Diameters in graphs

AbstractA diameter critical graph has the property that the addition of any edge decreases the diameter. All such graphs are determined for a given vertex connectivity and the edge number is given

Small representations of finite classical groups

Finite group theorists have established many formulas that express interesting properties of a finite group in terms of sums of characters of the group. An obstacle to applying these formulas is lack of control over the dimensions of representations of the group. In particular, the representations of small dimensions tend to contribute the largest terms to these sums, so a systematic knowledge of these small representations could lead to proofs of important conjectures which are currently out of reach. Despite the classification by Lusztig of the irreducible representations of finite groups of Lie type, it seems that this aspect remains obscure. In this note we develop a language which seems to be adequate for the description of the "small" representations of finite classical groups and puts in the forefront the notion of rank of a representation. We describe a method, the "eta correspondence", to construct small representations, and we conjecture that our construction is exhaustive. We also give a strong estimate on the dimension of small representations in terms of their rank. For the sake of clarity, in this note we describe in detail only the case of the finite symplectic groups.Comment: 18 pages, 9 figures, accepted for publications in the proceedings of the conference on the occasion of Roger Howe's 70th birthday (1-5 June 2015, Yale University, New Haven, CT

Amenability of groups and GG-sets

This text surveys classical and recent results in the field of amenability of groups, from a combinatorial standpoint. It has served as the support of courses at the University of G\"ottingen and the \'Ecole Normale Sup\'erieure. The goals of the text are (1) to be as self-contained as possible, so as to serve as a good introduction for newcomers to the field; (2) to stress the use of combinatorial tools, in collaboration with functional analysis, probability etc., with discrete groups in focus; (3) to consider from the beginning the more general notion of amenable actions; (4) to describe recent classes of examples, and in particular groups acting on Cantor sets and topological full groups

Convolution surfaces with varying radius: Formulae for skeletons made of arcs of circles and line segments

International audienceWe develop closed form formulae for the computation of the defining fields of convolutions surfaces. The formulae are obtained for power inverse kernels with skeletons made of line segments or arcs of circle. To obtain the formulae we use Creative Telescoping and describe how this technique can be used for other families of kernels and skeleton primitives. We apply the new formulae to obtain convolution surfaces around $\mathcal{G}^1$ skeletons, some of them closed curves. We showcase how the use of arcs of circles greatly improves the visualization of the surface around a general curve compared with a segment based approach

Theory of Graphs

VII, 270 tr.; 25 cm