Georgia Institute of Technology College of Sciences

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.

In this talk we will connect functional analysis and analytic function theory by studying the compact linear operators on Bergman spaces. In particular, we will show how it is possible to obtain a characterization of the compact operators in terms of more geometric information associated to the function spaces. We will also point to several interesting lines of inquiry that are connected to the problems in this talk. This talk will be self-contained and accessible to any mathematics graduate student.

I will be presenting the paper by S. Mendelson titled 'Rademacher Averages and Phase Transitions in Glivenko–Cantelli Classes'. Fat-shattering dimension and its use in characterizing GC classes will be introduced. Then a new quantity based on the growth rate of the Rademacher averages will be introduced. This parameter enables one to provide improved complexity estimates for the agnostic learning problem with respect to any norm. A phase transition phenomenon that appears in the sample complexity estimates, covering numbers estimates, and in the growth rate of the Rademacher averages will be discussed. Further (i) estimates on the covering numbers of a class when considered as a subset of spaces and (ii) estimate the fat-shattering dimension of the convex hull of a given class will be discussed.