Permutation Groups (13w5033)

Arriving in Banff, Alberta Sunday, July 21 and departing Friday July 26, 2013

Organizers

(University of Southern California)

(The University of Western Australia)

(Universität Münster)

(Ecole Polytechnique Federale de Lausanne)

Objectives

The aim of this workshop is to bring together leading researchers in these related areas as well as those whose research centres on permutation groups. This is a follow up to meetings on the subject in Oberwolfach and Banff. In both instances bringing together the leading researchers in different areas related to permutation groups and their applications as well exposing top graduate students and younger researchers has led to significant new results. We expect the same from this meeting.

The theory of permutation groups is a classical area of algebra.
It originates in the middle of nineteenth century, with very considerable
contributions by most of the major figures in algebra over the last
two centuries, including Galois, Mathieu, Jordan, Frobenius, Burnside,
Schur and Wielandt. In the last twenty years, the direction of the subject
has changed substantially. The classification of finite simple groups has had
many applications, many of these through thorough investigation of
relevant permutation actions. This in turn led to invigoration of
the subject of permutation groups, with interesting new questions
arising and techniques developed for tackling them. Interestingly,
some topics arose in more than one context, forming new connections.
The concept of exceptionality was first suggested by work on covers
of curves; it then appeared independently in homogeneous factorizations
of graphs, and more recently it has found applications in investigations
of line-transitive linear spaces. The concept of derangements in groups
(that is, fixed-point-free permutations) and their proportions is classical;
it has applications to images of rational points for maps between curves
over finite fields, in probabilistic group theory and in investigating
convergence rates of random walks on groups. Recently a conjecture
of Boston and Shalev on the proportion of derangements in simple groups
actions has been settled; interestingly, this conjecture fails
in the slightly more general case of almost simple groups, through
examples of exceptional actions mentioned above. This area continues
to be very lively. The exciting results of Helfgott on expansion in groups,
extended recently by Pyber and Szab'o, and independently by Breuillard,
Green and Tao, have shown not only that Cayley graphs of bounded rank finite simple
Lie type groups are expanders, but have also yielded proof of the Weiss Conjecture of
1978 for locally primitive graphs involving only bounded rank composition factors.
The topic of fixed point ratios and minimal degrees of elements
in permutation groups is classical, going back over 100 years,
but there hasbeen significant progress in the last fifteen years both for finite and
algebraic groups and has had applications in arithmetic algebraic geometry, besides
leading to significant insights in group theory -- a striking example is the
solution of Wielandt's conjecture on the characterization of subnormal subgroups.
The question of base size of permutation actions is of importance in computational group theory as well as in the study
of the graph isomorphism problem. Recent research has thrown much light on the base sizes of actions of almost simple groups in particular.
The concept of quasiprimitive permutation groups is also classical, but there has been revived interest in the subject through investigations of groups of graphs and designs. Algebraic graph theory has developed greatly over the last ten to twenty years; there are interesting connections
to association schemes and representation theory. Some of these in turn found an application in the study of derangements mentioned
above, as well in the study of random walks on groups.