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## Extremal problems in combinatorial geometry and Ramsey theory

Fonte: Massachusetts Institute of Technology Publicador: Massachusetts Institute of Technology
Tipo: Tese de Doutorado Formato: 223 p.; 10549429 bytes; 10546048 bytes; application/pdf; application/pdf
ENG
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The work presented in this thesis falls under the broad umbrella of combinatorics of Erd's type. We describe diverse facets of interplay between geometry and combinatorics and consider several questions about existence of structures in various combinatorial settings. We make contributions to specific problems in combinatorial geometry, Ramsey theory and graph theory. We first study extremal questions in geometric graph theory, that is, the existence of collections of edges with a specified crossing pattern in drawings of graphs in the plane with sufficiently many edges. Among other results, we prove that any drawing of a graph on n vertices and Cn edges, where C is a sufficiently large constant, contains each of the following crossing patterns: (1) three pairwise crossing edges, (2) two edges that cross and are crossed by k other edges, (3) an edge crossed by four other edges. In the latter, we show that C = 5.5 is the best possible constant, which, through Szekely's method, gives the best known value for a constant in the well known "Crossing Lemma" due to Ajtai, Chvatal, Leighton, Newborn and Szemeredi. After relaxing graph planarity in several ways, we proceed to study ... the maximum number of edges in a drawing of a graph on n vertices without self-crossing copy of C4...

## The bidimensionality theory and its algorithmic applications

Fonte: Massachusetts Institute of Technology Publicador: Massachusetts Institute of Technology
Tipo: Tese de Doutorado Formato: 219 p.; 11756236 bytes; 11770363 bytes; application/pdf; application/pdf
ENG
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45.52%
Our newly developing theory of bidimensional graph problems provides general techniques for designing efficient fixed-parameter algorithms and approximation algorithms for NP- hard graph problems in broad classes of graphs. This theory applies to graph problems that are bidimensional in the sense that (1) the solution value for the k x k grid graph (and similar graphs) grows with k, typically as Q(k²), and (2) the solution value goes down when contracting edges and optionally when deleting edges. Examples of such problems include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex- removal parameters, dominating set, edge dominating set, r-dominating set, connected dominating set, connected edge dominating set, connected r-dominating set, and unweighted TSP tour (a walk in the graph visiting all vertices). Bidimensional problems have many structural properties; for example, any graph embeddable in a surface of bounded genus has treewidth bounded above by the square root of the problem's solution value. These properties lead to efficient-often subexponential-fixed-parameter algorithms, as well as polynomial-time approximation schemes, for many minor-closed graph classes. One type of minor-closed graph class of particular relevance has bounded local treewidth...

## Stringy K-theory and the Chern character

Jarvis, Tyler J.; Kaufmann, Ralph; Kimura, Takashi
Tipo: Artigo de Revista Científica
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For a finite group G acting on a smooth projective variety X, we construct two new G-equivariant rings: first the stringy K-theory of X, and second the stringy cohomology of X. For a smooth Deligne-Mumford stack Y we also construct a new ring called the full orbifold K-theory of Y. For a global quotient Y=[X/G], the ring of G-invariants of the stringy K-theory of X is a subalgebra of the full orbifold K-theory of the the stack Y and is linearly isomorphic to the orbifold K-theory'' of Adem-Ruan (and hence Atiyah-Segal), but carries a different, quantum,'' product, which respects the natural group grading. We prove there is a ring isomorphism, the stringy Chern character, from stringy K-theory to stringy cohomology, and a ring homomorphism from full orbifold K-theory to Chen-Ruan orbifold cohomology. These Chern characters satisfy Grothendieck-Riemann-Roch for etale maps. We prove that stringy cohomology is isomorphic to Fantechi and Goettsche's construction. Since our constructions do not use complex curves, stable maps, admissible covers, or moduli spaces, our results simplify the definitions of Fantechi-Goettsche's ring, of Chen-Ruan's orbifold cohomology, and of Abramovich-Graber-Vistoli's orbifold Chow. We conclude by showing that a K-theoretic version of Ruan's Hyper-Kaehler Resolution Conjecture holds for symmetric products. Our results hold both in the algebro-geometric category and in the topological category for equivariant almost complex manifolds.; Comment: Exposition improved and additional details provided. To appear in Inventiones Mathematicae

## Algebraic cobordism theory attached to algebraic equivalence

Krishna, Amalendu; Park, Jinhyun
Tipo: Artigo de Revista Científica
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Based on the algebraic cobordism theory of Levine and Morel, we develop a theory of algebraic cobordism modulo algebraic equivalence. We prove that this theory can reproduce Chow groups modulo algebraic equivalence and the semi-topological $K_0$-groups. We also show that with finite coefficients, this theory agrees with the algebraic cobordism theory. We compute our cobordism theory for some low dimensional varieties. The results on infinite generation of some Griffiths groups by Clemens and on smash-nilpotence by Voevodsky and Voisin are also lifted and reinterpreted in terms of this cobordism theory.; Comment: 30 pages. A version of this article was accepted to appear in J. K-theory

## Parametrized K-Theory

Michel, Nicolas
Tipo: Artigo de Revista Científica