List of abstracts
The lectures will present the structural characterizations of several graph classes defined by distance properties, which have in part a general algebraic and geometric flavour and can be interpreted as subdirect decomposition. Some of these classes of graphs give raise to cubical, simplicial, or polyhedral complexes with interesting structural properties. Other graph classes are related to some constructions in general metric spaces or occur in the definition of some basic combinatorial structures.
The graphs we feature in the first place are the median graphs and their various kinds of generalizations, e.g., weakly modular graphs, or fiber-complemented graphs, or l1-graphs. Several kinds of l1-graphs admit natural geometric realizations as polyhedral complexes. Particular instances of these graphs also occur in other geometric contexts, for example, as dual polar graphs, basis graphs of Delta-matroids, tope graphs, lopsided sets, or plane graphs with vertex degrees and face sizes bounded from below. Several other classes of graphs, e.g., Helly graphs (as injective objects), or bridged graphs, or tree-like graphs occur in the investigation of graphs satisfying some basic properties of the distance function, such as the Helly property for balls, or the convexity of balls or of the neighborhoods of convex sets, etc.
(Extended) keywords: median graphs and cubings, isometric subgraphs of hypercubes and lopsided sets, l1-graphs, bridged and weakly bridged graphs and systolic and weakly systolic complexes, Helly graphs, basis graphs of matroids and Delta-matroids, weakly modular graphs, fiber complemented graphs, retracts of Cartesian products of chordal, bridged, and weakly bridged graphs.
We present recent applications of curvature notions to Discrete Topology, Polytope Combinatorics and Tiling Theory, using geometric inequalities for CAT Spaces, the Cartan-Hadamard-Alexandrov Theorem, Hyperbolic Dehn Surgery, and Morse Theory.
Classical Morse Theory studies a smooth manifold by analyzing smooth functions defined on it. Discrete Morse theory studies a simplicial complex (for example, a PL triangulation of a smooth manifold) by analizing weakly-increasing functions defined on its face poset. We shed light on the gap between the two notions, in connection with the desire of finding a minimal discrete Morse function algorithmically. We also highlight what we believe to be a dangerous 'systematic error' in the current methodology: The test examples typically used to test algorithms for discrete Morse theory are mostly ``too simple'', both from a topological and a combinatorial point of view.
We will investigate the ways in which Out(Fn) can act on conjugacy classes of a free group Fm, and conclude that if n < m < n(n-1)/2 then such an action factors through the finit group of order 2. In the process we will develop some representation theory of Out(Fn) as well as some facts about groups closely related to alternating groups acting on finite graphs.
A quasi-automorphism of a colored graph is a bijection f of the set of vertices with the property, that both f and its inverse respect all but finitely many colored edges of each color. Thus the set of all quasi-automorphisms of a fixed graph forms a subgroup of the group of all permutations of the set of vertices. We will discuss the groups of quasi-automorphims of quasi-automorphisms of the rooted binary tree, which turns out out to be a very interesting group for several reasons. One reason is the strong relationship to Thompson's groups. In this talk we will focus on the importance of these groups for the study of complexity in decision problems in group theory.
Many problems in geometry and combinatorics contain a natural symmetry such that one can try to apply equivariant algebraic topology methods to handle them. This talk will contain a very quick survey on a few methods and examples to illustrate that. In particular we will explain a recent result, a colored Tverberg theorem for manifolds, which is joint work with Pavle Blagojević and Günter Ziegler.