A base of a permutation group G acting on Ωis a subset of Ωwhose pointwise stabiliser in G is trivial. Bases have their origins in computational group theory, where they were used to efficiently store permutation groups of large degree into a small amount of computer memory. The minimal base size of G is denoted by b(G). When b(G)=1, we say that G has a regular orbit on Ω.
A wellknown conjecture made by Pyber in 1993 states that there is an absolute constant c such that if G acts primitively on Ω, then b(G) < c log G / log n, where Ω= n. Pyber's conjecture was established in 2016 by Duyan, Halasi and Maroti, following on from contributions from a variety of authors.
In this talk, I will cover some of the history and uses of bases, and discuss Pyber's conjecture, as well as present some results on the determination of the constant c for bases of almost quasisimple linear groups. I will also outline some recent work on determining which irreducible modules of linear groups contain a regular orbit.
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Vorträge 2018
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 Mai

24. Mai17:00  18:00
Ort: 48436AG Algebra, Geometrie und Computer AlgebraMelissa Lee, Imperial College London: Bases of almost quasisimple groups and Pyber's conjecture
Kategorie: AG Algebra, Geometrie und Computeralgebra 
17. Mai17:00  18:00
Ort: 48436AG Algebra, Geometrie und Computer AlgebraZhicheng Feng, TU Kaiserslautern: On the inductive blockwise Alperin weight condition for special linear and unitary groups
Kategorie: AG Algebra, Geometrie und ComputeralgebraThe inductive blockwise Alperin weight condition is a system of conditions whose verification for all nonabelian finite simple groups would imply the blockwise Alperin weight conjecture. This talk will explain how one can obtain the weights of a special linear or unitary group X from the general linear or unitary group and verify the inductive blockwise Alperin weight condition for any unipotent block of X when the prime does not divide the order of Z(X).

07. Mai17:00  18:00
Ort: 48519AG Algebra, Geometrie und Computer AlgebraThomas Gobet, University of Sydney: Dual and classical generators of Artin groups of spherical type
Kategorie: AG Algebra, Geometrie und ComputeralgebraThe nstrand braid group has two Garside structures, a classical and a dual one, leading to different solutions to the word problem in the group. The corresponding Garside monoids are the classical braid monoid and the BirmanKoLee braid monoid. These monoids exist for any Artin group of spherical type (i.e., attached to a finite Coxeter group).
The aim of the talk is to explain how to express the generators of an Artin group of spherical type coming from the dual braid monoid (and more generally the simples, that is, the divisors of the Garside element of the dual monoid) in terms of the classical generators: we give a closed formula to express a simple of the dual braid monoid, using combinatorial objects called csortable elements (introduced by Reading). It gives a word of shortest possible length in the classical generators representing a simple element of the dual braid monoid. It has as an immediate consequence that the simples of the dual braid monoids are Mikado braids. 
03. Mai17:00  18:00
Ort: 48436AG Algebra, Geometrie und Computer AlgebraKeivan MallahiKarai, Jacobs University, Bremen: Kirillov's orbit method and the polynomiality of the essential dimension of pg
Kategorie: AG Algebra, Geometrie und ComputeralgebraThe faithful dimension of a finite group G is defined to be the smallest dimension of a faithful complex representation of G. Aside from its intrinsic interest, the problem of determining the faithful dimension of finite groups is intimately related to the notion of essential dimension, introduced by Buhler and Reichstein.
The problem of determining the faithful dimension of families of pgroups arising from F_ppoints of a nilpotent algebraic group defined over the field of rational numbers has been studied in some special cases, e.g. the Heisenberg and the full uppertriangular unipotent group.
In this paper, we will use Kirillov’s orbit method to address this problem for a large family of groups. It will be shown that this function is always a piecewise polynomial function along certain “numbertheoretically defined” sets, while, in some specific cases, it is given by a uniform polynomial in p.
This talk is based on a joint work with Mohammad Bardestani and Hadi Salmasian.  Januar

25. Januar17:00  18:00
Ort: 48436AG Algebra, Geometrie und Computer AlgebraPetra Schwer, KIT Karlsruhe: Reflection length in affine Coxeter groups
Kategorie: AG Algebra, Geometrie und ComputeralgebraAffine Coxeter groups have a natural presentation as reflection groups on some affine space. Hence the set R of all its reflections, that is all conjugates of its standard generators, is a natural (infinite) set of generators. Computing the reflection length of an element in an affine Coxeter group means that one wants to determine the length of a minimal presentation of this element with respect to R. In joint work with Joel Brewster Lewis, Jon McCammond and T. Kyle Petersen we were able to provide a simple formula that computes the reflection length of any element in any affine Coxeter group. In this talk I would like to explain this formula, give its simple uniform proof and allude to the geometric intuition behind it.

18. Januar17:00  18:00
Ort: 48436AG Algebra, Geometrie und Computer AlgebraFlorian Eisele, University of London: A counterexample to the first Zassenhaus conjecture
Kategorie: AG Algebra, Geometrie und ComputeralgebraThere are many interesting problems surrounding the unit group U(RG) of the ring RG, where R is a commutative ring and G is a finite group. Of particular interest are the finite subgroups of U(RG). In the seventies, Zassenhaus conjectured that any u in U(ZG) is conjugate, in the group U(QG), to an element of the form +/g, where g is an element of the group G. This came to be known as the "Zassenhaus conjecture". In recent joint work with L. Margolis, we were able to construct a counterexample to this conjecture. In this talk I will give an introduction to the various conjectures surrounding finite subgroups of U(RG), and how they can be reinterpreted as questions on the (non)existence of certain R(GxH)modules, where H is another finite group. This establishes a link with the representation theory of finite groups, and I will explain how, plocally, our example is made up of certain ppermutation modules.