7,940 research outputs found

    Profinite Galois descent in K(h)-local homotopy theory

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    We investigate the category of K(h)-local spectra through the action of the Morava stabiliser group. Using condensed mathematics, we give a model for the continuous action of this profinite group on the ∞-category of K(h)-local modules over Morava E-theory, and explain how this gives rise to descent spectral sequences computing the Picard and Brauer groups of K(h)-local spectra. In the second part, we focus on the computation of these spectral sequences at height one, showing that they recover the Hopkins-Mahowald-Sadofsky computation of the Picard group, and giving a complete computation of the Brauer group relative to p-completed complex K-theory

    Kan extensions in probability theory

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    n this thesis we will discuss results and ideas in probability theory from a categorical point of view. One categorical concept in particular will be of interest to us, namely that of Kan extensions. We will use Kan extensions of ‘ordinary’ functors, enriched functors and lax natural transformations to give categorical proofs of some fundamental results in probability theory and measure theory. We use Kan extensions of ‘ordinary’ functors to represent probability monads as codensity monads. We consider a functor representing probability measures on countable spaces. By Kan extending this functor along itself, we obtain a codensity monad describing probability measures on all spaces. In this way we represent probability monads such as the Giry monad, the Radon monad and the Kantorovich monad. Kan extensions of lax natural transformations are used to obtain a categorical proof of the Carath´eodorody extensions theorem. The Carath´eodory extension theorem is a fundamental theorem in measure theory that says that premeasures can be extended to measures. We first develop a framework for Kan extensions of lax natural transformations. We then represent outer and inner (pre)measures by certain lax and colax natural transformations. By applying the results on extensions of transformations a categorical proof of Carath´eodory’s extension theorem is obtained. We also give a categorical view on the Radon–Nikodym theorem and martingales. For this we need Kan extensions of enriched functors. We start by observing that the finite version of the Radon–Nikodym theorem is trivial and that it can be interpreted as a natural isomorphism between certain functors, enriched over CMet, the category of complete metric spaces and 1-Lipschitz maps. We proceed by Kan extending these, to obtain the general version of the Radon–Nikodym theorem. Concepts such as conditional expectation and martingales naturally appear in this construction. By proving that these extended functors preserve certain cofiltered limits, we obtain categorical proofs of a weaker version of a martingale convergence theorem and the Kolmogorov extension theorem

    Evolutionary ecology of obligate fungal and microsporidian invertebrate pathogens

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    The interactions between hosts and their parasites and pathogens are omnipresent in the natural world. These symbioses are not only key players in ecosystem functioning, but also drive genetic diversity through co-evolutionary adaptations. Within the speciose invertebrates, a plethora of interactions with obligate fungal and microsporidian pathogens exist, however the known interactions is likely only a fraction of the true diversity. Obligate invertebrate fungal and microsporidian pathogen require a host to continue their life cycle, some of which have specialised in certain host species and require host death to transmit to new hosts. Due to their requirement to kill a host to spread to a new one, obligate fungal and microsporidian pathogens regulate invertebrate host populations. Pathogen specialisation to a single or very few hosts has led to some fungi evolving the ability to manipulate their host’s behaviour to maximise transmission. The entomopathogenic fungus, Entomophthora muscae, infects houseflies (Musca domestica) over a week-long proliferation cycle, resulting in flies climbing to elevated positions, gluing their mouthparts to the substrate surface, and raising their wings to allow for a clear exit from fungal conidia through the host abdomen. These sequential behaviours are all timed to occur within a few hours of sunset. The E. muscae mechanisms used in controlling the mind of the fly remain relatively unknown, and whether other fitness costs ensue from an infection are understudied.European Commissio

    On the Global Topology of Moduli Spaces of Riemannian Metrics with Holonomy Sp(n)\operatorname{Sp}(n)

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    We discuss aspects of the global topology of moduli spaces of hyperkähler metrics. If the second Betti number is larger than 44, we show that each connected component of these moduli spaces is not contractible. Moreover, in certain cases, we show that the components are simply connected and determine the second rational homotopy group. By that, we prove that the rank of the second homotopy group is bounded from below by the number of orbits of MBM-classes in the integral cohomology. \\ An explicit description of the moduli space of these hyperkähler metrics in terms of Torelli theorems will be given. We also provide such a description for the moduli space of Einstein metrics on the Enriques manifold. For the Enriques manifold, we also give an example of a desingularization process similar to the Kummer construction of Ricci-flat metrics on a Kummer K3K3 surface.\\ We will use these theorems to provide topological statements for moduli spaces of Ricci-flat and Einstein metrics in any dimension larger than 33. For a compact simply connected manifold NN we show that the moduli space of Ricci flat metrics on N×TkN\times T^k splits homeomorphically into a product of the moduli space of Ricci flat metrics on NN and the moduli of sectional curvature flat metrics on the torus TkT^k

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Semi-simplicial Set Models for Distributed Knowledge

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    In recent years, a new class of models for multi-agent epistemic logic has emerged, based on simplicial complexes. Since then, many variants of these simplicial models have been investigated, giving rise to different logics and axiomatizations. In this paper, we present a further generalization, where a group of agents may distinguish two worlds, even though each individual agent in the group is unable to distinguish them. For that purpose, we generalize beyond simplicial complexes and consider instead simplicial sets. By doing so, we define a new semantics for epistemic logic with distributed knowledge. As it turns out, these models are the geometric counterpart of a generalization of Kripke models, called "pseudo-models". We identify various interesting sub-classes of these models, encompassing all previously studied variants of simplicial models; and give a sound and complete axiomatization for each of them

    A note on the cohomology of pp-adic analytic group actions

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    We prove that given an analytic action of a compact pp-adic Lie group on a Banach space over a field of positive characteristic, one can detect either the simultaneous vanishing or the simultaneous finite-dimensionality of all of the continuous cohomology groups from the corresponding statement for the restriction to a pro-pp procyclic subgroup. We also formulate a conjecture generalizing this result, in which the base field is allowed to have mixed characteristic and the subgroup is allowed to be nilpotent. Finally, we formulate an analogous conjecture about Lie algebra cohomology and relate this to a theorem of Kostant.Comment: 11 page

    Higher Geometric Structures on Manifolds and the Gauge Theory of Deligne Cohomology

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    We study smooth higher symmetry groups and moduli \infty-stacks of generic higher geometric structures on manifolds. Symmetries are automorphisms which cover non-trivial diffeomorphisms of the base manifold. We construct the smooth higher symmetry group of any geometric structure on MM and show that this completely classifies, via a universal property, equivariant structures on the higher geometry. We construct moduli stacks of higher geometric data as \infty-categorical quotients by the action of the higher symmetries, extract information about the homotopy types of these moduli \infty-stacks, and prove a helpful sufficient criterion for when two such higher moduli stacks are equivalent. In the second part of the paper we study higher U(1)\mathrm{U}(1)-connections. First, we observe that higher connections come organised into higher groupoids, which further carry affine actions by Baez-Crans-type higher vector spaces. We compute a presentation of the higher gauge actions for nn-gerbes with kk-connection, comment on the relation to higher-form symmetries, and present a new String group model. We construct smooth moduli \infty-stacks of higher Maxwell and Einstein-Maxwell solutions, correcting previous such considerations in the literature, and compute the homotopy groups of several moduli \infty-stacks of higher U(1)\mathrm{U}(1)- connections. Finally, we show that a discrepancy between two approaches to the differential geometry of NSNS supergravity (via generalised and higher geometry, respectively) vanishes at the level of moduli \infty-stacks of NSNS supergravity solutions.Comment: 102 pages; comments welcom

    Notes on Factorization Algebras and TQFTs

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    These are notes from talks given at a spring school on topological quantum field theory in Nova Scotia during May of 2023. The aim is to introduce the reader to the role of factorization algebras and related concepts in field theory. In particular, we discuss the relationship between factorization algebras, En\mathbb{E}_n-algebras, vertex algebras, and the functorial perspective on field theories.Comment: 42 pages. Comments welcome

    On the metaphysics of F1\mathbb F_1

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    In the present paper, dedicated to Yuri Manin, we investigate the general notion of rings of S[μn,+]\mathbb S[\mu_{n,+}]-polynomials and relate this concept to the known notion of number systems. The Riemann-Roch theorem for the ring Z\mathbb Z of the integers that we obtained recently uses the understanding of Z\mathbb Z as a ring of polynomials S[X]\mathbb S[X] in one variable over the absolute base S\mathbb S, where 1+1=X+X21+1=X+X^2. The absolute base S\mathbb S (the categorical version of the sphere spectrum) thus turns out to be a strong candidate for the incarnation of the mysterious F1\mathbb F_1.Comment: Dedicated to Yuri Manin, 14 Figure
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