Main Speakers

Invited Speakers

Contributed Speakers

The fundamental group of a random clique complex

Abstract: Random objects often have desirable properties for which explicit examples are hard to construct. For example often random graphs are good expanders and have strong Ramsey properties. The most well-studied model of random graphs is the Erdos-Renyi model G(n,p). In the G(n,p) model one generates a random graph with n vertices by adding each possible edge with independent probability p. Properties of random graphs are often studied asymptotically, ie by having n tend to infinity and the probability parameter p depend on n. A pioneering result of Erdos and Renyi establishes the threshold, ie the critical p(n), for a random graph to be connected with probability tending to one.

In this talk we will study a model of random simplicial complexes introduced recently by M. Kahle. This model is known as the random clique complex model. Here a random complex is generated by first generating a random graph G in the Erdos-Renyi model and subsequently adding the faces spanned by complete subraphs (cliques) of G. Unlike in the graph setup, one can study several interesting topological properties of random complexes. We will focus on properties of the fundamental group of a random clique complex. This is joint work with M. Farber and D. Horak.

Directed Topology - dicoverings. Top versus dTop.

Abstract: Directed topology is a new mathematical area inspired by and with applications to concurrency. A topological space is directed by choosing a a subset of its paths, called the directed paths. Maps have to be continuous and respect this choice - directed paths are mapped to directed paths. In concurrency, algorithms and tools to investigate programs without loops are quite well developed in this geometric setting, and it is obvious from a topological viewpoint to try to get rid off loops by taking a (directed) covering. This talk will give examples and illustrate in what respect the usual definition actually works and where new ideas have to come in. In particular, the focus has to be on lifting properties, not the usual discrete fibration properties, which in the undirected setting imply lifting. The usual focus on connected spaces has to be modified, and again
not in an obvious way. This tour through directed coverings will highlight many of the unusual and surprising features of directed topology.

Hopf invariants for sectional category with applications to
Topological Robotics

Abstract: Topological complexity is a numerical homotopy invariant of spaces. It was defined by Farber as part of his topological study of the motion planning problem in Robotics. After reviewing the definition and basic properties, we will introduce refined homotopy-theoretic tools for the estimation of topological complexity, and more generally sectional category.
These generalized Hopf invariants satisfy a sort of product formula, generalizing an observation originally due to N. Iwase. We will give applications to calculating the topological complexity of two-cell complexes and to the analogue for topological complexity of Ganea's conjecture on Lusternik-Schnirelmann category. This is joint work with Jesús González and Lucile Vandembroucq. (pdf)

Directed algebraic topology of higher-dimensional automata

Abstract: Higher-dimensional automata constitute one of the most expressive models for concurrent systems. By definition, an HDA is a precubical set (i.e., a cubical set without degeneracies) with labels on edges. An important practical problem in concurrency theory is the fact that models of systems can easily become very large. This is called the state explosion problem. In this talk, I will discuss topological abstraction of higher-dimensional automata, i.e, the replacement of an HDA by a smaller one that is weakly equivalent from the point of view directed algebraic topology and models the same system.