Analysis Seminar

Almanac


14/10/09 A/Professor Guo Hua Zhang, Fudan University and University of New South Wales

Lowering topological entropy in topological dynamics (PDF)

            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.    

07/10/09 Doctor Nurulla Azamov, Flinders University

New approach to abstract scattering theory

            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 $W_\pm$ and the scattering operator $\mathbf S,$ and after that one shows existence a.e. of the wave matrices $w_\pm(\lambda)$ and of the scattering matrix $A(\lambda).$ Further, one proves the stationary formula for the scattering matrix for a.e. $\lambda.$ 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.     

12/08/09 Professor Melvin Faierman, University of New South Wales

Estimates for solutions of a parameter elliptic multi-order system of differential equations (PDF)

            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.      

05/08/09 Professor Tuomas  Hytönen, University of Helsinki

Pseudo-localization of singular integrals in LP

            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 Lp 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.


29/07/09 Professor Dorina I. Mitrea, University of Missouri, Columbia

On the regularity of the Dirichlet Green potential in convex domains

            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.

22/07/09 Doctor Andrea Carbonaro, University of Genova

Comparison of spaces of Hardy type

            In the first part of the talk I will speak about a Hardy space theory for locally doubling metric measure spaces that I developed in two joint papers with G. Mauceri and Stefano Meda.
            In the second part of the talk I will compare our Hardy spaces with the local Hardy spaces of Goldberg type recently defined by M. Taylor in the setting of Riemannian manifolds with bounded geometry.


24/06/09 Andrew Morris, Australian National University

Local Hardy Spaces of Differential Forms (PDF)


06/05/09 Doctor Sergey Ajiev, UNSW

On concentration, deviation and Dvoretzky's theorem for function and other spaces

           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. 

22/04/09 Dmitriy Zanin, Flinders University

Orbits in fully symmetric spaces

           Semiroup 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.

08/04/09 Doctor Steven Lord, University of Adelaide

The Riemann Zeta Function, The Laplacian, and How to Recover Lebesgue Integration from the Pole of a Zeta Function (Part 2)

           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".

01/04/09 Professor Shijun Zheng, Georgia Southern University

Spectral Multipliers for Schroedinger Operators

           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.

04/03/09 Professor Vladimir Peller, Michigan State University

Analytic approximation of rational matrix functions

           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

25/02/09 Professor Evgeny M. Semenov, Voronezh State University

Strictly singular and compact operators (PDF)

          

26/11/08 Doctor Robert Taggart, Australian National University

Regularity for de Rham complexes on Lipschitz domains

           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.


12/11/08 Professor Valentin Ya. Golodets, UNSW

The geometric dimension of an equivalence relation and finite extensions of countable groups

           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 An, such that, if G belongs to An,  then geom-dim(G)=n and, if K is a finite extension of G then K also belongs to An.
           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.


15/10/08 Professor Fyodor A. Sukochev, UNSW

Derivations in algebras of operator-valued functions (PDF)

            It is well known that any derivation acting on a von Neumann algebra is inner. In particular, there are no nontrivial derivations on a commutative von Neumann algebra M= L1(0; 1).
            Consider an arbitrary semifinite von Neumann algebra M and the algebra S(M) of all measurable operators affiliated with M (the algebra S(M) was first introduced by I.E. Segal in 1953 and is a cornerstone of noncommutative integration theory). Recently, there have appeared a number of publications treating the question whether every derivation on S(M) is inner.
            In 2006, it was established by A. Ber, V. Chilin and F.S. that if M = L1(0; 1), then the algebra S(M) (which in this case may be identified with the algebra S(0;1) of all Lebesgue measurable functions on (0; 1)) has non-trivial derivations, in particular, the classical derivations d=dt extends to S(0; 1).
            Our main objective in this talk is to present a recent result by A. Ber (Tashkent), B. de Pagter (Delft) and F.S. that if M = L1(0; 1)¹xB(H) (that is M is a von Neumann tensor product of the algebra L1(0; 1) and the algebra B(H) of all bounded linear operators on an infinite dimensional separable Hilbert space), then every
derivation on S(M) is still inner.


08/10/08 Doctor Astrid an Huef, UNSW

Strength of convergence in the orbit space of a transformation group

            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.


01/10/08 Doctor Nirmalendu Chaudhuri, University of Wollongong

On derivation of Euler-Lagrange equations for incompressible energy-minimizers

            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=(f^i_j), where f^i_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.


24/09/08 Doctor James McCoy, University of Wollongong

A new class of fully nonlinear curvature flows

            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.


03/09/08 Doctor Sergey Ajiev, UNSW

Retractions and projections for Chebyshev subsets of function and sequence spaces

             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.

13/08/08 Professor John Quigg, Arizona State University

An application of nonabelian duality to higher-rank graph coverings

             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.


06/08/08 Professor Melvin Faierman, UNSW

On the Essential Spectrum of an Operator Arising in Magnetohydrodynamics (PDF)

             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.

30/07/08 Doctor Patrik Wahlberg, University of Newcastle

A transformation of almost periodic pseudodifferential operators to Fourier multiplier operators on vector-valued functions (PDF)

             We treat pseudodifferential operators on $\mathbf R^d$ in the Kohn--Nirenberg quantization, where the symbol $a(\cdot,\xi)$ is almost periodic (a.p.) for each $\xi \in \mathbf R^d$, and belongs to a H{\"o}rmander class $S_{\rho,\delta}^m$. We study the symbol transformation $a \mapsto U(a)$
$$
U(a)(\xi)_{\lambda,\lambda'} = M_x ( a(x,\xi-\lambda') e^{- 2 \pi i x (\lambda'-\lambda)} )
$$
where $M_x$ denotes the mean value for a.p. functions, which was introduced, for operator kernels rather than symbols, by E. Gladyshev. $U(a)(\xi)$ can be considered a matrix indexed by $(\lambda,\lambda') \in \Lambda \times \Lambda$ where $\Lambda$ is the set of frequencies that occur in $\{ a(\cdot,\xi) \}_{\xi \in \mathbf R^d}$. Thus $U(a)$ may be considered the operator-valued symbol of a Fourier multiplier operator that acts on vector-valued functions.

             Using results by M. A. Shubin, we show that the transformation respects operator composition, $U(a \#_0 b) (\xi) = U(a)(\xi) \cdot U(b)(\xi)$, where
$a(x,D) \circ b(x,D) = (a \#_0 b)(x,D)$. Moreover, $a(x,D) \geq 0$ if and only if $U(a)(D) \geq 0$. Positivity and boundedness on Besicovitch-Sobolev spaces of $a(x,D)$ are encoded in the matrix $U(a)(0)$.


25/06/08 Doctor Alessandro Ottazzi, Università di Genova

Rigidity of Carnot groups

             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.


11/06/08 Doctor Chris Meaney, Macquarie University

Salem and the Rademacher-Menshov Theorem

             Salem's proof of the Rademacher-Menshov Theorem shows that one constant works for all orthogonal expansions in all L^2-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.

04/06/08 Doctor Andrew Hassell, Australian National University

Classical systems with hyperbolic trapped sets and dispersive  estimates for PDE

             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).


28/05/08 Doctor Raed Raffoul, UNSW

A New Approach to the Orbit Method for Compact Lie Groups II

             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.

21/05/08 Doctor Sergey Ajiev, UNSW

Generalized embedding theorems for vector-valued Besov and Lizorkin-Triebel spaces

             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.


30/04/08 Professor Sergey Neshveyev, University of Oslo

Quantum random walks and their boundaries

             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.

23/04/08 Doctor Quôc Thông Lê Gia, UNSW

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.


09/04/08 Doctor Sergey Ajiev, UNSW

Approximation properties and X-bases of vector-valued
Besov and Lizorkin-Triebel spaces

             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.


28/03/08 Professor James Byrnes, Prometheus Inc.

Unimodular Polynomials: Many Problems, Some Solutions

             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.



26/03/08 Doctor Mikhail Neklyudov, UNSW

Beale-Kato-Majda type condition for Burgers equation II

             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 joint work with Ben Goldys.


19/03/08 Doctor Robert Taggart, UNSW

Maximal theorems for contraction semigroups in vector-valued Lebesgue spaces

             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.


12/03/08 Professor Vladimir Peller, Michigan State University

Differentiability of operator functions

             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.


05/03/08 Doctor Raed Raffoul, UNSW

A New Approach to the Orbit Method for Compact Lie Groups

             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.

13/02/08 Doctor Sergey Ajiev, UNSW

Certain X-bases and analogs of Jackson theorem

             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.


30/01/08 Professor Kyewon Koh Park,  Ajou University

Analysis of Entropy Zero Systems

            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. 


12/12/07 Doctor Oleg Ageev,  UNSW

Furstenberg's conjecture: New spectral approach once more

            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.


05/12/07 Professor Xiaoping Shen,  Ohio University and CSE, UNSW

Energy concentration problem and its connection to wavelet theory

           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. 

28/11/07 Doctor  Andrea Carbonaro,  Università di Genova

Spectral multipliers for Laplacians associated to some Dirichlet forms

           It has been conjectured that all the generators of symmetric diffusion semigroups have a bounded holomorphic functional calculus in Lp in the sector of angle arcsin|2/p-1|,  1<p<\infty.
           We shall show that generators associated to some weighted Dirichlet forms on R^d admit a bounded holomorphic Lp 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.

07/11/07 Professor Alexander Isaev,  Australian National University

Proper group actions in complex geometry (PDF)

           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.

31/10/07 Doctor Benjamin Warhurst,  UNSW

ODE's and Carnot groups

           I will discuss how to construct a Carnot group from certain ODE's.

24/10/07 Professor Norman Wildberger,  UNSW

Infinities and infinitesimals

           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.


10/10/07 Doctor Quôc Thông Lê Gia, UNSW 

Domain decomposition methods for interpolation by spherical basis functions on spheres

           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.

19/09/07 Doctor Aleksandar Ignjatovic, CSE, UNSW 

Some inner product spaces of uncountable dimension and their applications

           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", http://www.springerlink.com/content/d361x28401571112/fulltext.pdf
.

12/09/07 Professor Sergey I. Piskarev, Lomonosov Moscow State University (Canceled due to unforeseen circumstances)

A general approximation scheme for attractors of abstract parabolic problems (PDF)

           In this talk we consider the semilinear problems of the form u'=Au+f(u), where A generates an exponentially decaying compact analytic semigroup in a Banach space E and f is globally Lipschitz and bounded map from E^\alpha into E (E^\alpha=D((-A)^\alpha) with the graph norm). These assumptions
ensure that the problem has a global attractor. Under a very general approximation scheme we prove that the dynamics of such problem behaves upper semicontinuously.
           We also show that, if all equilibrium solutions of this problem are hyperbolic, then there is an odd number of such equilibrium solutions. Additionally, if we also assume that every global solution converges as t tends to plus or minus infinity, (e.g. gradient semigroups with isolated equilibria), then we prove, under this approximation scheme, that the attractors also behave lower semicontinuously.
           This general approximation scheme includes the finite element method, projection and finite difference methods.


05/09/07 Mister Jens Keiner, UNSW

Computing with expansions in Gegenbauer polynomials

           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/eps))$ arithmetic operations where $eps$ 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.


29/08/07 Doctor Sergey Ajiev, UNSW

Certain non-classical properties of function, sequence and other Banach spaces

           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.


22/08/07 Doctor Mikhail Neklyudov, UNSW

Beale-Kato-Majda type condition for Burgers equation

           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. 


15/08/07 Professor Anthony Weston, UNSW and Canisius College, Buffalo NY, USA

Determining lower bounds on the maximal p-negative
type of finite metric trees (PDF,PS)


08/08/07 Professor Melvin Faierman, UNSW

The Calderon approach to an Elliptic Boundary problem (PS)


22/06/07 Professor Douglas Lind, University of Washington, WA, USA

Dynamical Zeta Functions

           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.



22/06/07 Doctor Hab. Alexandre Danilenko, ILTPE, Ukraine

Simplicity concepts for ergodic actions

          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.

13/06/07 Doctor Hab. Alexandre Danilenko, ILTPE, Ukraine

On spectral multiplicities in ergodic theory

           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.

06/06/07 Doctor Hab. Alexandre Danilenko, ILTPE, Ukraine

(C,F)-actions in ergodic theory

           This is about the 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 $III$).

30/05/07 Doctor Aleksandar Ignjatovic, (CSE) UNSW

Local Approximations Based on Orthogonal Differential Operators

           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; the preprint is available at
http://www.cse.unsw.edu.au/~ignjat/diff/LocalApprox.pdf.

23/05/07 Doctor Sergey Ajiev, UNSW

Extrapolation of functional calculus of Dirac operators and applications

           Several rather general sufficient conditions for the extrapolation of the calculus of generalized Dirac operators from L2 to Lp 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.


16/05/07 Professor Igor Shparlinski, Macquarie University

On Some Characteristics of Uniformity of Distribution and Their Applications

           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. 

09/05/07 Professor Alan McIntosh, Australian National University

Hardy spaces of differential forms on Riemannian manifolds

            Let M be a complete Riemannian manifold. Assuming the doubling  condition on the volume of balls, we define Hardy spaces Hp of  differential forms on M and give various characterizations of them,  including a molecular decomposition. As a consequence, we derive the  Hp-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.

02/05/07 Doctor Andrew Hassell, Australian National University

Estimating the number of bound states of quantum systems

            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.

18/04/07 Doctor Patrik Wahlberg, University of Newcastle

Weyl product algebras and modulation spaces (PDF)

            J. Sjoestrand introduced 1994 a new symbol class for pseudo-differential calculus with no explicit reference to derivatives. It was later realized that this symbol class is the modulation space $M^{\infty,1}$. The modulation spaces, invented by Feichtinger in 1983, are Banach spaces that quantify the asymptotic decay of tempered distributions in the phase space.
            In this talk we discuss algebraic properties of the Weyl product acting on modulation spaces. For a certain class of weight functions $\omega$ we prove that the weighted modulation space $M_{(\omega)}^{p,q}$ is an algebra under the Weyl product if $p \in[1,\infty]$ and $1\leq q \leq\min(p,p')$. For the remaining cases
$p\in [1,\infty]$ and $\min(p,p')<q\leq \infty$ we show that the unweighted spaces $M^{p,q}$ are not algebras under the Weyl product.
            The talk will contain a background and a description of our results.
            This is a joint work with A. Holst, Lund, and J. Toft, Vaexjoe, Sweden.

04/04/07 Sci. Professor  Ian Sloan, FAA, UNSW

Breaking the curse of dimensionality for integration over the product of many spheres (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 $S^2 \subset \mathbb{R}^3$.
            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" \gamma_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.


28/03/07 Mister  Jens Keiner, UNSW

Nonequispaced Fast Fourier Transforms on the Sphere

            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.


21/03/07 Doctor Hendrik Grundling, UNSW

Generalising Group Algebras

           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]


14/03/07 Professor Vladimir I. Bogachev, Moscow State University

On the individual ergodic theorem in the Kozlov--Treshchev form (PDF)

       The talk concerns the recent observation due to Kozlov and Treshchev  that in the situation of the classic individual ergodic theorem for an ergodic semiflow  $(T_t)_{t\ge 0}$ on a probability space $(\Omega,P)$, for any bounded measurable function $f$, the  averages 
$$
\int_0^{\infty} f(T_{ts}x) \nu(ds)
$$
converge almost surely to the expectation of $f$ as $t\to +\infty$, whenever $\nu$ is an absolutely continuous probability measure on $[0,+\infty)$. This result can be formulated as certain convergence of the images of $\nu$ under the mappings $s\mapsto T_{ts}x$ from $[0,+\infty$ to $\Omega$, i.e., natural measures on the trajectories of the dynamical system. Several interesting questions arise concerning the mode of convergence and various restrictions on $f$ and $\nu$.  
        The discussion will involve some basic concepts from real analysis and probability (Fourier transform, weak convergence, $L^p$-spaces), no special background is required, in particular, no acquaintance with ergodic theory is assumed. 

07/03/07 Professor Michael Baake, University of Bielefeld

Repeat distributions from unequal crossovers

       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.

28/02/07 Doctor Oleg Ageev, UNSW

Some more  dynamical characterizations of amenability and property (T)

       Recently we have 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 the typical unitary representations. We will also discuss the equivalence of the weak* Rokhlin property (W*RP for short) and amenability.

21/02/07 Professor Valentin Golodets, UNSW

Non-Bernoulli actions of amenable groups

       (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.

07/02/07  Doctor Quôc Thông Lê Gia, UNSW

Quadrature formulas and localized linear polynomial operators on the sphere

               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.

24/01/07  Professor Florian Luca, Instituto de Matemáticas, Universidad Nacional Autónoma de México

On the maximal order of the "factorisatio numerorum" (PDF)

               Let m(n) be the number of ordered factorizations of n in factors >1. We improve on some claims of P. Erdős 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 \rho of the equation \zeta(\rho)=2.
              This is joint work with M. Klazar.


10/01/07  Professor Vadim A. Kaimanovich, International University of Bremen

Amenability and isoperimetric properties of equivalence relations

              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. 

06/12/06  Professor Gregory Hjorth, University of Melbourne

The Poisson boundary

           I will discuss Kaimonivich's work on the Poisson boundary of a group. As time allows, we can lead in to mixing properties and an application to the theory of Borel equivalence relations.

29/11/06  Doctor  Bartek Trojan, University of Sydney

Hua-harmonic functions and asymptotic expansions

           Let D be a homogeneous Siegel domain of type II. We prove that every bounded Hua-harmonic function f on D is pluriharmonic. The proof is based on asymptotic expansion of f.

22/11/06  Doctor  Quôc Thông Lê Gia, UNSW

Local approximation on the sphere using shifts of a positive definite kernel

           In this talk, we consider the local interpolation problem on the sphere S^n \subset R^{n+1} using scattered data inside a spherical cap of small angular radius. The interpolant are constructed using shifts of a positive definite kernel on the sphere. Error estimates in terms of Sobolev norms H^s(S^n) of the target function and the local mesh norm of the scattered data sets will be discussed. 
           This is joint work with Kerstin Hesse and Ian Sloan.

15/11/06  Doctor  Subramaniam Murugesh, UNSW

Integrable reductions in the kinematics of 2+1 dimensional composite fibres: Constant Divergence

           The idea of enhancing the strength and mechanical prop erties of materials by reinforcing them with inextensible fibres has been in practice for the
past few decades. This is of engineering applications in the construction of light composite bodies. The system consists of a matrix fluid material reinforced by elastic fibres. Under plane deformations the kinematic conditions are of key importance.
           We show that for the non-steady fluid, under a condition of constant divergence, the system is reducible to equations in the integrable mKdV hierarchy. In particular, we establish that the fibres, which are convected with the fluid, constitute generalised tractrices for base curves whose curvatures are given by solutions of equations in the mKdV hierarchy. We show explicitly the fibre foliations for certain special solutions of the mKdV equation.


08/11/06  Professor Dr.  Zoltan Balogh, Universitaet Bern and UNSW                    

Hausdorff measures, characteristic sets and iterated function systems in the Heisenberg group

           The problem of comparison between Hausdorff measures and dimension in terms of the Euclidean and the Carnot-Caratheodory  metric will be considered in the simplest setting of the first Heisenberg group.
           The solution of this problem leads naturally to the notion of horizontal fractals defined as invariant sets of iterated function systems on the Heisenberg group.

18/10/06  Doctor  Sergey Ajiev, UNSW                     

Quantitative perturbed  Lyapunov type theorems for function and sequence spaces

           Established more than 60 years ago,  the classical Lyapunov convexity theorem has found many interesting applications and generalizations.  For instance, it permits the admission of the measurable control functions for the wide class of Lyapunov and equivalent optimisation problems.
           We estimate from the above and below the Lyapunov constants describing the influence of the atomic components of the vector measures with the values in Besov, Lebesgue, Lizorkin-Triebel, Sobolev and sequence spaces.

11/10/06  Doctor  Oleg Ageev, UNSW                     

Spectral invariants in modern ergodic theory

           This talk will mainly be based on the materials of my invited lecture at the International Congress of Mathematicians, Madrid, Spain, 2006.

20/09/06 Doctor Dmitry Demskoi, UNSW 

On the Liouville type equations and their applications to soliton theory


           The Liouville-type equations were first studied in classical works of Liouville, Darboux, Goursat, Vessiot, and etc. The modern theory is based on two key ingredients: pseudoconstants and Laplace invariants. The presentation concerns application of pseudoconstants to constructing higher symmetries and Backlund-type transformations of equations integrable by inverse scattering transform. Several new examples of such transformations are presented. In particular we give Backlund transformations for coupled three-component KdV-type systems. The problems arising in classification of Liouville-type systems will also be discussed.


13/09/06 Doctor Hisatoshi Yuasa, UNSW (Visiting Fellow)

Invariant measures for the subshifts arising from non-primitive substitutions

           A map from a finite alphabet to the set of nonempty words over the alphabet is called a substitution. Since W. Gottschalk's work in 1963, the ergodic and
topological properties of the subshift arising from a primitive substitution has been extensively studied by many authors.
          In particular, it is well-known that the subshift is minimal and uniquely ergodic. In this talk, a new class of non-primitive substitutions is introduced. Those dynamical properties of the associated subshift are also discussed which are the recurrence property called almost minimality and a uniqueness in a certain sense of sigma-finite (not necessarily finite) invariant measures.
          A key to the uniqueness of invariant measures is to represent the subshift as an adic transformation, or a Bratteli-Vershik system.


06/09/06 Professor Valentin Golodets, UNSW

An equivalence relation, a finite index subrelation and their costs (PDF)

           The cost of an equivalence relation is an important numerical invariant in ergodic theory introduced recently by H.Levitt. It allowed to discover some new properties of Borel actions of non-amenable countable groups preserving a finite measure. A theory of the costs were developed recently by Gaboriau, Hjorth, Kechris, Miller and other authors.
          In this talk we discuss the following problem. Let $E \subset F$ be aperiodic countable Borel equivalence relations on a standard Borel space $(X,\mu)$ where
$\mu$ is a finite $F$-invariant measure.  We prove that if $[F:E]= n < \infty$ then
\[ \phantom {XXXXXXXXXX} C_\mu(E) - \mu(X) = n(C_\mu(F) - \mu(X)),\phantom{XXXXXXXXXXX} (*) \]
where $C_\mu(E)$ and $C_\mu(F)$ are costs of $E$ and $F$ respectively.
          To prove this equality we develop some results of the cost theory. We also use index cocycles of Feldman, Sutherland and Zimmer.
          This result has some applications. In particulary , we can prove that $F$ is amenable  if and only if $E$ is amenable.
          This research was done in cooperation with Tony Dooley 

30/08/06  Doctor  Sergey Ajiev, UNSW                     

Local and similar geometric constants for function and sequence spaces

            Anisotropic Besov, Lebesgue, Lizorkin-Triebel and Sobolev spaces endowed with various norms  and related Lebesgue and sequence spaces, including those with the mixed norm are considered.
         We either evaluate, or obtain two-sided estimates for various geometric constants of local or alike nature, which either play an important role in the fixed point theory for uniformly Lipschitzian mappings (self-Jung and Lifshits constants), useful in applications (Jung and compact Jung constants) or are of independent
interest (B, G),  or of historical interest, such as the Jourdan-von Neumann constant. Mutual relations between the constants and alternative simple and case-specific
proofs of some known results and their natural complements and corollaries are discussed too.
         The analysis of our models relies on the usage of symmetries in the form of some ideas from the group theory, counterparts of the Jacobi identity and the study of the Birkhoff-Fortet orthogonality.   



23/08/06  Professor  Melvin Faierman, UNSW

An Elliptic Boundary Problem involving Fourier Multipliers

           The classical form of Mikhlin's multiplier theorem requires that the multiplier act as an operator between Hilbert spaces. In this talk we discuss an elliptic boundary problem for which this classical version does not give the required results.
         We also discuss, in relation to the boundary problem, more recent results
on Fourier multipliers. 

16/08/06  Mister Raed Raffoul, UNSW

Functional Calculus and Coadjoint Orbits

          Associated to any unitary representation of a compact Lie group is a commutative Banach algebra of "operants", first constructed by Edward Nelson in 1968  to facilatate a rigorous interpretation of the Feynman operator calculus.
         The joint spectrum of the generators of this operant algebra remarkably characterises irreducible representations, it is the image of  the symplectic geometry moment map arising from the Hamiltonian action of G on the projectivisation of its representation space which, most of the time, equals the convex hull of the coadjoint orbit through the highest weight -  a result due to Norman Wildberger and independantly D. Arnal and J. Ludwig.
         The construction of this algebra, which deserves to be better known, and some of its ramifications will be discussed. 

09/08/06  Doctor Quoc Thong Le Gia, UNSW

Spectral approximation of Navier-Stokes equations on the unit sphere
             using vector spherical harmonics

           The Navier-Stokes equations (NSEs) on the unit sphere plays a major role in weather forecasting models. In this work, we will  discuss the spectral approximation method for NSEs on the global scale using vector spherical harmonics. Assuming the Gevrey regularity of the function that describes the external force in
the NSEs, we derive L2-error estimates between the exact solution and the approximate solution.
           This is joint work with
Ian Sloan and M. Ganesh. 

03/08/06  Professor  Ian H. Sloan, FAA, UNSW

The best of both worlds: radial basis function and polynomial approximation
         on the sphere. Part II

           Many researchers have discussed approximation by radial basis functions on a sphere, using scattered data.  Usually there is no polynomial component   in such approximations if, as assumed here, the kernel that generates the radial functions is (strictly) positive definite.
          On the other hand, the utility of polynomials for approximating slowly
varying components is well known -- an extreme case is the NASA model of the earth's gravitational potential, which represents the potential by a purely polynomial approximation of high degree.
          In this joint work with Alvise Sommariva we propose a hybrid approximation, in which there is a radial basis functions component to handle the rapidly varying  and localised aspects, but also a polynomial component to handle the more slowly varying and global parts.
          In this lecture I discuss the beautiful convergence theory (including a doubled rate of convergence for sufficiently smooth functions), which makes use of the ``native space" associated with the positive definite kernel (with no polynomial involvement in the definition). A numerical experiment for a simple model with a geophysical flavour establishes the potential value of the hybrid approach.  

27/07/06  Professor  Herbert Amann Universitaet Zuerich, Switzerland 

Diffusion methods in image processing

          In this talk we will explain the role of nonlinear heat equation methods  in image processing, concentrating on the well-known and widely used Perona-Malik equation. We will try to show motivations which lead to this equation, explain its advantages and drawbacks and propose a time-regularized version of it which is well-posed in the analytic sense.
          By showing results of numerical experiments we provide evidence for the superiority of this time regularization versus the standard space regularization technique.
          The talk will be directed towards a non-specialized audience.


26/07/06  Professor  Anthony Weston, Canisius College, Buffalo NY, USA                

Group Structure on Banach Spaces

          Group structures on Banach spaces that - for example - generalise  addition  have not been widely studied despite some extraordinary initial work in this  direction by Per Enflo (1970).
         The purpose of this talk will be to dig belowthe surface of Enflo's initial study and isolate a new technique to "linearise" uniform homeomorphisms and - in so doing - recast old results and extract new results in the uniform theory of Banach spaces.
         Papers on this work  may be downloaded from the website http://www.canisius.edu/topos/weston.asp
.

24/07/06  Professor  Mahadevan Ganesh, UNSW and Colorado  School of Mines, USA                

Matrix-free interpolation and scattered data approximations on the sphere

         We give an explicit construction for a system of interpolation nodes and the corresponding interpolation basis  for a space that allows a discrete fast  Fourier transform type matrix-free formula for  interpolating functions on the sphere. We prove that the quality of the spherical interpolation operator is same as that of the classical spectral interpolation operator for two dimensional periodic functions.
         We also construct a minimal quadrature rule for the  space  (with number of points equal to   the dimension of the space), and describe an associated  interpolation operator.
         Finally, given a large dataset in the latitudinal and   longitudinal directions, with a  longitudinal symmetry condition, we construct a quasi--interpolatory approximation of the function representing the scattered data. The quasi--interpolatory operator gives near best approximation to every continuous function on the sphere.
         We demonstrated our matrix-free operators with several benchmark numerical experiments.


19/07/06  Professor  Ian Sloan, FAA, UNSW                 

The best of both worlds: radial basis function and polynomial approximation on the sphere

         Many researchers have discussed approximation by radial basis functions on a sphere, using scattered data.  Usually there is no polynomial component in such approximations if, as here, the kernel that generates the radial functions is (strictly) positive definite.
     On the other hand, the utility of polynomials for approximating slowly varying components is well known -- an extreme case is the NASA model of the earth's gravitational potential, which represents  the potential by a purely polynomial approximation of high degree.
     In this joint work with Alvise Sommariva we propose a hybrid approximation, in which there is a radial basis functions component to handle the rapidly varying and localised aspects, but also a polynomial component to handle the more slowly varying and global parts.
     The convergence theory (including a doubled rate of convergence for sufficiently smooth functions) make use of the "native space" associated with the positive definite kernel (with no polynomial involvement in the definition).  A numerical experiment for a simple model with a geophysical flavour establishes the potential value of the hybrid approach.


12/07/06  Doctor  Sergey Ajiev, UNSW                     

Measures of nooncompactness and related moduli and normal structure constants

in the nonlocal geometry of function and sequence spaces.

        This is the second talk in the series devoted to the local and nonlocal geometry of Besov, Lebesgue, Lizorkin-Triebel, Sobolev and sequence spaces. One considers the Hausdorff, Kuratovski, and  separation measures of noncompactness, the corresponding moduli and the constants D and W on  various spaces, including
the anisotropic spaces of differentiable functions defined on an open
subset of the Euclidean space, and calculates or evaluates the constants.
       One of the tools is the strengthened version of the estimates related to the weak continuity of the generalised duality mapping. The questions of sharpness, open problems and applications to Nonlinear Analysis and PDE are discussed.   

05/07/06  Doctor  Oleg Ageev, UNSW                     

Spectral rigidity of group actions and Kazhdan property (T)

       We will discuss new properties of subclasses of representations  naturally corresponding to dynamical systems. It can be viewed as modern applications of analytical and topological theories, mainly, the operator theory (C*-algebras), the representation theory of finite or countable discrete groups, and the theory of analytic sets, to the typical dynamical systems. Surprisingly  enough, different dynamical systems have the same  spectral characteristics, known as spectral rigidity.

28/06/06  Professor Douglas Lind, University of Washington

Entropy for algebraic actions of the integral Heisenberg group and noncommutative Mahler measure

       I'll first give gentle introduction, intended for a general audience, to entropy for the joint action of several commuting group automorphisms. The     answer involves the traditional Mahler measure of a polynomial in several commuting variables. Next I'll turn this on its head, and use entropy to define Mahler measure for polynomials in several noncommuting variables, using the integral Heisenberg group as the main example. This leads to a whole host of interesting unsolved problems. This is joint work with Klaus Schmidt, begun while we were in Sydney at the conference on algebraic dynamics in February, 2005.

21/06/06 Professor Tony Dooley, UNSW

A counter example to the finite extension conjecture



14/06/06 Doctor Sergey Ajiev, UNSW

Extension of Hölder maps between Besov, Lebesgue and Lizorkin-Triebel spaces with mixed norm and counterparts of the Jacobi identity

      The applicabiity of numerical methods relies on the theory of approximation, incorporating the fundamental principles of S. N. Bernstein and D. Jackson.  The latter  depends heavily on the existence of a convenient  extension of a function or a mapping. The talk is devoted to identifying the range of the Hölder maps from a metric or a function space to a function space, which can be extended from an arbitrary subset to the whole space with the preservation of both the norm and the class of the maps.
      The applications of the Bernstein and Jackson principles are discussed.

23/03/05 Professor Zeev Ditzian

Multidimensional averages



09/03/05 Doctor Johann S Brauchart, University of Technology Graz, Austria

Points on the sphere and some Proof Techniques

       In dealing with points on an unit sphere in an Euclidean space $\mathbb{R}^{d+1}$, $d\geq2$, which are 'good' in some sense,
one faces certain technical difficulties to answer questions like
       What are the explicit coordinates of such points? How good do such points approximate the uniform measure on the sphere ---> 'equidistribution' ?
How can this be quantified ---> 'discrepancy(-bounds)' ?  What is the distance of two closest points--> well-separation' ?
What is their 'Riesz-$s$-energy' when these points are thought to interact through a Riesz-potential $1/r^s$. ?
If these points are used as nodes for a numerical integration rule what can be said about the error of integration ---> 'worst-case error bounds' ?
We want to consider some of these questions and look mostly at the techniques used to prove results.


03/11/04 Professor Valentin Golodets

Orbital equivalence in ergodic theory: old and new problems II

     This is the second talk on this topic, and will deal with costs of an equivalence relation and a subrelation with a finite index.
     We use an approach of Colin Sutherland to prove this relation. This approach was realized by Tony Dooley and V. Golodets.

27/10/04 Doctor Nils Byrial Andersen, UNSW

Paley-Wiener theorems

     We study (the classical) Paley-Wiener theorems for the Dunkl, Fourier and Hankel transforms on the real line. In particular, we give an unified
proof/approach. We also give some applications in sampling theory.


20/10/04 Professor Valentin Golodets

Orbital equivalence in ergodic theory: old and new problems I

    Some or all of the following topics will be considered.
    Orbital properties of amenable group actions. Results of H.A. Dyefor abelian group actions, the theorem of Connes, Feldman and Weiss for amenable group actions.
    Countable equivalence relations and the theorem of Feldman and Moore.The cost of an equivalence relation of G. Levitt. Gaboriau's theory about the costs and its properties. Computations of costs for some special cases: a free join of two equivalence relations, an amalgamated free join of two equivalence relations and other cases.

29/09/04 Professor Ruyun Ma, Northwest Normal University, China

Multi-Point Boundary Value Problems of Ordinary Differential

     We show the existence of positive solutions and the global behavior of positive solutions of the nonlinear multi-point boundary value problem
$u''+f(t,u)=0$,
$u(0)=0$, $u(1)=\alpha u(\eta)$,
where $\eta \in (0,1)$.
This is achieved by the Fixed-Point Index and Global Continuation Principle of Leray-Schauder.
   The boundary condition reduces to the Dirichlet boundary condition $u(0)=0$, $u(1)=0$, if $\alpha=0$, and to the Robin boundary condition  $u(0)=0$, $u'(1)=0$
if $\alpha=1$ and $\eta$ approaches $1$. 

22/10/04 Doctor Sergey Ajiev, Australian National University

 Some geometric properties of function spaces and their applications

     Some local and non-local geometric properties of anisotropic Besov and Lizorkin-Triebel type spaces with either Lorentz or mixed Lebesgue basic (quasi) norm of differentiable functions defined on an open set are considered. They include superreflexivity, complementability, and the description of the type, cotype and asymptotic behavior of the moduli of uniform convexity and smoothness for some equivalent norms.
     The range of the applications is represented by some properties of unconditionally convergent series, a variant of a "Hahn-Banach theorem"
for bilinear forms, and sharpness of some boundedness results for singular and supersingular operators between function spaces. 

01/09/04 Professor Thong Le Gia,Texas A&M and UNSW

Approximation of linear partial differential equations on spheres

     Partial differential equations on spheres have many applications in weather forecasting, geography and meteorology. In this talk,
I will outline methods of approximation and error estimates in the L^2(S^n), L^infinity(S^n) and Sobolev norms for an elliptic PDE and the heat equation on spheres.

25/08/04 Professor Tony Dooley, UNSW 

Critical dimension: an approach to non-singular entropy