Georgia Institute of Technology College of Sciences

A subdivision of a graph G, also known as a topological G and denoted by TG, is a graph obtained from G by replacing certain edges of G with internally vertex-disjoint paths. This dissertation has two parts. The first part studies a structural problem and the second part studies an extremal problem. In the first part of this dissertation, we focus on TK_5, or subdivisions of K_5. A well-known theorem of Kuratowski in 1932 states that a graph is planar if, and only if, it does not contain a subdivision of K_5 or K_{3,3}. Wagner proved in 1937 that if a graph other than K_5 does not contain any subdivision of K_{3,3} then it is planar or it admits a cut of size at most 2. Kelmans and, independently, Seymour conjectured in the 1970s that if a graph does not contain any subdivision of K_5 then it is planar or it admits a cut of size at most 4. In this dissertation, we give a proof of the Kelmans-Seymour conjecture. We also discuss several related results and problems. The second part of this dissertation concerns subdivisions of large cliques in C_4-free graphs. Mader conjectured that every C_4-free graph with average degree d contains TK_l with l = \Omega(d). Komlos and Szemeredi reduced the problem to expanders and proved Mader's conjecture for n-vertex expanders with average degree d < exp( (log n)^(1/8) ). In this dissertation, we show that Mader's conjecture is true for n-vertex expanders with average degree d < n^0.3, which improves Komlos and Szemeredi's quasi-polynomial bound to a polynomial bound. As a consequence, we show that every C_4-free graph with average degree d contains a TK_l with l = \Omega(d/(log d)^c) for any c > 3/2. We note that Mader's conjecture has been recently verified by Liu and Montgomery.

The properties of euclidean space seem natural and obvious to us, to thepoint that it took mathematicians over two thousand years to see analternative to Euclid’s parallel postulate. The eventual discovery ofhyperbolic geometry in the 19th century shook our assumptions, revealingjust how strongly our native experience of the world blinded us fromconsistent alternatives, even in a field that many see as purelytheoretical. Non-euclidean spaces are still seen as unintuitive and exotic,but with direct immersive experiences we can get a better intuitive feel forthem. The latest wave of virtual reality hardware, in particular the HTCVive, tracks both the orientation and the position of the headset within aroom-sized volume, allowing for such an experience. We use this nacenttechnology to explore the three-dimensional geometries of theThurston/Perelman geometrization theorem. This talk focuses on oursimulations of H³ and H²×E.

The flow polytope associated to an acyclic graph is the set of all nonnegative flows on the edges of the graph with a fixed netflow at each vertex. We will examine flow polytopes arising from permutation matrices, alternating sign matrices and Tesler matrices. Our inspiration is the Chan-Robins-Yuen polytope (a face of the polytope of doubly-stochastic matrices), whose volume is equal to the product of the first n Catalan numbers (although there is no known combinatorial proof of this fact!). The volumes of the polytopes we study all have nice product formulas.

It is well known that periodic orbits give all the information about dynamical systems, at least for expanding maps, for which the periodic orbits are dense. This turns out to be true in dimensions 1 and 2, and false in dimension 4 or higher.We will present a proof that two $C^\infty$ expanding maps of the circle, which are topologically equivalent are $C^\infty$ conjugate if and only if the derivatives or the return map at periodic orbits are the same.