4,613 research outputs found
Differential KO-theory: constructions, computations, and applications
We provide a systematic and detailed treatment of differential refinements of
KO-theory. We explain how various flavors capture geometric aspects in
different but related ways, highlighting the utility of each. While general
axiomatics exist, no explicit constructions seem to have appeared before. This
fills a gap in the literature in which K-theory is usually worked out leaving
KO-theory essentially untouched, with only scattered partial information in
print. We compare to the complex case, highlighting which constructions follow
analogously and which are much more subtle. We construct a pushforward and
differential refinements of genera, leading to a Riemann-Roch theorem for
-theory. We also construct the corresponding
Atiyah-Hirzebruch spectral sequence (AHSS) and explicitly identify the
differentials, including ones which mix geometric and topological data. This
allows us to completely characterize the image of the Pontrjagin character.
Then we illustrate with examples and applications, including higher tangential
structures, Adams operations, and a differential Wu formula.Comment: 105 pages, very minor changes, comments welcom
Effective statistical physics of Anosov systems
We present evidence indicating that Anosov systems can be endowed with a
unique physically reasonable effective temperature. Results for the two
paradigmatic Anosov systems (i.e., the cat map and the geodesic flow on a
surface of constant negative curvature) are used to justify a proposal for
extending Ruelle's thermodynamical formalism into a comprehensive theory of
statistical physics for nonequilibrium steady states satisfying the
Gallavotti-Cohen chaotic hypothesis.Comment: 38 pages, 17 figures. Substantially more details in sections 4 and 6;
new and revised figures also added. Typos and minor errors (esp. in section
6) corrected along with minor notational changes. MATLAB code for
calculations in section 16 also included as inline comment in TeX source now.
The thrust of the paper is unaffecte
Convex Hull Realizations of the Multiplihedra
We present a simple algorithm for determining the extremal points in
Euclidean space whose convex hull is the nth polytope in the sequence known as
the multiplihedra. This answers the open question of whether the multiplihedra
could be realized as convex polytopes. We use this realization to unite the
approach to A_n-maps of Iwase and Mimura to that of Boardman and Vogt. We
include a review of the appearance of the nth multiplihedron for various n in
the studies of higher homotopy commutativity, (weak) n-categories,
A_infinity-categories, deformation theory, and moduli spaces. We also include
suggestions for the use of our realizations in some of these areas as well as
in related studies, including enriched category theory and the graph
associahedra.Comment: typos fixed, introduction revise
The categorical limit of a sequence of dynamical systems
Modeling a sequence of design steps, or a sequence of parameter settings,
yields a sequence of dynamical systems. In many cases, such a sequence is
intended to approximate a certain limit case. However, formally defining that
limit turns out to be subject to ambiguity. Depending on the interpretation of
the sequence, i.e. depending on how the behaviors of the systems in the
sequence are related, it may vary what the limit should be. Topologies, and in
particular metrics, define limits uniquely, if they exist. Thus they select one
interpretation implicitly and leave no room for other interpretations. In this
paper, we define limits using category theory, and use the mentioned relations
between system behaviors explicitly. This resolves the problem of ambiguity in
a more controlled way. We introduce a category of prefix orders on executions
and partial history preserving maps between them to describe both discrete and
continuous branching time dynamics. We prove that in this category all
projective limits exist, and illustrate how ambiguity in the definition of
limits is resolved using an example. Moreover, we show how various problems
with known topological approaches are now resolved, and how the construction of
projective limits enables us to approximate continuous time dynamics as a
sequence of discrete time systems.Comment: In Proceedings EXPRESS/SOS 2013, arXiv:1307.690
Labelled transition systems as a Stone space
A fully abstract and universal domain model for modal transition systems and
refinement is shown to be a maximal-points space model for the bisimulation
quotient of labelled transition systems over a finite set of events. In this
domain model we prove that this quotient is a Stone space whose compact,
zero-dimensional, and ultra-metrizable Hausdorff topology measures the degree
of bisimilarity such that image-finite labelled transition systems are dense.
Using this compactness we show that the set of labelled transition systems that
refine a modal transition system, its ''set of implementations'', is compact
and derive a compactness theorem for Hennessy-Milner logic on such
implementation sets. These results extend to systems that also have partially
specified state propositions, unify existing denotational, operational, and
metric semantics on partial processes, render robust consistency measures for
modal transition systems, and yield an abstract interpretation of compact sets
of labelled transition systems as Scott-closed sets of modal transition
systems.Comment: Changes since v2: Metadata updat
Discrete homotopies and the fundamental group
We generalize and strengthen the theorem of Gromov that every compact
Riemannian manifold of diameter at most D has a set of generators g_1,...,g_k
of length at most 2D and relators of the form g_ig_m = g_j . In particular, we
obtain an explicit bound for the number k of generators in terms of the number
"short loops" at every point and the number of balls required to cover a given
semilocally simply connected geodesic space. As a consequence we obtain a
fundamental group finiteness theorem (new even for Riemannian manifolds) that
implies the fundamental group finiteness theorems of Anderson and Shen-Wei. Our
theorem requires no curvature bounds, nor lower bounds on volume or 1-systole.
We use the method of discrete homotopies introduced by the first author and V.
N. Berestovskii. Central to the proof is the notion of the homotopy critical
spectrum that is closely related to the covering and length spectra. Discrete
methods also allow us to strengthen and simplify the proofs of some results of
Sormani-Wei about the covering spectrum
q-deformations of two-dimensional Yang-Mills theory: Classification, categorification and refinement
We characterise the quantum group gauge symmetries underlying q-deformations
of two-dimensional Yang-Mills theory by studying their relationships with the
matrix models that appear in Chern-Simons theory and six-dimensional N=2 gauge
theories, together with their refinements and supersymmetric extensions. We
develop uniqueness results for quantum deformations and refinements of gauge
theories in two dimensions, and describe several potential analytic and
geometric realisations of them. We reconstruct standard q-deformed Yang-Mills
amplitudes via gluing rules in the representation category of the quantum group
associated to the gauge group, whose numerical invariants are the usual
characters in the Grothendieck group of the category. We apply this formalism
to compute refinements of q-deformed amplitudes in terms of generalised
characters, and relate them to refined Chern-Simons matrix models and
generalized unitary matrix integrals in the quantum beta-ensemble which compute
refined topological string amplitudes. We also describe applications of our
results to gauge theories in five and seven dimensions, and to the dual
superconformal field theories in four dimensions which descend from the N=(2,0)
six-dimensional superconformal theory.Comment: 71 pages; v2: references added; final version to be published in
Nuclear Physics
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