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Extremal problems in combinatorial geometry and Ramsey theory

Radoičić, Radoš, 1978-
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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45.52%
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

Hajiaghayi, MohammadTaghi
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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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
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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45.54%
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
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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45.52%
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
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 01/04/2013
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45.53%
In nature, one observes that a K-theory of an object is defined in two steps. First a "structured" category is associated to the object. Second, a K-theory machine is applied to the latter category to produce an infinite loop space. We develop a general framework that deals with the first step of this process. The K-theory of an object is defined via a category of "locally trivial" objects with respect to a pretopology. We study conditions ensuring an exact structure on such categories. We also consider morphisms in K-theory that such contexts naturally provide. We end by defining various K-theories of schemes and morphisms between them.; Comment: 31 pages

Twisted and untwisted K-theory quantization, and symplectic topology

Savelyev, Yasha; Shelukhin, Egor
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 27/08/2015
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45.53%
A prequantization space $(P,\alpha)$ is a principal $S^1$-bundle with a connection one-form $\alpha$ over a symplectic manifold $(M,\omega),$ with curvature given by the symplectic form. In particular $\alpha$ is a contact form. Using the theory of $Spin ^{c} $ Dirac quantization, we set up natural K-theory invariants of structure group $Cont _{0} (P, \alpha) $ fibrations of prequantization spaces. We further construct twisted K-theory invariants of Hamiltonian $(M, \omega)$ fibrations. As an application we prove that the natural map $BU(r) \to BU$ of classifying spaces factors as $BU(r) \to B \mathcal{Q}(r \to BU,$ where $\mathcal{Q}(r)=Cont_0(S^{2r-1},\alpha_{std})$ for the standard contact form on the odd-dimensional sphere. As a corollary we show that the natural map $BU(r) \rightarrow B\mathcal{Q}(r)$ induces a surjection on complex K-theory and on integral cohomology, strengthening a theorem of Spacil \cite{SpacilThesis,CasalsSpacil} for rational cohomology, and that it induces an injection on integral homology. Furthermore, we improve a theorem of Reznikov, showing the injectivity on homotopy groups of the natural map $BU (r) \to B \text{Ham} (\mathbb{CP} ^{r-1}, \omega )$, in the stable range. Finally, we produce examples of non-trivial $\mathcal{Q}(r)$ and $\text{Ham} (\mathbb{CP} ^{r-1}...

Algebraic K-theory is stable and admits a multiplicative structure for module objects

Devalapurkar, Sanath
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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45.53%
After recognizing higher homotopy coherences, algebraic K-theory can be regarded as a functor from stable $\infty$-categories to $\infty$-categories. We establish the stability theoremm which states that the algebraic K-theory of a stable $\infty$-category is actually a stable $\infty$-category itself. This is a generalization of the statement that algebraic K-theory is a functor from spectra to spectra. We then prove a result which provides a simpler interpretation of the algebraic K-theory of ring spectra. In order to do this, we compute the algebraic K-theory of an $\infty$-category of modules, and establish that it is an $\infty$-category of modules itself. This result, known as the multiplicativity theorem, vastly generalizes results obtained by Elmendorf and Mandell. Since the algebraic K-theory of a ring spectrum $R$ is the algebraic K-theory of the $\infty$-category of perfect modules over $R$, this provides a simpler interpretation of the algebraic K-theory of ring spectra. Using this result, we prove an $\infty$-categorical counterpart of the derived Morita context for flat rings, which shows that algebraic K-theory is a homotopy coherent version of Morita theory.; Comment: 12 pages. Any updates to this version of the paper will be available from the author's webpage. Comments are welcome!

Connective algebraic K-theory

Dai, Shouxin; Levine, Marc
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 02/12/2012
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45.53%
We examine the theory of connective algebraic K-theory, CK, defined by taking the -1 connective cover of algebraic K-theory with respect to Voevodsky's slice tower in the motivic stable homotopy category. We extend CK to a bi-graded oriented duality theory (CK', CK) in case the base scheme is the spectrum of a field k of characteristic zero. The homology theory CK' may be viewed as connective algebraic G-theory. We identify CK' theory in bi-degree (2n, n) on some finite type k-scheme X with the image of K_0(M(X,n)) in K_0(M(X, n+1)), where M(X,n) is the abelian category of coherent sheaves on X with support in dimension at most n; this agrees with the (2n,n) part of the theory defined by Cai. We also show that the classifying map from algebraic cobordism identifies CK' with the universal oriented Borel-Morel homology theory \Omega_*^{CK}:=\Omega_*\otimes_L\Z[\beta] having formal group law u+v-\beta uv with coefficient ring \Z[\beta]. As an application, we show that every pure dimension d finite type k scheme has a well-defined fundamental class [X] in \Omega_d^{CK}(X), and this fundamental class is functorial with respect to pull-back for lci morphisms. Finally, the fundamental class maps to the fundamental class in G-theory after inverting \beta...

The Eta-invariant and Pontryagin duality in K-theory

Savin, A. Yu.; Sternin, B. Yu.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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45.53%
The topological significance of the spectral Atiyah-Patodi-Singer eta-invariant is investigated under the parity conditions of P. Gilkey. We show that twice the fractional part of the invariant is computed by the linking pairing in K-theory with the orientation bundle of the manifold. The Pontrjagin duality implies the nondegeneracy of the linking form. An example of a nontrivial fractional part for an even-order operator is presented. This result answers the question of P. Gilkey (1989) concerning the existence of even-order operators on odd-dimensional manifolds with nontrivial fractional part of eta-invariant.; Comment: 24 pages, 1 figure; final version; see http://www.kluweronline.com/issn/0001-4346/contents

Algebraic K-theory of strict ring spectra

Rognes, John
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 24/03/2014
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45.54%
We view strict ring spectra as generalized rings. The study of their algebraic K-theory is motivated by its applications to the automorphism groups of compact manifolds. Partial calculations of algebraic K-theory for the sphere spectrum are available at regular primes, but we seek more conceptual answers in terms of localization and descent properties. Calculations for ring spectra related to topological K-theory suggest the existence of a motivic cohomology theory for strictly commutative ring spectra, and we present evidence for arithmetic duality in this theory. To tie motivic cohomology to Galois cohomology we wish to spectrally realize ramified extensions, which is only possible after mild forms of localization. One such mild localization is provided by the theory of logarithmic ring spectra, and we outline recent developments in this area.; Comment: Contribution to the proceedings of the ICM 2014 in Seoul

A Theory of Adjoint Functors--with some Thoughts about their Philosophical Significance

Ellerman, David
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 15/11/2005
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45.54%
The question "What is category theory" is approached by focusing on universal mapping properties and adjoint functors. Category theory organizes mathematics using morphisms that transmit structure and determination. Structures of mathematical interest are usually characterized by some universal mapping property so the general thesis is that category theory is about determination through universals. In recent decades, the notion of adjoint functors has moved to center-stage as category theory's primary tool to characterize what is important and universal in mathematics. Hence our focus here is to present a theory of adjoint functors, a theory which shows that all adjunctions arise from the birepresentations of "chimeras" or "heteromorphisms" between the objects of different categories. Since representations provide universal mapping properties, this theory places adjoints within the framework of determination through universals. The conclusion considers some unreasonably effective analogies between these mathematical concepts and some central philosophical themes.; Comment: 58 pages. Forthcoming in: What is Category Theory? Giandomenico Sica ed., Milan: Polimetrica

On determinant functors and $K$-theory

Muro, Fernando; Tonks, Andrew; Witte, Malte
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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45.53%
In this paper we introduce a new approach to determinant functors which allows us to extend Deligne's determinant functors for exact categories to Waldhausen categories, (strongly) triangulated categories, and derivators. We construct universal determinant functors in all cases by original methods which are interesting even for the known cases. Moreover, we show that the target of each universal determinant functor computes the corresponding $K$-theory in dimensions 0 and 1. As applications, we answer open questions by Maltsiniotis and Neeman on the $K$-theory of (strongly) triangulated categories and a question of Grothendieck to Knudsen on determinant functors. We also prove additivity theorems for low-dimensional $K$-theory and obtain generators and (some) relations for various $K_{1}$-groups.; Comment: 73 pages. We have deeply revised the paper to make it more accessible, it contains now explicit examples and computations. We have removed the part on localization, it was correct but we didn't want to make the paper longer and we thought this part was the less interesting one. Nevertheless it will remain here in the arXiv, in version 1. If you need it in your research, please let us know

The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in hermitian K-theory

Berrick, A. J.; Karoubi, M.; Schlichting, M.; Østvær, P. A.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.53%
Let X be a noetherian scheme of finite Krull dimension, having 2 invertible in its ring of regular functions, an ample family of line bundles, and a global bound on the virtual mod-2 cohomological dimensions of its residue fields. We prove that the comparison map from the hermitian K-theory of X to the homotopy fixed points of K-theory under the natural Z/2-action is a 2-adic equivalence in general, and an integral equivalence when X has no formally real residue field. We also show that the comparison map between the higher Grothendieck-Witt (hermitian K-) theory of X and its \'etale version is an isomorphism on homotopy groups in the same range as for the Quillen-Lichtenbaum conjecture in K-theory. Applications compute higher Grothendieck-Witt groups of complex algebraic varieties and rings of 2-integers in number fields, and hence values of Dedekind zeta-functions.; Comment: 17 pages, to appear in Adv. Math

K-Theory in Quantum Field Theory

Freed, Daniel S.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 18/06/2002
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45.54%
We survey three different ways in which K-theory in all its forms enters quantum field theory. In Part 1 we give a general argument which relates topological field theory in codimension two with twisted K-theory, and we illustrate with some finite models. Part 2 is a review of pfaffians of Dirac operators, anomalies, and the relationship to differential K-theory. Part 3 is a geometric exposition of Dirac charge quantization, which in superstring theories also involves differential K-theory. Parts 2 and 3 are related by the Green-Schwarz anomaly cancellation mechanism. An appendix, joint with Jerry Jenquin, treats the partition function of Rarita-Schwinger fields.; Comment: 56 pages, expanded version of lectures at "Current Developments in Mathematics"

Comparison between algebraic and topological K-theory of locally convex algebras

Cortiñas, Guillermo; Thom, Andreas
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.54%
This paper is concerned with the algebraic K-theory of locally convex algebras stabilized by operator ideals, and its comparison with topological K-theory. We show that the obstruction for the comparison map between algebraic and topological K-theory to be an isomorphism is (absolute) algebraic cyclic homology and prove the existence of an 6-term exact sequence. We show that cyclic homology vanishes in the case when J is the ideal of compact operators and L is a Frechet algebra with bounded app. unit. This proves the generalized version of Karoubi's conjecture due to Mariusz Wodzicki and announced in his paper "Algebraic K-theory and functional analysis", First European Congress of Mathematics, Vol. II (Paris, 1992), 485--496, Progr. Math., 120, Birkh\"auser, Basel, 1994. We also consider stabilization with respect to a wider class of operator ideals, called sub-harmonic. We study the algebraic K-theory of the tensor product of a sub-harmonic ideal with an arbitrary complex algebra and prove that the obstruction for the periodicity of algebraic K-theory is again cyclic homology. The main technical tools we use are the diffeotopy invariance theorem of Cuntz and the second author (which we generalize), and the excision theorem for infinitesimal K-theory...

Homotopy invariance of higher K-theory for abelian categories

Mochizuki, Satoshi; Sannai, Akiyoshi
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.53%
The main theorem in this paper is that the base change functor from a noetherian abelian category to its noetherian polynomial category induces an isomorphism on K-theory. The main theorem implies the well-known fact that A^1-homotopy invariance of K'-theory for noetherian schemes.; Comment: arXiv admin note: substantial text overlap with arXiv:1104.4240

Unified Foundations for Mathematics

Burgin, Mark
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 10/03/2004
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45.57%
There are different meanings of foundation of mathematics: philosophical, logical, and mathematical. Here foundations are considered as a theory that provides means (concepts, structures, methods etc.) for the development of whole mathematics. Set theory has been for a long time the most popular foundation. However, it was not been able to win completely over its rivals: logic, the theory of algorithms, and theory of categories. Moreover, practical applications of mathematics and its inner problems caused creation of different generalization of sets: multisets, fuzzy sets, rough sets etc. Thus, we encounter a problem: Is it possible to find the most fundamental structure in mathematics? The situation is similar to the quest of physics for the most fundamental "brick" of nature and for a grand unified theory of nature. It is demonstrated that in contrast to physics, which is still in search for a unified theory, in mathematics such a theory exists. It is the theory of named sets.

Almost ring theory - sixth release

Gabber, Ofer; Ramero, Lorenzo
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.55%
We develop almost ring theory, which is a domain of mathematics somewhere halfway between ring theory and category theory (whence the difficulty of finding appropriate MSC-class numbers). We apply this theory to valuation theory and to p-adic analytic geometry. You should really have a look at the introductions (each chapter has one).; Comment: This is the sixth - and assuredly final - release of "Almost ring theory". It is about 230 page long; it is written in AMSLaTeX and uses XYPic and a few not so standard fonts. Any future corrections (mainly typos, I expect) will be found on my personal web page: http://www.math.u-bordeaux.fr/~ramero/

Homotopy Type Theory: Univalent Foundations of Mathematics

Program, The Univalent Foundations
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 03/08/2013
Relevância na Pesquisa
45.57%
Homotopy type theory is a new branch of mathematics, based on a recently discovered connection between homotopy theory and type theory, which brings new ideas into the very foundation of mathematics. On the one hand, Voevodsky's subtle and beautiful "univalence axiom" implies that isomorphic structures can be identified. On the other hand, "higher inductive types" provide direct, logical descriptions of some of the basic spaces and constructions of homotopy theory. Both are impossible to capture directly in classical set-theoretic foundations, but when combined in homotopy type theory, they permit an entirely new kind of "logic of homotopy types". This suggests a new conception of foundations of mathematics, with intrinsic homotopical content, an "invariant" conception of the objects of mathematics -- and convenient machine implementations, which can serve as a practical aid to the working mathematician. This book is intended as a first systematic exposition of the basics of the resulting "Univalent Foundations" program, and a collection of examples of this new style of reasoning -- but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant.; Comment: 465 pages. arXiv v1: first-edition-257-g5561b73...

Categorical Foundations for K-Theory

Michel, Nicolas
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 14/11/2011
Relevância na Pesquisa
45.54%
Recall that the definition of the $K$-theory of an object C (e.g., a ring or a space) has the following pattern. One first associates to the object C a category A_C that has a suitable structure (exact, Waldhausen, symmetric monoidal, ...). One then applies to the category A_C a "$K$-theory machine", which provides an infinite loop space that is the $K$-theory K(C) of the object C. We study the first step of this process. What are the kinds of objects to be studied via $K$-theory? Given these types of objects, what structured categories should one associate to an object to obtain $K$-theoretic information about it? And how should the morphisms of these objects interact with this correspondence? We propose a unified, conceptual framework for a number of important examples of objects studied in $K$-theory. The structured categories associated to an object C are typically categories of modules in a monoidal (op-)fibred category. The modules considered are "locally trivial" with respect to a given class of trivial modules and a given Grothendieck topology on the object C's category.; Comment: 176 + xi pages. This monograph is a revised and augmented version of my PhD thesis. The official thesis is available at http://library.epfl.ch/en/theses/?nr=4861