Recent trends and future directions in vertex-transitive graphs

A graph is said to be vertex-transitive if its automorphism group acts transitively on the vertex set. Some recent developments and possible future directions regarding two famous open problems, asking about existence of Hamilton paths and existence of semiregular automorphisms in vertex-transitive graphs, are discussed, together with some recent results on arc-transitive graphs and half-arc-transitive graphs, two special classes of vertex-transitive graphs that have received particular attention over the last decade

Tropical Geometry: new directions

The workshop "Tropical Geometry: New Directions" was devoted to a wide discussion and exchange of ideas between the leading experts representing various points of view on the subject, notably, to new phenomena that have opened themselves in the course of the last 4 years. This includes, in particular, refined enumerative geometry (using positive integer q-numbers instead of positive integer numbers), unexpected appearance of tropical curves in scaling limits of Abelian sandpile models, as well as a significant progress in more traditional areas of tropical research, such as tropical moduli spaces, tropical homology and tropical correspondence theorems

PROPERTIES OF H-SETS, KATETOV SPACES AND H-CLOSED EXTENSIONS WITH COUNTABLE REMAINDER

In this work we obtain results related to H-sets, Katetov spaces and H-closed extensions with countable remainder. As we shall see, these three areas are closely related but the results of each section carry their own definite flavor. Our first results concern finding cardinality bounds of H-sets in Urysohn spaces. In particular, a Urysohn space X is constructed which has an H-set A with |A| > 2 2ψ(X), where ψ(X) is the closed pseudocharacter of the space X. The space provides a counterexample to Fedeli's question in [16]. In addition, it is demonstrated that there is no θ-continuous map from a compact Hausdorff space to the space X with the H-set A as the image, giving a Urysohn counterexample to Vermeer's conjecture in [51]. Finally, it is shown that the cardinality of an H-set in a Urysohn space X is bounded by 2χ(X(s)), where χ(X) is the character of X and X(s) is the semiregularization of X. This refines Bella's result in [4] that the cardinality of such an H-set is bounded by 2χ(X). The next section concerns the relationship of H-sets and Katetov spaces. We recall that a Katetov space can be embedded as an H-set in some space. Herrlich showed in [23] that the space of rational numbers, Q, is not Katetov. Later Porter and Vermeer [41] refined this result with the fact that countable Katetov spaces are scattered. We obtain a similar refinement of Herrlich's result, and a generalization under an additional set-theoretic assumption. Our results include that a countable crowded space cannot be embedded as an H-set and that, assuming the Continuum Hypothesis, neither can the minimal η1 space. Chapter 4 investigates necessary and sufficient conditions for a space to have an H-closed extension with countable remainder. For countable spaces we are able to give two characterizations of those spaces admitting an H-closed extension with countable remainder. The general case appears more difficult, however, we arrive at a necessary condition - a generalization of Cech completeness, and several sufficient conditions for a space to have an H-closed extension with countable remainder. In particular, using the notation of Csaszar in [11], we show that a space X is a Cech g-space if and only if X is Gδ in σX or equivalently if EX is Cech complete. An example of a space which is a Cech f -space but not a Cech g-space is given answering a couple of questions of Csaszar. We show that if X is a Cech g-space and R(EX), the residue of EX, is Lindelof, then X has an H-closed extension with countable remainder. Finally, we investigate some natural extensions of the residue to the class of all Hausdorff spaces

Fractal, group theoretic, and relational structures on Cantor space

Cantor space, the set of infinite words over a finite alphabet, is a type of metric space with a `self-similar' structure. This thesis explores three areas concerning Cantor space with regard to fractal geometry, group theory, and topology. We find first results on the dimension of intersections of fractal sets within the Cantor space. More specifically, we examine the intersection of a subset E of the n-ary Cantor space, C[sub]n with the image of another subset Funder a random isometry. We obtain almost sure upper bounds for the Hausdorff and upper box-counting dimensions of the intersection, and a lower bound for the essential supremum of the Hausdorff dimension. We then consider a class of groups, denoted by V[sub]n(G), of homeomorphisms of the Cantor space built from transducers. These groups can be seen as homeomorphisms that respect the self-similar and symmetric structure of C[sub]n, and are supergroups of the Higman-Thompson groups V[sub]n. We explore their isomorphism classes with our primary result being that V[sub]n(G) is isomorphic to (and conjugate to) V[sub]n if and only if G is a semiregular subgroup of the symmetric group on n points. Lastly, we explore invariant relations on Cantor space, which have quotients homeomorphic to fractals in many different classes. We generalize a method of describing these quotients by invariant relations as an inverse limit, before characterizing a specific class of fractals known as Sierpiński relatives as invariant factors. We then compare relations arising through edge replacement systems to invariant relations, detailing the conditions under which they are the same

Proceedings of the 3rd International Workshop on Optimal Networks Topologies IWONT 2010

Peer Reviewe

Lower algebraic K-theory of certain reflection groups

For a finite volume geodesic polyhedron P in hyperbolic 3-space, with the property that all interior angles between incident faces are integral submultiples of Pi, there is a naturally associated Coxeter group generated by reflections in the faces. Furthermore, this Coxeter group is a lattice inside the isometry group of hyperbolic 3-space, with fundamental domain the original polyhedron P. In this paper, we provide a procedure for computing the lower algebraic K-theory of the integral group ring of such Coxeter lattices in terms of the geometry of the polyhedron P. As an ingredient in the computation, we explicitly calculate some of the lower K-groups of the dihedral groups and the product of dihedral groups with the cyclic group of order two.Comment: 35 pages, 2 figure

Extremality and stationarity of collections of sets : metric, slope and normal cone characterisations

Variational analysis, a relatively new area of research in mathematics, has become one of the most powerful tools in nonsmooth optimisation and neighbouring areas. The extremal principle, a tool to substitute the conventional separation theorem in the general nonconvex environment, is a fundamental result in variational analysis. There have seen many attempts to generalise the conventional extremal principle in order to tackle certain optimisation models. Models involving collections of sets, initiated by the extremal principle, have proved their usefulness in analysis and optimisation, with non-intersection properties (or their absence) being at the core of many applications: recall the ubiquitous convex separation theorem, extremal principle, Dubovitskii Milyutin formalism and various transversality/regularity properties. We study elementary nonintersection properties of collections of sets, making the core of the conventional definitions of extremality and stationarity. In the setting of general Banach/Asplund spaces, we establish nonlinear primal (slope) and linear/nonlinear dual (generalised separation) characterisations of these non-intersection properties. We establish a series of consequences of our main results covering all known formulations of extremality/ stationarity and generalised separability properties. This research develops a universal theory, unifying all the current extensions of the extremal principle, providing new results and better understanding for the exquisite theory of variational analysis. This new study also results in direct solutions for many open questions and new future research directions in the fields of variational analysis and optimisation. Some new nonlinear characterisations of the conventional extremality/stationarity properties are obtained. For the first time, the intrinsic transversality property is characterised in primal space without involving normal cones. This characterisation brings a new perspective on intrinsic transversality. In the process, we thoroughly expose and classify all quantitative geometric and metric characterisations of transversality properties of collections of sets and regularity properties of set-valued mappings.Doctor of Philosoph