Georgia Institute of Technology College of Sciences

We study some finite quotients of the A_n Milnor fibre which coincide with the Stein surfaces that appear in Fintushel and Stern's rational blowdown construction. We show that these Stein surfaces have no exact Lagrangian submanifolds by using the already available and deep understanding of the Fukaya category of the A_n Milnor fibre coming from homological mirror symmetry. On the contrary, we find Floer theoretically essential monotone Lagrangian tori, finitely covered by the monotone tori that we studied in the A_n Milnor fibre. We conclude that these Stein surfaces have non-vanishing symplectic cohomology. This is joint work with M. Maydanskiy.

A hereditary chip-firing model is a chip-firing game whose dynamics are induced by an abstract simplicial complex \Delta on the vertices of a graph, where \Delta may be interpreted as encoding graph gluing data. These models generalize two classical chip-firing games: The Abelian sandpile model, where \Delta is disjoint collection of points, and the cluster firing model, where \Delta is a single simplex. Two fundamental properties of these classical models extend to arbitrary hereditary chip-firing models: stabilization is independent of firings chosen (the Abelian property) and each chip-firing equivalence class contains a unique recurrent configuration. We will present an explicit bijection between the recurrent configurations of a hereditary chip-firing model on a graph G and the spanning trees of G and, if time permits, we will discuss chip-firing via gluing in the context of binomial ideals and metric graphs.

Over the past 40 years, researchers have made many connections between the dimension of posets and the issue of planarity for graphs and diagrams, but there appears to be little work connecting dimension to structural graph theory. This situation has changed dramatically in the last several months. At the Robin Thomas birthday conference, Gwenael Joret, made the following striking conjecture, which has now been turned into a theorem: The dimension of a poset is bounded in terms of its height and the tree-width of its cover graph. In this talk, I will present the proof of this result. The general contours of the argument should be accessible to graph theorists and combinatorists (faculty and students) without deep knowledge of either dimension or tree-width. The proof of the theorem was accomplished by a team of six researchers: Gwenael Joret, Piotr Micek, Kevin Milans, Tom Trotter, Bartosz Walczak and Ruidong Wang.

We will give an overview of open book foliation method by emphasizing the aspect that it is a generalization of Birman-Menasco's braid foliation theory. We explain how surfaces in open book reflects topology and (contact) geometry of underlying 3-manifolds, and will give several applications. This talk is based on joint work with Keiko Kawamuro.