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2003/01 Michelle Vail.
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Error Estimates for Spaces Arising from
Approximation by Translates of a Basic Function (PhD Thesis)
We look at aspects of error analysis for interpolation by
translates of a basic function. In particular, we
consider ideas of localisation and how they can be used to
obtain improved error estimates. We shall consider
certain seminorms and associated spaces of functions which
arise in the study of such interpolation methods. These
seminorms are naturally given in an indirect form, that is
in terms of the Fourier Transform of the function rather
than the function itself. Thus, they do not lend
themselves to localisation. However, work by Levesley and
Light rewrites these seminorms in a direct form and thus
gives a natural way of defining a local seminorm. Using
this form of local seminorm we construct associated local
spaces. We develop bounded, linear extension operators
for these spaces and demonstrate how such extension
operators can be used in developing improved error
estimates. Specifically, we obtain improved $L_2$
estimates for these spaces in terms of the spacing of the
interpolation points. Finally, we begin a discussion of
how this approach to localisation compares with
alternatives.
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2003/02 Paola Frediani and Frank Neumann.
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Etale Homotopy Types of Moduli Stacks
of Algebraic Curves with Symmetries
Using the machinery of etale homotopy theory \`{a} la
Artin-Mazur we determine the etale homotopy types of
moduli stacks over $\bar{\Q}$ parametrizing families of
algebraic curves of genus $g \geq 2$ endowed with an
action of a finite group $G$ of automorphisms, which comes
with a fixed embedding in the mapping class group
$\Gamma_g$, such that in the associated complex analytic
situation the action of $G$ is precisely the
differentiable action induced by this specified embedding
of $G$ in $\Gamma_g$
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2003/03 John Hunton and Mikhail Shchukin.
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The $K$-theory of $C^*$-algebras with finite dimensional
irreducible representations
We study the $K$-theory of unital $C^*$-algebras $A$
satisfying the condition that all irreducible
representations are finite and of dimension at most some
integer $n$. When the algebra $A$ is $n$-homogeneous, {\em
i.e.}, all irreducible representations are exactly of
dimension $n$, we show $K_*(A)$ is the topological
$K$-theory of a related compact Hausdorff space,
generalising the classical Gelfand-Naimark theorem. For
general $A$ we give a spectral sequence computing $K_*(A)$
from a sequence of topological $K$-theories of related
spaces. For $A$ generated by two idempotents, this becomes
a 6-term long exact sequence.
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2003/04 Frank Neumann and Ulrich Stuhler.
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Moduli Stacks of Vector Bundles and Frobenius Morphisms
We describe the action of the different Frobenius
morphisms on the etale cohomology ring of the moduli stack
of algebraic vector bundles of fixed rank and determinant
on an algebraic curve over a finite field in
characteristic p and analyse special situations like
vector bundles on the projective line and relations with
infinite Grassmannians.
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2003/05 Phillipe Caldero and Robert
Marsh.
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A multiplicative property of quantum flag minors
II
Let U^+ be the plus part of the quantized enveloping
algebra of a simple Lie algebra of type A_n and let B^* be
the dual canonical basis of U^+. Let b, b' be in B^* and
suppose that one of the two elements is a q-commuting
product of quantum flag minors. We show that b and b' are
multiplicative if and only if they q-commute.
Available as:
gzipped PostScript (.ps.gz)
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2003/06 Jeremy Levesley.
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Perturbed kernel approximation on homogeneous
manifolds
Current methods for proving the existence of unique
interpolants for kernel approximation require the kernels
to be conditionally positive definite. The subsequent
convergence rates for interpolants require the same
condition. In this paper we show how this requirement may
be relaxed, if the kernel being used to approximate with
is a smooth perturbation of a positive definite kernel.
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2003/07 Volodymyr Mazorchuk and Catharina Stroppel.
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Translation and shuffling of projectively presentable
modules and a categorification of a parabolic Hecke module
We investigate the category of $\p$-presentable modules
in the principal block of the Bernstein-Gelfand-Gelfand
category $\Oo$. This category is equivalent to the module
category of a properly stratified algebra. We describe the
socles and endomorphism rings of standard objects in this
category. We consider translation and shuffling functors
and their action on the standard modules. Finally, we
study a graded version of this category, in particular,
giving a graded version of the properly stratified
structure, and using graded versions of translation
functors to categorify a parabolic Hecke module.
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2003/08 Teijo Arponen.
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2-Tensor invariants in numerical integration
In geometric integration of ordinary differential equations (ODEs) one
needs concepts which are both geometric and algebraic. In
this paper we start from the algebraic point of view: we
introduce the 2-tensor invariants attached to an
ODE. These generalize the well known notion of a
symplectic structure.
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2003/10 Claire
Irving.
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Embeddings and Immersions of Real Projective
Spaces
In this document, rigorous definitions of the concepts of immersions
and embeddings of manifolds are set out. Whitney's
immersion and embedding theorems are proved and stronger
versions of the theorems are quoted. The real projective
plane is considered in detail; an immersion and an
embedding of $\mathbb{R}P^{2}$ are given, along with the
results that $\mathbb{R}P^{2}$ cannot be immersed into
$\mathbb{r}^{2}$ and cannot be embedded into
$\mathbb{R}^{3}$. A non-immersion theorem for
$\mathbb{R}P^{2^{r}}$ is proved using Stiefel-Whitney
classes and a non-embedding result for $\mathbb{R}P^{n}$,
where $n$ is not a power of 2, is proved using cohomology
and cohomology operations. Finally, $K$-theory is
introduced and used to prove two theorems, one giving
dimensions of Euclidean space into which $\mathbb{R}P^{n}$
cannot be embedded or immersed and the other giving
possible immersions for $\mathbb{R}P^{n}$.
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2003/12 R. J. Marsh and K. Rietsch.
-
Parametrizations of flag varieties.
For the flag variety G/B of a reductive algebraic group G we define a
certain (set-theoretical) cross-section phi from G/B to G, which depends
on a choice of reduced expression for the longest element in the Weyl
group. This cross-section is continuous along the components of Deodhar's
decomposition of G/B and assigns to any flag gB a representative phi(gB)
in G which comes with a natural factorization into simple root subgroups
and simple reflections. We introduce a generalization of the Chamber
Ansatz of Berenstein, Fomin and Zelevinsky and use it to obtain formulas
for the factors of phi(gB). Our results then allow us parameterize
explicitly the components of the totally nonnegative part of the flag
variety as defined by Lusztig. This gives a new proof of Lusztig's
conjectured cell decomposition of this set.
Available as:
gzipped PostScript (.ps.gz)
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2003/13 Steffen Koenig.
-
Filtrations, Stratifications and Applications
These are lecture notes from a course given at ICRA
X in Toronto in 2002. Filtrations and stratifications define classes of
algebras such as quasi-hereditary, cellular and stratified algebras. The
use of their properties is illustrated by a number of examples, including
Schur-Weyl duality, Soergel's structure theorem of the Bernstein-Gelfand-
Gelfand category, Enright's conjecture, and identities of decomposition
numbers.
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2003/14 P. Houston, I. Perugia and D. Schotzau.
-
Discontinuous Galerkin Methods for Maxwell's Equations
We present discontinuous Galerkin methods for the numerical
discretization of time-harmonic eddy current problems in
conducting and insulating materials. The involved
operators are discretized by using suitable variants of
the classical interior penalty technique; here, the
divergence-free constraint on the electric field in
insulating materials is imposed either by using a
regularization approach or by a Lagrange multiplier technique.
Available as:
gzipped PostScript (.ps.gz)
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2003/15 P. Houston, I. Perugia and D. Schotzau.
-
Nonconforming Mixed Finite Element Approximations to
Time-Harmonic Eddy Current Problems
We present nonconforming mixed finite element methods for
the discretization of
time-harmonic eddy current problems. These
methods are based on a discontinuous Galerkin approach, where
the unknowns are approximated by completely discontinuous
piecewise polynomials. In particular, we consider
a stabilized mixed formulation
involving equal-order elements,
and a non-stabilized variant employing mixed-order
elements.
Available as:
gzipped PostScript (.ps.gz)
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2003/16 P. Houston, I. Perugia and D. Schotzau.
-
Energy Norm A Posteriori Error Estimation for Mixed Discontinuous Galerkin
Approximations of the Maxwell Operator
In this paper we develop the {\em a posteriori} error estimation of
mixed discontinuous Galerkin finite element approximations
of the Maxwell operator. In particular, by employing suitable Helmholtz
decompositions of the error, together with the conservation properties
of the underlying method, computable upper bounds on the error,
measured in terms of a natural (mesh-dependent) energy norm,
are derived. Numerical experiments testing the performance
of our {\em a posteriori} error bounds for problems with both smooth
and singular analytical solutions are presented.
Available as:
gzipped PostScript (.ps.gz)
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2003/17 P. Houston, I. Perugia and D. Schotzau.
-
Mixed Discontinuous Galerkin Approximation of the Maxwell Operator:
Non-stabilized Formulation
A non-stabilized
mixed discontinuous Galerkin method for the discretization
of the time-harmonic Maxwell operator
on simplicial meshes is studied. In contrast to the stabilized
scheme introduced in [8], the proposed
formulation
contains no normal--jump stabilization; instead, it is based on discontinuous
mixed-order elements for the approximation of
the unknowns. Optimal a priori error bounds in the energy norm
are derived;
the error analysis relies on
suitable decompositions of discontinuous spaces and on
stability properties of the underlying conforming spaces.
The formulation is tested on a set of numerical examples in two space
dimensions.
Available as:
gzipped PostScript (.ps.gz)
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2003/18 P. Houston, D. Schotzau and T. P. Wihler.
-
Energy Norm A Posteriori Error Estimation for Mixed Discontinuous
Galerkin Approximations of the Stokes Problem
In this paper, we develop the a posteriori error
estimation of mixed discontinuous Galerkin finite element
approximations of the Stokes problem. In particular, computable
upper bounds on the error, measured in terms of a natural
(mesh--dependent) energy norm,
are derived. The proof of the a posteriori error bound is based
on rewriting the underlying method in a non-consistent form
by introducing appropriate lifting operators, and employing
a decomposition result for the discontinuous spaces.
A series of numerical experiments highlighting
the performance of the proposed a posteriori error estimator
on adaptively refined meshes are presented.
Available as:
gzipped PostScript (.ps.gz)
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2003/19 J. Levesley.
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Where there's a Will there's a Way -- the Research of Will Light
This paper describes the work of Will Light, who died on December 8th,
2002. It highlights his contributions in the study of
minimal norm projections, tensor product approximation
(including the convergence of the Diliberto-Straus
algorithm), proximality, radial and ridge function
approximation, both via quasi-interpolation and
interpolation. My aim is not only to describe the impact
of Will's work, but also to convey some of his impact as a person.
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2003/20 Gerhard Rosenberger, Martin Scheer and Richard M. Thomas.
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Finite generalized tetrahedron groups with a cubic relator
An ordinary tetrahedron group is a group with a presentation
of the
form
%
\[\langle x,y,z \mid
x^{e_1}=y^{e_2}=z^{e_3}=(xy^{-1})^{f_1}=(yz^{-1})^{f_2}=(zx^{-
1})^{f_3}=1
\rangle\]
%
where $e_i\ge2$ and $f_i\ge2$ for all $i$. Following Vinberg, we
call groups defined by a presentation of the form
%
\[\langle x,y,z \mid
x^{e_1}=y^{e_2}=z^{e_3}=R_1(x,y)^{f_1}=R_2(y,z)^{f_2}=R_3(z,x)^{f_3}=1
\rangle,\]
%
where each $R_i(a,b)$ is a cyclically reduced word involving both $a$
and
$b$, generalized tetrahedron groups. These groups appear in
many
contexts, not least as subgroups of generalized triangle groups.
In this paper, we build on previous work of Coxeter, Edjvet, Fine,
Howie,
Levin, Metaftsis, Roehl, Rosenberger, Stille, Thomas, Tsaranov and
Vinberg
(amongst others) to give a classification of the finite generalized
tetrahedron groups with a cubic relator.
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2003/21 Vincent Schmitt.
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Flatness, preorders and general metric spaces.
This paper studies a general notion of flatness in the
enriched context:
p-flatness where the parameter p stands for a class of presheaves.
One obtains a completion of a category A by considering the category
Fp(A) of p-flat presheaves over A. This completion is related to the
free-cocompletion under a class of colimits defined by Kelly.
For a category A, for p = pz the class of all presheaves, Fpz(A) is the
Cauchy-completion
of A. Two classes Pu and Pd of interest for general metric spaces are
considered. The pu and pd flatness are investigated and the associated
completions are characterized for general metric spaces (enrichments over R)
and preorders (enrichments over B).
We get this way two non-symmetric completions for metric spaces
and retrieve the ideal completion for preorders.
Available as:
gzipped PostScript (.ps.gz)
-
2003/22 M. Felisatti.
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Multiplicative K-theory and K-theory of functors
In this article we give a construction of Max Karoubi's multiplicative
K-theory as the K-theory of an appropriate functor between
two categories. We use this construction to explain why
the two definitions of relative multiplicative K-theory
for a compact pair of manifolds we give in the article agree.
-
2003/23 Steffen
Koenig, Oleksandr Khomenko and Volodymyr Mazorchuk.
-
Finitistic dimension and tilting modules for stratified algebras
For quasi-hereditary or standardly stratified algebras there are
well-known upper bounds for the global or finitistic dimensions, but the
precise values of these dimensions are usually not known. Recently, the
projective and injective dimensions of tilting modules have been related
to the global or finitistic dimensions, in particular through a conjecture
of Mazorchuk and Parker. This paper contributes new techniques and classes
of examples, where the conjecture is true.
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2003/24 Rick Thomas.
-
A course on permutation groups
This is the set of lecture notes to accompany a course of
lectures given at the University of Helsinki in June 2003.
This was one of a series of algebra courses organized in
conjunction with the Finite Model Theory project.
The audience consisted of a mixture of staff and postgraduate
students from a variety of backgrounds. To try and make the course
completely self-contained, the only assumed prerequisite
knowledge consisted of some elementary concepts from general algebra
(such as the notion of an equivalence relation and some
basic linear algebra) and some elementary group theory.
The facts from group theory we assumed are listed in Section 2
(this is essentially just a check-list of what we were taking
for granted).
These notes were drafted before the course was given and
edited as we went along. The point of producing a complete
set is for the participants (and anyone else who is interested)
to have a record of what was delivered in the course.
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2003/26 Karin Erdmann, Miles Holloway, Nicole Snashall, Oeyvind Solberg and Rachel Taillefer.
-
Support Varieties for Selfinjective Algebras
Support varieties for any finite dimensional algebra over a field were
introduced by N. Snashall and O. Solberg, using graded subalgebras of the
Hochschild cohomology. We mainly study these varieties for selfinjective
algebras under appropriate finite generation hypotheses. Then many of the
standard results from the theory of support varieties for finite groups
generalize to this situation. In particular, the complexity of the module
equals the dimension of its corresponding variety, all closed homogeneous
varieties occur as the variety of some module, the variety of an
indecomposable module is connected, periodic modules are lines and for
symmetric algebras a generalization of Webb's theorem is true.
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2003/27 A. A. Baranov.
-
Classification of direct limits of involution simple
associative algebras
A classification of (countable) direct limits
of finite dimensional involution simple associative algebras
over an algebraically closed field of arbitrary characteristic
is obtained. The set of invariants consists of two supernatural
(or Steinitz) numbers, two real parameters, and type of the sequence.
-
2003/28 Y.A. Bahturin, A.A. Baranov and A.E. Zalesskii.
-
Simple Lie subalgebras of locally finite associative algebras
We prove that any simple Lie subalgebra of a locally finite
associative algebra is either finite dimensional or
isomorphic to the commutator algebra of the
Lie algebra of skew symmetric elements
of some involution simple locally finite associative algebra. The
ground field is assumed to be algebraically closed of characteristic
0. This result can be viewed as a classification theorem for simple Lie
algebras that can be embedded in locally finite associative
algebras. We also establish a link between this class of Lie algebras
and that of Lie algebras graded by finite root systems.
-
2003/29 A.A. Baranov and I.D. Suprunenko.
-
Modular branching rules for 2-column diagram
representations of general linear groups
The article is devoted to finding branching rules for the
irreducible representations of the general linear groups in
positive characteristic associated with 2-column partitions and
the restrictions of these representations to the special linear
groups.
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