Theses - Mathematics
http://hdl.handle.net/1721.1/7842
Thu, 06 Aug 2015 15:36:31 GMT2015-08-06T15:36:31ZSome problems in Graph Ramsey Theory
http://hdl.handle.net/1721.1/97767
Some problems in Graph Ramsey Theory
Grinshpun, Andrey Vadim
A graph G is r-Ramsey minimal with respect to a graph H if every r-coloring of the edges of G yields a monochromatic copy of H, but the same is not true for any proper subgraph of G. The study of the properties of graphs that are Ramsey minimal with respect to some H and similar problems is known as graph Ramsey theory; we study several problems in this area. Burr, Erdös, and Lovász introduced s(H), the minimum over all G that are 2- Ramsey minimal for H of [delta](G), the minimum degree of G. We find the values of s(H) for several classes of graphs H, most notably for all 3-connected bipartite graphs which proves many cases of a conjecture due to Szabó, Zumstein, and Zürcher. One natural question when studying graph Ramsey theory is what happens when, rather than considering all 2-colorings of a graph G, we restrict to a subset of the possible 2-colorings. Erdös and Hajnal conjectured that, for any fixed color pattern C, there is some [epsilon] > 0 so that every 2-coloring of the edges of a Kn, the complete graph on n vertices, which doesn't contain a copy of C contains a monochromatic clique on n[epsilon] vertices. Hajnal generalized this conjecture to more than 2 colors and asked in particular about the case when the number of colors is 3 and C is a rainbow triangle (a K3 where each edge is a different color); we prove Hajnal's conjecture for rainbow triangles. One may also wonder what would happen if we wish to cover all of the vertices with monochromatic copies of graphs. Let F = {F₁, F₂, . . .} be a sequence of graphs such that Fn is a graph on n vertices with maximum degree at most [delta]. If each Fn is bipartite, then the vertices of any 2-edge-colored complete graph can be partitioned into at most 2C[delta] vertex disjoint monochromatic copies of graphs from F, where C is an absolute constant. This result is best possible, up to the constant C.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.; This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.; Cataloged from student-submitted PDF version of thesis.; Includes bibliographical references (pages 149-156).
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/1721.1/977672015-01-01T00:00:00ZConformal loop ensembles and the Gaussian free field
http://hdl.handle.net/1721.1/97319
Conformal loop ensembles and the Gaussian free field
Watson, Samuel Stewart, 1986-
The study of two-dimensional statistical physics models leads naturally to the analysis of various conformally invariant mathematical objects, such as the Gaussian free field, the Schramm-Loewner evolution, and the conformal loop ensemble. Just as Brownian motion is a scaling limit of discrete random walks, these objects serve as universal scaling limits of functions or paths associated with the underlying discrete models. We establish a new convergence result for percolation, a well-studied discrete model. We also study random sets of points surrounded by exceptional numbers of conformal loop ensemble loops and establish the existence of a random generalized function describing the nesting of the conformal loop ensemble. Using this framework, we study the relationship between Gaussian free field extrema and nesting extrema of the ensemble of Gaussian free field level loops. Finally, we describe a coupling between the set of all Gaussian free field level loops and a conformal loop ensemble growth process introduced by Werner and Wu. We prove that the dynamics are determined by the conformal loop ensemble in this coupling, and we use this result to construct a conformally invariant metric space.
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.; Cataloged from PDF version of thesis.; Includes bibliographical references (pages 173-178).
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/1721.1/973192015-01-01T00:00:00ZRewriting rules applied to the formation of strategy in games,
http://hdl.handle.net/1721.1/95542
Rewriting rules applied to the formation of strategy in games,
Lewis, Clayton
Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1968.; Bibliography: leaf 27.
Mon, 01 Jan 1968 00:00:00 GMThttp://hdl.handle.net/1721.1/955421968-01-01T00:00:00ZAn Erlanger program for combinatorial geometries.
http://hdl.handle.net/1721.1/95532
An Erlanger program for combinatorial geometries.
Kung, Joseph Pee Sin
Thesis. 1978. Ph.D.--Massachusetts Institute of Technology. Dept. of Mathematics.; MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE.; Vita.; Bibliography: leaves 132-137.
Sun, 01 Jan 1978 00:00:00 GMThttp://hdl.handle.net/1721.1/955321978-01-01T00:00:00Z