Pure Mathematics Seminars
16th January 2007
No Seminar
23rd January 2007
Gerald Williams (Kent)
"Free subgroups in generalizations of Coxeter groups"
Abstract:
We consider groups defined by presentations that generalize those of Coxeter groups,
in particular the so-called Pride groups,
and the `groups defined by periodic paired relations'.
We show that under a suitable non-positive curvature condition the groups
contain a
non-abelian free subgroup.
30th January 2007
Andreas Enge (Ecole Polytechnique)
"An L(1/3) algorithm for the discrete
logarithm problem in low degree curves"
(joint work with Pierrick Gaudry)
Abstract:
Starting with the work by Adleman, DeMarrais and Huang more than ten
years ago, it has been well-established that the discrete logarithm
problem in Jacobians of curves of high genus over finite fields is
easier to solve than in elliptic curves of the same size. Letting
$L (\alpha, c) = e^{(c + o (1)) (g \log q)^\alpha (\log (g \log q))^{1 -\alpha}}$
denote the subexponential function with respect to the curve genus $g$ and the
cardinality of the finite field $q$, the algorithms for computing discrete
logarithms in high genus curves have a complexity of $L (1/2, c)$.
This is in contrast on one hand to elliptic curves, for which so far
only exponential algorithms are known except for some rare special
cases, but also to the computation of discrete logarithms in finite
prime fields, for which the number field sieve yields a more efficient
algorithm in $L (1/3, c)$.
We present the first subexponential algorithm with an exponent $\alpha
< 1/2$ to attack the discrete logarithm problem in a class of
algebraic curves. The curves are characterised by their total degree
being relatively low with respect to their genus. The algorithm has a
complexity of $L (1/3, c)$ for computing the group structure and
discrete logarithms in the Jacobian.
6th February 2007
Karin Erdmann (Oxford)
"Schensted correspondence and Littelmann paths"
Abstract:
This is joint work with J.A. Green and M. Schocker.
We study the Littelmann path model for the case gl_n.
In this case, the paths become words, and we work with the combinatorics of words.
13th February 2007
Gavin Brown (Kent)
"Key varieties and surgery in birational geometry"
Abstract:
I will sketch the ingredients that go into the classification of complex
algebraic varieties (that is, the solutions of polynomial equations over the
complex numbers), concentrating especially on the methods available to
transform one variety into another. While these blowups, flips and flops
arise (in Mori theory, at least) from rather abstract considerations, I am
interested in writing them concretely, using simple equations and group actions.
The punchline is that there are some classes of straightforward-but-large
varieties, the _key_ varieties, that house whole families of flips, etc.
(in much the same way as the "cooling tower" quadric surface contains
the various conic curves as sections).
20th February 2007
Benjamin Klopsch (Royal Holloway)
“Anosov diffeomorphisms and strongly hyperbolic elements in arithmetic subgroups of SL_n(R)”
Abstract:
I will talk about some ongoing work, motivated by a long standing problem in the theory of
dynamical systems. In particular, I will explain how p-adic methods lead to the construction of elements
in SL_n(Z) whose eigenvalues e_1, …, e_n generate a free abelian subgroup of rank n-1 in the multiplicative
group of positive real numbers. This is a special instance of a more general theorem, asserting the existence
of strongly hyperbolic elements in arithmetic subgroups of SL_n(R).
27th February 2007
Beth Holmes(Cambridge)
"Which elements are in the same subgroup?"
Abstract:
If we are given a set of elements from a group then we can
ask whether any of them lie in the same proper subgroup. Considering this
leads to several related questions. We will look at three of these,
namely subgroup coverings, pairwise generating sets and spread.
6th March 2007
Alan Camina (UEA)
"Automorphisms of finite linear spaces"
Abstract:
We look at recent work on classifying those quasisimple groups
of large rank which can act line-transitively. The emphasis will be on the
techniques which enable us to use the knowledge of the structure of the
groups rather than on technical things about the classical linear groups.
All terms WILL be explained and they are not difficult.
13th March 2007
Oliver Baues (TU Karlsruhe)
"Constructions of aspherical manifolds"
Abstract:
A manifold is called aspherical if its
universal covering space is contractible.
This is the case, for example, if the universal
covering is homeomorphic to an Euclidean space.
Given an abstract group $\Gamma$,
there is the basic question if it is possible to
construct compact aspherical smooth manifolds with
fundamental group $\Gamma$, and also to
understand the geometric properties of such
manifolds. Ideally, one would like to classify
them up to homeomorphism or up to
diffeomorphism.
For example, 'most' polycyclic groups $\Gamma$ appear
as fundamental groups of so called solvmanifolds.
Another type of examples which appear in geometry
are the fundamental groups of locally symmetric
spaces. We would like to discuss a method
which allows to build 'mixed' examples from
these basic building blocks. This construction
corresponds to the notion of group extension
on the level of the fundamental group, and it
has many interesting geometric properties.
20th March 2007
Victor Flynn (New College, Oxford)
"Cycles of Covers"
Abstract:
When a direct attack fails to determine all rational points on a curve C,
one can try to find an associated collection of curves, whose rational points cover
those on C, and attempt to resolve these curves. I shall describe a special case of
this technique, when it is possible to apply the process repeatedly. I shall give an
example to show that some curves can even be resistant to arbitrarily many repeated
applications.
27th March 2007
Tuvi Etzion (Technion, Haifa)
"Correction of Multi-dimensional Cluster Errors"
Abstract:
In current memory devices for advanced storage systems the information is
stored in two or more dimensions. In such systems errors usually take the form
of multi-dimensional bursts. Usually, a cluster of errors either will be affected
by the position in which the error event occurred or will be of an arbitrary shape.
In this talk we will give a brief introduction to error-correcting codes
and burst-correcting codes and distinguish between one-dimensional codes
and multi-dimensional codes. We improve the known lower bound (Reiger bound) on
the redundancy of two-dimensional codes. We will consider another lower
bound implied by the number of possible burst patterns. We show variety of constructions
of two-dimensional codes with redundancy close to the the lower bound. In particular
we give constructions, where the shape of the burst is rectangle, Lee sphere,
or an arbitrary shape. Most of these constructions can be easily generalized
for multi-dimensional codes. The techniques we are using are mostly a combination
of algebra and combinatorics.
ALL WELCOME !
This page lists this term's Pure Mathematics Seminars. All are welcome.
All seminars will take place in the McCrea Building, Room 219, at 4.00pm,
unless stated otherwise. Tea is served before the seminar at 3.30 in Room
237 of the McCrea Building.
Seminars of the past years.
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