Almanac
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