Programme of Arjeh Cohen 60, 27-28-29 May 2009

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Wednesday 27

Arrival and coffee 
10:20-10:30 Opening
10:30-11:25  Hanspeter Kraft Finite Field Extensions, Covariant Dimension and a Result of Hermite's
  (Basel) A classical result from 1861 due to Hermite says that every separable equation of degree 5 can be transformed into an equation of the form x5 + b x3 + c x + d = 0. Later this was generalized to equations of degree 6 by Joubert.
We show that both results can be understood as an explicit analysis of certain covariants of the symmetric groups S_5 and S_6. Moreover, these result have to be understood within the framework of "compressions" of finite group actions and the concept of covariant and essential dimension, introduced by Buehler and Reichstein in 1997.
Relinde Jurrius
Classifying polynomials of linear codes
The weight enumerator of a linear code is a classifying polynomial associated with the code. Besides its intrinsic importance as a mathematical object, it is used in the probability theory around codes. For example, the weight enumerator of a binary code is very useful if we want to study the probability that a received message is closer to a different codeword than to the codeword sent. (Or, rephrased: the probability that a maximum likelihood decoder makes a decoding error.)
We will generalize the weight enumerator in two ways, which lead to polynomials which are better invariants for a code. A procedure for the determination of these polynomials is given. We will show that the two generalisations determine each other, and that they connect to the Tutte polynomial of a matroid, thus linking coding theory and matroid theory. A complete overview of the connections will be given.
Koen Struyve
Polygons with valuation and R-buildings
We present an `equivalence' between certain generalized polygons with valuation, and R-buildings of dimension 2, which are non-discrete generalizations of affine buildings of rank 3. For the discrete case, this completes earlier work by H. Van Maldeghem.
12:30-13:55  Lunch
Wilberd van der Kallen Power reductivity
  (Utrecht) We discuss finite generation of invariants (First Fundamental Theorem of Invariant Theory) when the base ring is noetherian. Power reductivity is one of the original formulations of geometric reductivity. When the base ring is not a field it matters a lot which formulation is chosen.
15:00-15:25 Stefan Maubach
k[z]-automorphisms and coordinates in two variables
Recently there have been a few results on polynomial automorphisms fixing one variable. Given time, I will give a short overview of some of these results and present one new one. A component of a polynomial automorphism, like x+y2, is called a coordinate (or sometimes variable). Now suppose you have a polynomial p in k(z)[x,y] which is a coordinate over k(z). If you additionally assume that p in k[z][x,y], then a priori it does not have to be a coordinate in the three variables x,y,z over k. (If (f,g) is the automorphism over k(z) of which f is a component, g may have fractions even though f does not have them. Life is more difficult.) However, in this talk I will prove a result which implies that f must be a coordinate, and even a k[z]-coordinate (i.e. fixing z). The actual result is that there is a surprising equivalence between (1) f is a k[z]-coordinate of k[z][x,y], and (2) k[x,y,z]/(f) isomorphic to k[x,y] and f(x,y,a) is a coordinate in k[x,y] for some a in k. This is joint work of A. van den Essen, E. Edo, and me.
Tea break
Gabriele Nebe Integral forms for algebraic groups
  (RWTH Aachen) Martin Kneser developed a method to classify the finitely many isometry classes of lattices in a genus of positive definite lattices based on the strong approximation theorem for orthogonal groups. His neighbouring method has a nice interpretation in the affine Bruhat-Tits building of the orthogonal group over a suitable complete field. This can be transfered to other algebraic groups to enumerate equivalence classes of integral forms in a genus. Together with Arjeh Cohen we aplied this method to enumerate the genera of maximal integral forms defined by the four Lie primitive subgroups of G2(C). A mass-formula helps to check completeness.
Free session
This slot will be filled with short communications (10 to 15 minutes) by whoever wants to present something, or pose a problem.
Thursday 28

9:00-9:55 Tomas Recio New "tapas" on automatic reasoning in geometry
  (Santander) The talk will revisit  some mathematical issues I have shared along the years, in one way or another, with Arjeh: computer algebra, elementary geometry, mathematics education. Following the "tapas" style, I will present some small, but hopefully tasty, portions of each of these topics, with automatic reasoning as a common goal.
10:00-10:25 Arjen Stolk
An algebraic approach to discrete tomography
Tomography deals with the problem of reconstructing an image from a series of projections. We consider problems in discrete tomography: the images are finite sets of numbers placed on a rectangular grid; the projections are sums along lines in a prescribed set of directions. Traditionally problems in discrete tomography have been approached using tools from discrete mathematics and combinatorics. We have developed a framework for studying these problems within the context of commutative algebra. It allows us to use algebraic machinery to supplement traditional techniques that have been applied to these problems.
10:30-10:55 Coffee break
11:00-11:25 Jos in 't panhuis
Lie algebras, extremal elements, and associated geometries


An extremal element of a Lie algebra L over a field K of characteristic not two is a nonzero element x for which [x,[x,L]] is a subset of the corresponding projective point Kx. This projective point is also called an extremal point. Over characteristic two the definitions are a bit more involved. Extremal elements play an important role in both classical and modern Lie algebra Theory. In complex Lie algebras they are the elements that are long-root vectors relative to some maximal torus.
     We consider a Lie algebra generated a set E of extremal points and we assume that the product of two extremal elements is never extremal. We show how, depending on the field in question, a set F of subsets of E can be chosen such that (E,F) is a partial linear space. If K is the field of two elements, then (E,F) is a Fischer space. Otherwise, (E,F) is a polar space.
11:30-12:25 David Wales Reductions in Brauer algebras and applications to BMW algebras of simply laced type
  (Caltech) I will discuss Brauer algebras of simply laced type and give some properties.  I will describe reductions and give a uniqueness property. This is used to show that the
BMW algebras of the same type are free and have the same rank as that  of the Brauer algebras.
12:30-13:55 Lunch
14:00-14:55 Ernest Shult Long root geometries
  (Kansas State University) Suppose Gamma is a parapolar space for which no point is collinear with exactly one point of any symplecton. Then Gamma is either (i) a metasymplectic space, E_{6,2}, E_{7,1}, E_{8,8}, or a Grassmannian of lines of a polar space, or (ii) it is a strong parapolar space of point diameter 2. What spaces lurk behind this second alternative? Connections with root filtration spaces are discussed.
15:00-15:25 Shona Yu
The cyclotomic Birman-Murakami-Wenzl algebras and affine tangles
The BMW algebras are closely tied with the Artin braid group of type A, Iwahori-Hecke algebras of the symmetric group, and are a deformation of the Brauer algebras. Their algebraic definition was originally inspired by the Kauffman link invariant and, geometrically, it is isomorphic to the Kauffman Tangle algebras. These algebras also feature in the theory of quantum groups, statistical mechanics and topological quantum field theory.
The cyclotomic BMW algebras are a generalization of the BMW algebras associated with type B knot theory and the cyclotomic Hecke algebras of type G(k,1,n) (aka Ariki-Koike algebras).
In this talk, I will show they are free and may be topologically realized as a certain cylindrical analogue of the Kauffman tangle algebras. In particular, bases which may be explicitly described both algebraically and diagrammatically in terms of affine tangles are given. These yield a cellular basis, in the sense of Graham and Lehrer. I will also mention an application of cyclotomic BMW algebras to invariants of links in the solid torus.
- Based on research completed during my Ph.D. (completed end of 2007 at the University of Sydney) and joint work with Stewart Wilcox.
15:30-15:55 Tea break
16:00-16:55 Willem de Graaf Classifying nilpotent orbits of theta-groups
  (Trento) I will present two algorithms for classifying nilpotent orbits of theta-groups, and report on experiences with their implementation in the computer algebra system GAP4. Also I will discuss two applications: finding principal automorphisms of simple Lie algebras, and deciding good index behaviour.
Free session
This slot will be filled with short communications (10 to 15 minutes) by whoever wants to present something, or pose a problem.
Anniversary dinner

Friday 29

9:00-9:55 Lajos Rónyai
Combinatorics and Gröbner bases
  (MTA SZTAKI Budapest) The theory of Gröbner bases is one of the favourite subjects of Arjeh. Discussions  with him played a major role in that my interest turned to the topic in the title. I would like to discuss some  applications of Gröbner basis techniques to combinatorial problems, including uniquely vertex colourable graphs (Hillar-Windfeldt), an application of the full Alon Nullstellensatz to additive number theory (Károlyi), and a solution of a conjecture of Babai and Frankl from extremal combinatorics. In the reverse direction, combinatorial methods to compute lexicographic Gröbner normal sets (standard monomials) will be considered.
Geertrui Van de Voorde
The intersection of a subline and a linear set
We define a linear set in the following way. Suppose q = q0t , with t ≥ 1. By ”field reduction”, the points of PG(n,q) correspond to (t−1)-dimensional subspaces of PG((n+1)t−1, q0), since a point of PG(n,q) is a 1-dimensional vector space over F_q, and so a t-dimensional vector space over Fq0. In this way, we obtain a partition D of the pointset of PG((n+1)t−1,q0) by (t−1)-dimensional subspaces, which forms a Desarguesian spread. Let D be the Desarguesian (t−1)-spread of PG((n+1)t−1,q0). If U is a subset of PG((n+1)t−1,q_0), then we define B(U):= {R ∈ D||U ∩ R != ∅}, and we identify the elements of B(U) with the corresponding points of PG(n,q0t). If U is subspace of projective dimension k of PG((n+1)t−1,q0), then B(U) is an Fq0-linear set of rank k + 1. Using this representation, we investigate the intersection of linear sets. For example, we show that a subline intersects a linear set of rank 3 in 0, 1, 2, 3 or q+1 points.
10:30-10:55 Coffee break
Dan Roozemond
Constructing Chevalley Bases in all Characteristics
Lie algebras are often used to study the algebraic groups from which
they originate, but they are interesting objects in their own right as
well. For (almost) every simple Lie algebra there exists a particular
basis with special properties, invented by Chevalley. It is an extremely
useful tool to study Lie algebras.
Algorithms exist and have been implemented to, given a Lie algebra in some way, compute its Chevalley basis. Unfortunately, these algorithms break down in some special cases and in particular over fields of characteristic 2 and 3. We show what problems arise in these cases and how they can be solved.
11:30-12:25 Bruce Cooperstein Subspaces and generation of Lie incidence geometries:  a survey
  (UC Santa Cruz) I will survey what is known about characterizing subspaces of Lie Incidence Geometries by their isomorphism type, for example, subspaces isomorphic to a Grassmannian, G(n,k).  Will also survey those Lie Incidence Geometries for which a minimal generating set has been determined.
12:30-13:55  Lunch
14:00-14:55 Bernhard Mühlherr Isomorphisms of Coxeter Groups
  (Giessen) In the first part of my talk I will give a survey on several results about isomorphisms of Coxter groups. Then I will explain a recent joint result with T. Marquis which reduces the solution of the isomorphism problem to one main conjecture. I will also report on recent progress towards a proof of this conjecture.
15:00-15:55 William Kantor Presentations of finite simple groups
  (University of Oregon) (with R. M. Guralnick, M. Kassabov, A. Lubotzky)
Theorem: All finite simple groups of rank n over Fq, with the possible exception of the Ree groups 2G2(q), have presentations with at most 10 generators, 50 relations
and bit-length O(log n+log q).
I will focus on a more  special situation: For each n>4, An and Sn have presentations with 3 generators, 7 relations and  bit-length O(log n).

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