Georgia Institute of Technology College of Sciences

We start studying open book foliations in this series of seminars. We will go through the theory and see how it is used in applications to contact topology.

Robertson and Seymour proved that graphs are well-quasi-ordered by the minor relation and the weak immersion relation. In other words, given infinitely many graphs, one graph contains another as a minor (or a weak immersion, respectively). An application of these theorems is that every property that is closed under deleting vertices, edges, and contracting (or "splitting off", respectively) edges can be characterized by finitely many graphs, and hence can be decided in polynomial time. In this thesis we are concerned with the topological minor relation. We say that a graph G contains another graph H as a topological minor if H can be obtained from a subgraph of G by repeatedly deleting a vertex of degree two and adding an edge incident with the neighbors of the deleted vertex. Unlike the relation of minor and weak immersion, the topological minor relation does not well-quasi-order graphs in general. However, Robertson conjectured in the late 1980's that for every positive integer k, the topological minor relation well-quasi-orders graphs that do not contain a topological minor isomorphic to the path of length k with each edge duplicated. This thesis consists of two main results. The first one is a structure theorem for excluding a fixed graph as a topological minor, which is analogous to a cornerstone result of Robertson and Seymour, who gave such structure for graphs that exclude a fixed minor. Results for topological minors were previously obtained by Grohe and Marx and by Dvorak, but we push one of the bounds in their theorems to the optimal value. This improvement is needed for the next theorem. The second main result is a proof of Robertson's conjecture. As a corollary, properties on certain graphs closed under deleting vertices, edges, and "suppressing" vertices of degree two can be characterized by finitely many graphs, and hence can be decided in polynomial time.

We discuss asymptotics of multiple orthogonal polynomials with respect to Nikishin systems generated by two measures (\sigma_1, \sigma_2) with unbounded supports (supp(\sigma_1) \subset \mathbb{R}_+, supp(\sigma_2) \subset \mathbb{R}_-); moreover, the second measure \sigma_2 is discrete. We focus on deriving the strong and weak asymptotic for a special system of multiple OP from this class with respect to two Pollaczek type weights on \mathbb{R}_+. The weak asymptotic for these polynomials can be obtained by means of solution of an equilibrium problem. For the strong asymptotic we use the matrix Riemann-Hilbert approach.

We study the Legendre elliptic curve E: y^2=x(x+1)(x+t) over the field F_p(t) and its extensions K_d=F_p(mu_d*t^(1/d)). When d has the form p^f+1, in previous work we exhibited explicit points on E which generate a group V of large rank and finite index in the full Mordell-Weil group E(K_d), and we showed that the square of the index is the order of the Tate-Shafarevich group; moreover, the index is a power of p. In this talk we will explain how to use p-adic cohomology to compute the Tate-Shafarevich group and the quotient E(K_d)/V as modules over an appropriate group ring.