Sergey Norin (McGill University)

Title: Brambles, stack number and topological overlap.

Abstract: A (strict) bramble in a graph G is a collection of subgraphs of G such that the union of any number of them is connected. The order of a bramble is the smallest size of a set of vertices that intersects each of the subgraphs in it. Brambles have long been part of the graph minor theory toolkit, in particular, because a bramble of high order is an obstruction to existence of a low width tree decomposition. We will discuss high dimensional analogues of brambles. In particular, we show that an d-dimensional bramble of high order in a d-dimensional simplicial complex X is an obstruction to existence of a low multiplicity continuous map from X to R^d (and more generally to any d-dimensional contractible complex). This can be seen as a qualitative variant of Gromov's topological overlap theorem. As an application, we construct the first explicit example of a graph family with bounded maximum degree and unbounded stack-number. Based in part on joint work with David Eppstein, Robert Hickingbotham, Laura Merker, Michał T. Seweryn and David R. Wood.