Georgia Institute of Technology College of Sciences

We discuss bi-parameter Calderon-Zygmund singular integrals from the point of view of modern probabilistic and dyadic techniques. In particular, we discuss their structure and boundedness via dyadic model operators. In connection to this we demonstrate, via new examples, the delicacy of the problem of finding a completely satisfactory product T1 theorem. Time permitting related non-homogeneous bi-parameter results may be mentioned.

We prove Ruelle's Entropy Inequality for C^1 maps. This is part of a reading seminar geared towards understanding of Smooth Ergodic Theory. (The study of dynamical systems using at the same time tools from measure theory and from differential geometry)It should be accesible to graduate students and the presentation is informal. The first goal will be a proof of the Oseledets multiplicative ergodic theorem for random matrices. Then, we will try to cover the Pesin entropy formula, invariant manifolds, etc.

In this thesis we study topology of symplectic fillings of contact manifolds supported by planar open books. We obtain results regarding geography of the symplectic fillings of these contact manifolds. Specifically, we prove that if a contact manifold $(M,\xi)$ is supported by a planar open book, then Euler characteristic and signature of any Stein filling of $(M,\xi)$ is bounded. We also prove a similar finiteness result for contact manifolds supported by spinal open books with planar pages. Moving beyond the geography of Stein fillings, we classify fillings of some lens spaces.In addition, we classify Stein fillings of an infinite family of contact 3-manifolds up to diffeomorphism. Some contact 3-manifolds in this family can be obtained by Legendrian surgeries on $(S^3,\xi_{std})$ along certain Legendrian 2-bridge knots. We also classify Stein fillings, up to symplectic deformation, of an infinite family of contact 3-manifolds which can be obtained by Legendrian surgeries on $(S^3,\xi_{std})$ along certain Legendrian twist knots. As a corollary, we obtain a classification of Stein fillings of an infinite family of contact hyperbolic 3-manifolds up to symplectic deformation.

The first result of this thesis is a partial result in the direction of Steinberg's Conjecture. Steinberg's Conjecture states that any planar graph without cycles of length four or five is three colorable. Borodin, Glebov, Montassier, and Raspaud showed that planar graphs without cycles of length four, five, or seven are three colorable and Borodin and Glebov showed that planar graphs without five cycles or triangles at distance at most two apart are three colorable. We prove a statement that implies the first of these theorems and is incomparable with the second: that any planar graph with no cycles of length four through six or cycles of length seven with incident triangles distance exactly two apart are three colorable. We are next concerned with the study of Pfaffian orientations. A theorem proved by William McCuaig and, independently, Neil Robertson, Paul Seymour, and Robin Thomas provides a good characterization for whether or not a bipartite graph has a Pfaffian orientation as well as a polynomial time algorithm for that problem. We reprove this characterization and provide a new algorithm for this problem. First, we generalize a preliminary result needed to reprove this theorem. Specifically, we show that any internally 4-connected, non-planar bipartite graph contains a subdivision of K3,3 in which each path has odd length. We then make use of this result to provide a much shorter proof of this characterization using elementary methods. In the final piece of the thesis we investigate flat embeddings. A piecewise-linear embedding of a graph in 3-space is flat if every cycle of the graph bounds a disk disjoint from the rest of the graph. We first provide a structural theorem for flat embeddings that indicates how to build them from small pieces. We then present a class of flat graphs that are highly non-planar in the sense that, for any fixed k, there are an infinite number of members of the class such that deleting k vertices leaves the graph non-planar.