. The norms include those defined in terms of differences, local approximations, functional calculus, as well as wavelet norms.

In a quantitative and uniform manner, we investigate the deviation and the concentration of measure and distance phenomena on subsets of finite-dimensional subspaces of the spaces under consideration following and comparing both classical and new approaches.

As an application, one establishes explicit estimates constituting the Dvoretzky theorem for finite-dimensional subspaces of all the spaces mentioned above by means of a modification of Schechtman's development of V. Milman's approach and its alternative.

Title: Orbits in fully symmetric spaces.
Speaker: Dmitriy Zanin (Flinders)
Abstract:
Semigroup of simultaneous contractions in L₁ and L_∞ is considered. In this talk we provide a criterion for the orbit of a particular element with respect to this semigroup to coincide with the closed convex hull of its extreme points. Sufficient condition was given by Braverman and Mekler more than 30 years ago. Some examples and applications are also given.

Title: The Riemann Zeta Function, The Laplacian, and How to Recover Lebesgue Integration from the Pole of a Zeta Function (Part 2).
Speaker: Dr. Steven Lord (Adelaide)
Abstract:
This talk is a continuation from the Departmental Seminar on Monday 6th April. At the start of the talk we will briefly review the contents of the first part.

The first part introduced a generalisation of the Riemann zeta function using the compact operators on a separable Hilbert space, and linked the residue of the zeta function of a compact operator to the Dixmier trace, a non-normal trace used as the foundation for the "noncommutative integral" in Alain Connes' theory of Noncommutative Geometry.

In this second part we consider specific zeta functions associated to Laplacians on compact Riemannian manifolds. For example, the zeta function associated to the Laplacian on the circle is just a multiple of the Riemann Zeta Function. We introduce zeta functions weighted by bounded operators and show how, in recent work, we solved a problem concerning the "noncommutative integral" that has been open for 20 years.

Namely, we recover the Lebesgue integral of any bounded (and then any) integrable function as the residue of a zeta function. If time permits, we will introduce the integral on the "noncommutative torus" and show, using the same technique, that it can be recovered from zeta functions associated to the "noncommutative Laplacian".

Title: Spectral Multipliers for Schroedinger Operators.
Speaker: rof. Shijun Zheng (Georgia Southern University)
Abstract:
We consider Hormander type spectral multiplier problem for Schroedinger operator with a critical potential in one and three dimensions. It is shown that the multiplier operator is bounded on Lp, Besov spaces and Triebel-Lizorkin spaces under the same sharp condition. We then derive Strichartz estimates for the corresponding wave equations.

Our work is partially motivated by the standing wave problem for the quintic wave equation in 3+1 spacetime dimensions.

Title: Analytic approximation of rational matrix functions.
Speaker: Prof. Vladimir Peller (Michigan State University)
Abstract:
I am going to speak about joint results with V. I. Vasyunin. We consider the problem of finding the very best (superoptimal) approximation of a given rational matrix functions by matrix functions analytic in the unit disk. The superoptimal approximant must also be rational.

It is a very important problem in control theory to estimate its degree. We have solved a problem that remained open for over 15 years.

We have obtained definitive results in the case of matrix functions of size 2 x 2.

Title: Strictly singular and compact operators.
Speaker: Prof. Evgeny M. Semenov (Voronezh State University)
Abstract: (PDF)

Title: Regularity for de Rham complexes on Lipschitz domains.
Speaker: Dr. Robert Taggart (ANU)
Abstract:
In a recent preprint, Martin Costabel and Alan McIntosh proved regularity results for the exterior derivative d, where d acts on l-forms with distributional coefficients restricted to (or with support contained in) bounded Lipschitz domains.

More recently, similar results have been proved, in joint work with McIntosh and the speaker, for unbounded Lipschitz domains.

This talk will review some of these results.

Title: The geometric dimension of an equivalence relation and finite extensions of countable groups.
Speaker: Prof. Valentin Ya. Golodets (UNSW)
Abstract:
We say that the geometric dimension of a countable group G is equal to a natural number n, if any free Borel action of G on a standard Borel space (X,m), preserving a probability measure m, induces an equivalence relation of geometric dimension n in the sense of Gaboriau.

Let G be as above and geom-dim(G)=n, and let K be a finite extension. Does geom-dim(K)=n?

We prove that, for any natural number n, there exists a big enough class of groups A_n, such that, if G belongs to A_n, then geom-dim(G)=n and, if K is a finite extension of G then K also belongs to A_n.

The important case n=1 is considered more explicitly. We prove that A₁ contains a big class of free products of amenable groups. In particular, all free groups and all finite free products of finite groups belong to A₁.

We use some constructions and results from combinatorial group theory, belonging to A. Karras, H. Neumann, John Stallings and others in combination with methods of orbit equivalence theory.

This is joint work with Anthony Dooley.

Title: Derivations in algebras of operator-valued functions.
Speaker: Prof. Fyodor A. Sukochev (UNSW)
Abstract:
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Title: Strength of convergence in the orbit space of a transformation group.
Speaker: Dr. Astrid an Huef (UNSW)
Abstract:
An action of a locally compact group G on a space X is proper if, thinking of the action as time evolution, a big push of time moves points far away from their original positions. There are two ways to quantify the extent of non-properness of an action: measure-theoretic accumulation and topological strength of convergence in the orbit space X/G. These two notions are linked via the representation theory of an associated C*-algebra. I will explain all of this using examples.

This is joint work with Robert Archbold from the University of Aberdeen.

Title: On derivation of Euler-Lagrange equations for incompressible energy-minimizers.
Speaker: Dr. Nirmalendu Chaudhuri (Wollongong)
Abstract:
In this talk we will discuss the local integrability of distributions q satisfying the system of equations Dq=div f for a given matrix field f=(fⁱ_j), where fⁱ_j are in the local Hardy space h1. As a consequence, we will discuss the existence and the local representation of the hydrostatic pressure and the derivation of Euler-Lagrange equations associated with incompressible, elastic energy-minimizing vector fields in R^n; partially resolving a long standing problem.

This is joint work with Aram Karakhanyan.

Title: A new class of fully nonlinear curvature flows.
Speaker: Dr. James McCoy (Wollongong)
Abstract:
This talk is about contraction by fully nonlinear curvature flows of convex hypersurfaces. As with previously considered flows, including the quasilinear mean curvature flow and fully nonlinear Gauss curvature flow, solutions exist for a finite time and contract to a point.

As with some other flows, including the mean curvature flow, under a suitable rescaling the solutions converge exponentially to spheres.

The main points of interest in this work are the allowance of nonsmooth initial data and that the only second derivative requirement on the speed is weaker than a requirement of convexity. We obtain new results in both cases of smooth and nonsmooth initial data.

This is joint work with Ben Andrews and Zheng Yu

Title: Retractions and projections for Chebyshev subsets of function and sequence spaces.
Speaker: Dr. Sergey Ajiev (UNSW)
Abstract:
Along with Lebesgue and sequence spaces with mixed norms, anisotropic Besov, Lebesgue, Lizorkin-Triebel and Sobolev spaces of differentiable functions defined on a domain and endowed with various norms are considered. We estimate the constants and determine the exponents for the local Hölder regularity of the Chebyshev centres, metric projections and some retractions for the closed convex subsets of these spaces. Attention is paid to the sharpness of some results.

Title: An application of nonabelian duality to higher-rank graph coverings.
Speaker: Prof. John Quigg (Arizona State University)
Abstract:
Recently, Pask, Raeburn, Rordam, and Sims have shown how to present AT-algebras (a broad class of well-known C*-algebras) using graphs of rank 2. The construction involves an infinite tower of coverings of graphs. This tower gives rise to an inverse system of finite groups, and I'll indicate how we've been able to show that the AT-algebra is a crossed product by a coaction (the dual of an action) of the inverse-limit pro-finite group.

This is joint work with David Pask and Aidan Sims.

Title: On the Essential Spectrum of an Operator Arising in Magnetohydrodynamics.
Speaker: Prof. Melvin Faierman (UNSW)
Abstract:
We consider a problem introduced by Descloux and Geymonat in 2-dimensional magnetohydrodynamics wherein all coefficients involved depend only upon one of the space variables. Because of this, we show how it is possible to completely characterize the essential spectrum of the induced Hilbert space operator by reducing the problem to one studied by Gohberg and Krein concerning systems of integral equations.

Title: A transformation of almost periodic pseudodifferential operators to Fourier multiplier operators on vector-valued functions.
Speaker: Dr. Patrik Wahlberg (Newcastle)
Abstract:
PDF

Title: Rigidity of Carnot groups.
Speaker: Dr. Alessandro Ottazzi (Università di Genova)
Abstract:
We are interested in contact mappings on nilpotent stratified Lie groups G (Carnot groups). If the group of contact mappings is infinite dimensional, we say that G is nonrigid, whereas we say that G is rigid otherwise.

We give a condition on the Lie algebra of G that implies nonrigidity. This condition allows us to construct new examples of nonrigid Carnot groups.

Title: Salem and the Rademacher-Menshov Theorem.
Speaker: Dr. Chris Meaney (Macquarie)
Abstract:
Salem's proof of the Rademacher-Menshov Theorem shows that one constant works for all orthogonal expansions in all L²-spaces. By changing the emphasis in Salem's proof we produce a lower bound for sums of vectors coming from bi-orthogonal sets of vectors in a Hilbert space. This inequality is applied to sums of columns of an invertible matrix and to Lebesgue constants.

Title: Classical systems with hyperbolic trapped sets and dispersive estimates for PDE.
Speaker: Dr. Andrew Hassell (ANU)
Abstract:
Consider the time-dependent Schrodinger equation on a complete noncompact Riemannian manifold M (for example, a manifold which looks like flat Euclidean space outside a compact set). This PDE has a dispersive character; that is, the solution cannot concentrate in a small region of space for more than a brief period of time. Various analytic estimates can be proved that give quantitative effect to this vague statement.

The precise form of these estimates depends on the dynamical properties of the associated classical system, namely geodesic flow on M (which is a Hamiltonian dynamical system). The sharpest form of the dispersive estimates are obtained when there is no trapped set, i.e. when all geodesics on the manifold M reach spatial infinity. I will talk about recent work of mine with Burq and Guillarmou, in which under suitable assumptions we can also obtain equally sharp estimates when trapping is present. The most important assumption is that the trapped set is hyperbolic (unstable).

Title: A New Approach to the Orbit Method for Compact Lie Groups II.
Speaker: Dr. Raed Raffoul (UNSW)
Abstract:
We use the Nelson algebra of operants, a construction generalising the symmetric algebra of a vector space which, in the setting of commutative Banach algebras, respects spectral theory in a very special way, to rederive the classical correspondence between unitary irreducible representations of a compact Lie group and orbits of the group on the dual of its Lie algebra.

Title: Generalized embedding theorems for vector-valued Besov and Lizorkin-Triebel spaces.
Speaker: Dr. Sergey Ajiev (UNSW)
Abstract:
The boundedness properties of the generalized Sobolev derivatives as operators in the anisotropic classes of Besov and Lizorkin-Triebel spaces of vector-valued functions with the mixed Lebesgue norm are discussed.

Paying special attention to the case of Besov spaces, we recover the vector-valued forms of the classical results in a numerically friendly manner relying on the characterizations of Besov-Nikol'skiy type considered earlier.

Title: Quantum random walks and their boundaries.
Speaker: Prof. Sergey Neshveyev (University of Oslo)
Abstract:
The spectrum of the center of an algebra can sometimes be interpreted as a boundary of a random walk, which is convenient for computations. It turns out that the algebra itself can often be considered as a noncommutative boundary. The theory was initiated by Biane in the early 90s, who showed that certain results on random walks on groups can be generalized to duals of compact Lie groups. Genuinely noncommutative phenomena arise from quantum groups and their actions. I will present main definitions and some examples.

Title: Approximation of pseudo-differential equations on the sphere using collocation.
Speaker: Dr. Quôc Thông Lê Gia (UNSW)
Abstract:
Approximation of pseudo-differential equations on the sphere using collocation Pseudo-differential equations on the unit sphere play an important role in geo-sciences, oceanography, and meteorology. Satellites provide global data coverage, and yield huge amounts of geophysical data, therefore numerical methods that allow fast processing of scattered data are of great interest.

In this work, we construct an approximation to the solution of a pseudo-differential equation on the unit sphere of the form Lu = f by collocation. Error estimates between the exact solution and the approximation in Sobolev norms are proved.

Title: Approximation properties and X-bases of vector-valued Besov and Lizorkin-Triebel spaces.
Speaker: Dr. Sergey Ajiev (UNSW)
Abstract:
Traditional approximation properties of anisotropic Besov and Lizorkin-Triebel spaces of vector-valued functions defined on an Euclidean space are studied.

We construct certain wavelet X-bases focusing on the existence of the orthogonal bases for the case of Besov spaces and establish some sharp generalizations of the Besov-Nikol'skiy type of the Jackson theorem.

Title: Unimodular Polynomials: Many Problems, Some Solutions.
Speaker: Prof. James Byrnes (Prometheus Inc.)
Abstract:
A question which naturally arises in both pure and applied mathematical analysis is: How close to constant can the modulus of a polynomial be on the unit circle if the coefficients of the polynomial all have the same modulus? While this question was indirectly considered by Gauss, the formal study of such polynomials was initiated by Hardy and furthered by Littlewood, Erdos and many others.

Several of my previous talks have focused on the applied aspects of this question, particularly applications to the design of antenna arrays. Here I concentrate on the purely mathematical side of this coin, giving some historical highlights, discussing some hard-won partial solutions, and pointing out many open problems.

Title: Beale-Kato-Majda type condition for Burgers equation.
Speaker: Dr. Mikhail Neklyudov (UNSW)
Abstract:
In this talk we consider Burgers equation in the torus and the whole space and show that there exists unique global solution if Beale-Kato-Majda type condition is satisfied. In particular, if initial condition and force has gradient form we get global existence and uniqueness of solution and establish new a priori estimate.

The talk is based on the joint work with Ben Goldys.

Title: Maximal theorems for contraction semigroups in vector-valued Lebesgue spaces.
Speaker: Dr. Robert Taggart (UNSW)
Abstract:
In this talk, we consider the extension of some classical theorems for contraction semigroups to the vector-valued L^p spaces. In particular, we generalise the maximal ergodic theorem of Hopf-Dunford-Schwartz and a maximal theorem for symmetric diffusion semigroups due to Stein-Cowling.

The tools used include subpositivity, three different functional calculi for generators of such semigroups and some deep results from harmonic analysis in the setting of UMD spaces.

As an application, it is shown how these new generalisations imply the pointwise convergence of solutions to certain evolution equations.

Title: Differentiability of operator functions.
Speaker: Prof. Vladimir Peller (Michigan State University)
Abstract:
I am going to consider the problem of differentiability (the existence of higher derivatives) of the map A to f(A), where A is an operator and f is a function.

I will deal with the cases of self-adjoint operators, unitary operators, and contractions on a Hilbert space. The main tool is the theory of double (and multiple) operator integrals.

Title: A New Approach to the Orbit Method for Compact Lie Groups.
Speaker: Dr. Raed Raffoul (UNSW)
Abstract:
We use the Nelson algebra of operants, a construction generalising the symmetric algebra of a vector space which, in the setting of commutative Banach algebras, respects spectral theory in a very special way, to rederive the classical correspondence between unitary irreducible representations of a compact Lie group and orbits of the group on the dual of its Lie algebra.

Title: Certain X-bases and analogs of Jackson theorem.
Speaker: Dr. Sergey Ajiev (UNSW)
Abstract:
For a Banach space X, the properties of several types of X-bases for Bochner-Lebesgue spaces are considered. Special attention is paid to the expansions of functions from vector-valued Besov and Lizorkin-Triebel spaces.

We compare the qualitative approach involving Franklin system with the quantitative one relying on certain direct approximation theorems and discuss some classical methods. Banach space X is not necessarily a UMD space.

Title: Analysis of Entropy Zero Systems.
Speaker: Prof. Kyewon Koh Park (Ajou University)
Abstract:
Motivated by the study of general group actions, we would like to investigate the complexity or randomness of entropy zero systems. Many entropy zero systems of general group actions have interesting chaotic behavior for subgroup actions.

We introduce several notions which we expect to be useful for the study of complexity of dynamical systems.

Title: Furstenberg's conjecture: New spectral approach once more.
Speaker: Dr. Oleg Ageev (UNSW)
Abstract:
Sufficiently recently Furstenberg's conjecture on 2-3 shift invariant measures was rewritten by the speaker in terms of the spectral invariants of triples of unitary operators/dynamical systems. It has revealed a bunch of closely related interesting questions which have been out of any attention of experts in dynamical systems.

I do intend to deliver most of them in full if the time permits.

Title: Energy concentration problem and its connection to wavelet theory.
Speaker: Prof. Xiaoping Shen (Ohio and UNSW, CSE)
Abstract:
It is well known that a non-trivial function cannot be compactly supported in time and frequency domains simultaneously. However, among
all possible band-limited functions with a given bandwidth, one can ask which function maximizes the fraction of energy over the prescribed time interval.

Prolate spheroidal wave functions (PSWFs) are special functions that lead to the optimal solutions of this concentration problem. This fact was unraveled by Slepian and his collaborators at Bell Lab in 1960s.

After a brief review, we discuss methods used to construct multiscale systems based on PSWFs. These systems enjoy multiscale structure similar to wavelets and preserve the high energy concentration property inherited from PSWFs. Approximation properties are proved theoretically and illustrated by numerical examples.

Title: Spectral multipliers for Laplacians associated to some Dirichlet forms.
Speaker: Doctor Andrea Carbonaro (Università di Genova)
Abstract:
It has been conjectured that all the generators of symmetric diffusion semigroups have a bounded holomorphic functional calculus in L_p in the sector of angle arcsin|2/p-1|, 1≤p&\leq infinity.

We shall show that generators associated to some weighted Dirichlet forms on R^d admit a bounded holomorphic L_p functional calculus in "pencil--like regions" of the complex plane which are strictly contained in the sector of angle arcsin|2/p-1|.

We consider weights that grow or decay at infinity exponentially. In particular the weighted measures are not doubling.

This is a joint work with G. Mauceri and S. Meda.

Title: Proper group actions in complex geometry.
Speaker: Prof. Alexander Isaev (ANU)
Abstract:
In their celebrated paper of 1939 Myers and Steenrod showed that the group of isometries of a Riemannian manifold acts properly on the manifold. This fact has many important consequences. In particular, it implies that the group of isometries is a Lie group in the compact-open topology. This result triggered extensive studies of closed subgroups of the isometry groups of Riemannian manifolds. The peak of activities in this area occurred in the 1950's-70's, with many outstanding mathematicians involved: Kobayashi, Nagano, Yano, H.-C. Wang, Egorov, to name a few. In particular, Riemannian manifolds whose isometry groups possess subgroups of sufficiently high dimensions were explicitly determined.

I will speak about proper actions in the complex-geometric setting. In this setting (real) Lie groups act properly by holomorphic transformations on complex manifolds. My general aim is to build a theory parallel to the theory that exists in the Riemannian case. In my lecture I will survey recent classification results for complex manifolds that admit proper actions of high-dimensional groups.

Title: ODE's and Carnot groups.
Speaker: Doctor Dr. Benjamin Warhurst (UNSW)
Abstract:
I will discuss how to construct a Carnot group from certain ODE's.

Title: Infinities and infinitesimals.
Speaker: A./Prof. Norman Wildberger (UNSW)
Abstract:
For several thousand years mathematicians have debated the role of infinities and infinitesimals in mathematics. Todays' analyst believes that one must talk about such things in the language of modern set theory, which relies on `axioms' that are incomprehensible to the uninitiated.

In this lecture, I will show you a concrete, understandable way to think about both concepts, without unnecessary philosophising: an infinity is a growth rate, and an infinitesimal is a decay rate. This allows a concrete non-standard analysis, and I will give some applications to first year calculus.

Title: Domain decomposition methods for interpolation by spherical basis functions on spheres.
Speaker: Dr. Quôc Thông Lê Gia (UNSW)
Abstract:
The interpolation problem on the unit sphere using scattered data (from ground stations or from satellites) have many applications in global models for geodesy and geopotential determination.

In this talk, we will discuss the interpolation problem on the unit sphere using spherical basis functions with illustrated numerical examples using MAGSAT satellite data. Domain decomposition methods are used to improve the speed and stability.

This is joint work with T.Tran and I.H.Sloan.

Title: Some inner product spaces of uncountable dimension and their applications.
Speaker: Dr. Aleksandar Ignjatovic (CSE, UNSW)
Abstract:
We present a family of inner product spaces associated in an unusual way with some families of orthogonal polynomials. These spaces have an uncountable dimension, and in them any two sine waves of different frequencies between zero and one are orthogonal. The scalar product in such spaces is defined through series of differential operators, rather than by an integral. We show that truncations of these series of differential operators define a scalar product in some finitely dimensional spaces spanned with sine waves of frequencies that correspond to the quadrature points of orthogonal polynomials.

Finally, we present some applications in signal processing for envelope and phase recovery. We will run Matlab implementations of these signal processing algorithms to show their interesting and useful features.

This is an extension of my research presented in the paper Local approximations based on differential operators that has just appeared in the Journal of Fourier Analysis and Applications.

Title: Computing with expansions in Gegenbauer polynomials.
Speaker: Mr. Jens Keiner (UNSW)
Abstract:
In this talk, we develop fast algorithms for computations involving finite expansions in Gegenbauer polynomials.

We develop an algorithm which converts an arbitrary linear combination of Gegenbauer polynomials up to degree n into an equivalent representation in a different family of Gegenbauer polynomials with generally O(n log(1/ε)) arithmetic operations where ε is a prescribed accuracy. The special cases where the source or target polynomials are the Chebyshev polynomials of first kind are particularly important. In combination with discrete cosine transforms, we get efficient methods for the evaluation of a given Gegenbauer expansion at prescribed nodes and for the projection of a given function onto a family of Gegenbauer polynomials, respectively.

Title: Certain non-classical properties of function, sequence and other Banach spaces.
Speaker: Dr Sergey Ajiev (UNSW)
Abstract:
Anisotropic Besov, Lebesgue, Lizorkin-Triebel and Sobolev spaces endowed with various norms and Lebesgue and sequence spaces, including those with the mixed norm are considered, sometimes, along with an arbitrary Banach spaces.

Mainly, we introduce and study, in the quantitative manner, certain non-classical forms of chaos of different orders. In particular, one establishes a number of the generalizations of the Khinchin-Kahane inequality and a result due to E. M. Stein. Upper estimates of related constants, as well as the limitation of such tools as the Hausdorff-Young inequality are discussed.

Title: Beale-Kato-Majda type condition for Burgers equation.
Speaker: Dr. Mikhail Neklyudov (UNSW)
Abstract:
We consider Burgers equation in the whole space and show that there exists unique global solution if Beale-Kato-Majda type condition is satisfied. In particular, if initial condition and force has gradient form we get global existence and uniqueness of solution and establish new a priori estimate.

This is joint work with A/Professor Ben Goldys.

Title: .
Speaker: Prof. Douglas Lind (University of Washington)
Abstract:
After a brief historical summary of zeta functions, I will describe the zeta function for a dynamical system. We will compute some examples, and derive the important product formula over periodic orbits. There is an amazing characterization of the zeta function of a mixing shift of finte type due to Kim, Ormes, and Roush, which I will discuss.

Title: .
Speaker: Dr. Alexandre Danilenko (ILTPE, Ukraine)
Abstract:
Concepts of near simplicity and near MSJ are introduced for weakly mixing measure preserving actions of a locally compact groups. They generalize Veech-del Junco-Rudolph notions of simplicity and MSJ. I will explain that the theory of near simple actions is more or less parallel to the theory of simple actions. Via the (C,F)-construction, we produce a near simple quasi-simple transformation which is disjoint from any simple map. This answers questions of Thovenot, Ryzhikov, Lemanczyk, del Junco about quasi-simple maps.

Title: On spectral multiplicities in ergodic theory.
Speaker: Dr. Alexandre Danilenko (ILTPE, Ukraine)
Abstract:
Recently Ageev proved (implicitly, via Baire category arguments) the existence of ergodic transformations with homogeneous spectrum of any given multiplicity. I will present a new short proof of his result. Then I will explain how to construct explicit examples and how to use them to produce transformations with non-trivial spectral multiplicities.

Title: (C,F)-actions in ergodic theory.
Speaker: Dr. Alexandre Danilenko (ILTPE, Ukraine)
Abstract:
This is about a recent progress related to the (C,F)-construction of funny rank-one actions for locally compact groups. I am going to discuss briefly a variety of examples and counterexamples produced via the (C,F)-techniques in every of the following categories: (i) probability preserving actions, (ii) infinite measure preserving actions, (iii) non-singular actions (Krieger's type III).

Title: Local Approximations Based on Orthogonal Differential Operators.
Speaker: Dr. Aleksandar Ignjatovic (CSE, UNSW)
Abstract:
We present some generalizations of the Neumann expansion of analytic functions (as a series of Bessel functions), which we call the chromatic expansions. Like truncations of a Taylor expansion, truncations of a chromatic expansion are local approximations; they converge uniformly for important classes of analytic functions. The coefficients of a chromatic expansion of an analytic function f(t) are of the form K_n[f](0), where K_n are linear differential operators, orthogonal with respect to a suitably defined scalar product. A family of such orthogonal operators K_n can be described using a three-term recurrence formula, akin to the recurrence formulas for families of orthogonal polynomials. We relate the class of analytic functions that can be represented by their chromatic expansions to the asymptotic growth rate of the recursion coefficients involved in such a corresponding recurrence. Unlike the derivatives of high order, the values of K_n[f](t) can be approximated in a numerically robust way using the values of discrete samples of f(t). This could make the chromatic approximations useful in practical applications, such as signal processing.

This talk is a summary of my paper "Local Approximations Based on Orthogonal Differential Operators" forthcoming in the Journal of Fourier Analysis and Applications.

Title: Extrapolation of functional calculus of Dirac operators and applications.
Speaker: Dr. Sergey Ajiev (UNSW)
Abstract:
Several rather general sufficient conditions for the extrapolation of the calculus of generalized Dirac operators from L² to L^p are presented. Using the resolvent approach and showing the irrelevance of the semigroup one, we extrapolate (with natural generalisations) the model considered by Axelsson, Keith and McIntosh in L² in order to generalise the setting of the Kato problem.

As applications, one obtains some embedding theorems, quadratic estimates and Littlewood-Paley-type theorems in terms of this calculus in Lebesgue spaces.

Among the auxiliary results are high order counterparts of the Hilbert identity, new forms of "off-diagonal" estimates, the study of the structure of the model in reflexive Banach spaces (especially, Lebesgue ones) and its interpolation properties, and up-to-date analogs of the Calderón-Zygmund theory. We do not use any stability. In particular, some coercivity conditions for bilinear forms in Banach spaces are used as substitutions for the ellipticity ones.

We also discuss the definitions of functional calculus and make an attempt to show how the algebraic and geometric structures come into and how the localisation problem is fought with.

Title: On Some Characteristics of Uniformity of Distribution and Their Applications.
Speaker: Prof. Igor Shparlinski (Macquarie University)
Abstract:
We consider some relatively new characteristics of uniformity of the distribution of sequences that are not widely known and show their connections to several classical measures like discrepancy and exponential sums.

They are connected to several problems from quite different areas such as choosing parameters of linear iteration processes for solving system of linear equations, choosing knots for polynomial interpolation, estimating the size of Varshamov codes correcting asymmetrical errors in binary channels.

Title: Hardy spaces of differential forms on Riemannian manifolds.
Speaker: Prof. Alan McIntosh (ANU)
Abstract:
Let M be a complete Riemannian manifold. Assuming the doubling condition on the volume of balls, we define Hardy spaces H^p of differential forms on M and give various characterizations of them, including a molecular decomposition. As a consequence, we derive the H^p-boundedness for Riesz transforms on M, generalizing previously known results. Further applications, in particular to functional calculus and Hodge decomposition, are given.

This is joint work with Pascal Auscher and Emmanuel Russ.

Title: Estimating the number of bound states of quantum systems.
Speaker: Dr. Andrew Hassell (Australian National University)
Abstract:
There is a heuristic in physics for estimating the number of bound states of a quantum system (or in mathematical terms, the number of negative eigenvalues of a self-adjoint operator) by regarding the eigenfunctions as disjoint `blobs' of phase space, each occupying a fixed volume. In the talk I will investigate the worth of this heuristic in the case of a simple quantum system, that of the Laplacian in R³ plus a potential function.

We find very precise asymptotics for the number of bound states in some cases, and see that the heuristic is a very good, but not perfect, guide to the actual situation.

This is joint work with Simon Marshall.

Title: .
Speaker: Dr. Patrik Wahlberg ( University of Newcastle)
Abstract: (PDF)

Title: Breaking the curse of dimensionality for integration over the product of many spheres.
Speaker: Prof. Ian Sloan (UNSW)
Abstract: (PDF)
This talk, describing joint work with Kerstin Hesse and Frances Kuo, presents a component-by-component approach to constructing a quasi-Monte Carlo (QMC) integration rule over the d-fold product of unit spheres S² \subset R³.

A recent paper of Kuo and Sloan established necessary and sufficient conditions for strong QMC tractability of the integration problem for the d-fold product of spheres, in a worst-case setting: as in the case of the d-dimensional cube, the necessary and sufficient condition is that the sum of the "weights" γ_j for j = 1,...,d must be bounded independently of d. If that condition holds, then there exists a QMC rule for which the worst-case error is bounded by cm^-1/2, where c>0 is independent of d, and m is the number of points in the QMC integration rule.

In the present work the QMC rule from the component-by-component construction is shown to have the same upper bound, under the same assumption on the weights and some assumptions on the smoothness of the function space and the number of points m.

The construction begins with the selection of a "spherical design" for the QMC integration rule over a single sphere. The algorithm then chooses a permutation of the m points for each sphere in the product, one sphere at a time, at each stage choosing the new permutation to minimise the worst-case error, while keeping all earlier permutations fixed.

Title: Nonequispaced Fast Fourier Transforms on the Sphere.
Speaker: Dr. Jens Keiner (UNSW)
Abstract:
Fast Fourier transforms on the sphere are of general interest for a variety of applications. On the sphere, spherical harmonics play the role of the usual Fourier basis. Unfortunately, this makes fast and stable transforms more challenging to implement. Moreover, in most applications, data sites are distributed
arbitrarily over the surface of the sphere for which a restriction to particular sets of nodes is not acceptable.

The main focus of this talk is to survey Fourier analysis on the sphere, related fast algorithms for Fourier transforms that don't rely on specific node distributions, and the NFFT 3 software library which, among others, implements these algorithms. The talk will also include some applications of these concepts, e.g. to fast summation of radial functions on the sphere or Fourier reconstruction from scattered data.

NFFT 3 is currently the only publicly available software library implementing usual multi-dimensional Fourier transforms, Fourier transforms on the sphere, and a lot more ... for arbitrary nodes.

Title: Generalising Group Algebras.
Speaker: Dr. Hendrik Grundling (UNSW)
Abstract:
We generalise group algebras to other algebraic objects with bounded Hilbert space representation theory - the generalised group algebras are called "host" algebras. The main property of a host algebra, is that its representation theory should be isomorphic (in the sense of the Gelfand-Raikov theorem) to a specified subset of representations of the algebraic object.

The main motivation behind this, comes from the analysis of infinite dimensional Lie groups and other non-locally compact groups (some of which occur in physics).

We will present both existence and uniqueness theorems for host algebras. Abstractly, this solves the question of when a set of Hilbert space representations is isomorphic to the representation theory of a C*-algebra.

In recent work on the topic we analyzed ordinary and multiplier (unitary) representations for non-locally compact Abelian groups. We obtained first the negative result if an Abelian group has a host algebra for its set of ordinary unitary representations, then it has a dense embedding into a locally compact group such that its representation theory factors through the embedding. Second, we obtained the positive result, that host algebras can exist for the multiplier representation theory associated to a fixed 2-cocycle of a non-locally compact Abelian group.

[This talk will be the one which was given at the conference to mark
Rick Loy's retirement at the ANU January 4 - 8, 2007]

Title: On the individual ergodic theorem in the Kozlov--Treshchev form.
Speaker: Prof. Vladimir I. Bogachev (Moscow State University)
Abstract:
PDF file.

Title: Repeat distributions from unequal crossovers.
Speaker: Prof. Michael Baake (Bielefeld)
Abstract:
It is a well-known fact that genetic sequences may contain sections with repeated units, called repeats, that differ in length over a population, with a length distribution of geometric type. A simple class of recombination models with single crossovers is analysed that result in equilibrium distributions of this type.

Due to the nonlinear and infinite-dimensional nature of these models, their analysis requires some nontrivial tools from measure theory and functional analysis, which makes them interesting also from a mathematical point of view. In particular, they can be viewed as quadratic, hence nonlinear, analogues of Markov chains.

Title: Some more dynamical characterizations of amenability and property (T).
Speaker: Dr. Oleg Ageev (UNSW)
Abstract:
Recently we calculated the discrete part of a typical group action of the Kazhdan groups. Now we have the same for every countable group. This implies one more characterization of property (T) in terms of the existence of the non-trivial finite dimensional subrepresentations of both the typical group actions and unitary representations. We will also discuss the equivalence of the weak Rokhlin property and amenability.

Title: Non-Bernoulli actions of amenable groups
Speaker: Prof. Valentin Golodets (UNSW)
Abstract:
(A. Dooley and V.Golodets.) Actions of the group Z with completely positive entropy (CPE for short) were introduced by A.N.Kolmogorov and generalised by Rokhlin and Sinai in 1961. These actions have nice mixing and spectral properties and arise in apllications.

Rudolph and Weiss (2000) suggested a new approach to study CPE actions for any amenable countable group and showed that CPE actions have a very strong mixing.

Dooley and Golodets (2002) proved that these CPE actions have a countable Lebesgue spectrum as in the case of the group Z.

The traditional problem in this field is the existence of a non-Bernoulli action with CPE for any amenable group. Such actions for Z were found by Ornstein-Shields, Feldman, Hoffman and Kalikow. In this talk we describe
a construction which allows to produce a non-Bernoulli CPE action for any countable amenable group which contains an element of the infinite order. We call such actions co-induced. This construction is related to but different from the standard induced action.

Title: Quadrature formulas and localized linear polynomial operators on the sphere
Speaker: Dr. Quôc Thông Lê Gia (UNSW)
Abstract:
We review existence theorems on quadrature formulas that satisfy Marcinkiewicz-Zygmund (M-Z) property on the sphere. Then we describe and compare numerical algorithms for construction of quadrature formulas on the sphere, exact for spherical polynomials of a high degree. Our formulas are based on scattered sites; and we are able to construct formulas exact for spherical polynomials of degree 178. We also demonstrate the use of these formulas in constructing localized,linear, quasi-interpolatory polynomial operators based on scattered sites. The approximation and localization properties of our operators are studied theoretically in deterministic as well as probabilistic settings. Numerical experiments are presented to demonstrate their superiority over traditional least squares and discrete Fourier projection polynomial approximations. This is joint work with H.N.Mhaskar, California State University at Los Angeles.

Title: On the maximal order of the "factorisatio numerorum"
Speaker: Prof. Florian Luca (Instituto de Matemáticas, Universidad Nacional Autónoma de México)
Abstract:
Let m(n) be the number of ordered factorizations of n in factors > 1. We improve on some claims of P. Erdös concerning the maximal order of the numbers m(n). The proofs use standard techniques in analytic number theory such as the prime number theorem, smooth numbers as well as a detailed analysis of the Riemann zeta function around the real zero ρ of the equation ζ(ρ) = 2.

This is joint work with M. Klazar.

Title: Amenability and isoperimetric properties of equivalence relations
Speaker: Prof. Vadim A. Kaimanovich (International University of Bremen)
Abstract:
The talk is devoted to a discussion of the relationship between two notions of amenability for equivalence relations: the global one (equivalent to hyperfiniteness) and the local one (based on leafwise isoperimetric properties). We give a complete answer to this problem, which, in particular, leads to a new transparent proof of the famous Connes-Feldman-Weiss theorem on equivalence of amenability and hyperfiniteness.