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