Homological Localisation of Model Categories

One of the most useful methods for studying the stable homotopy category is localising at some spectrum E. For an arbitrary stable model category we introduce a candidate for the E–localisation of this model category. We study the properties of this new construction and relate it to some well–known categories

The de Rham homotopy theory and differential graded category

This paper is a generalization of arXiv:0810.0808. We develop the de Rham homotopy theory of not necessarily nilpotent spaces, using closed dg-categories and equivariant dg-algebras. We see these two algebraic objects correspond in a certain way. We prove an equivalence between the homotopy category of schematic homotopy types and a homotopy category of closed dg-categories. We give a description of homotopy invariants of spaces in terms of minimal models. The minimal model in this context behaves much like the Sullivan's minimal model. We also provide some examples. We prove an equivalence between fiberwise rationalizations and closed dg-categories with subsidiary data.Comment: 47 pages. final version. The final publication is available at http://www.springerlink.co

Higher Structures in M-Theory

The key open problem of string theory remains its non-perturbative completion to M-theory. A decisive hint to its inner workings comes from numerous appearances of higher structures in the limits of M-theory that are already understood, such as higher degree flux fields and their dualities, or the higher algebraic structures governing closed string field theory. These are all controlled by the higher homotopy theory of derived categories, generalised cohomology theories, and $L_\infty$-algebras. This is the introductory chapter to the proceedings of the LMS/EPSRC Durham Symposium on Higher Structures in M-Theory. We first review higher structures as well as their motivation in string theory and beyond. Then we list the contributions in this volume, putting them into context.Comment: 22 pages, Introductory Article to Proceedings of LMS/EPSRC Durham Symposium Higher Structures in M-Theory, August 2018, references update

The homotopy theory of simplicial props

The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. In this paper, the second in a series on "higher props," we show that the category of all small colored simplicial props admits a cofibrantly generated model category structure. With this model structure, the forgetful functor from props to operads is a right Quillen functor.Comment: Final version, to appear in Israel J. Mat

The Dold-Kan Correspondence and Coalgebra Structures

By using the Dold-Kan correspondence we construct a Quillen adjunction between the model categories of non-cocommutative coassociative simplicial and differential graded coalgebras over a field. We restrict to categories of connected coalgebras and prove a Quillen equivalence between them.Comment: 24 pages. Accepted by the Journal of Homotopy and Related Structures. Online 28 November 201

Group actions on Segal operads

We give a Quillen equivalence between model structures for simplicial operads, described via the theory of operads, and Segal operads, thought of as certain reduced dendroidal spaces. We then extend this result to give an Quillen equivalence between the model structures for simplicial operads equipped with a group action and the corresponding Segal operads.Comment: Revised version. Accepted to Isr J Mat

Notes on factorization algebras, factorization homology and applications

These notes are an expanded version of two series of lectures given at the winter school in mathematical physics at les Houches and at the Vietnamese Institute for Mathematical Sciences. They are an introduction to factorization algebras, factorization homology and some of their applications, notably for studying $E_n$-algebras. We give an account of homology theory for manifolds (and spaces), which give invariant of manifolds but also invariant of $E_n$-algebras. We particularly emphasize the point of view of factorization algebras (a structure originating from quantum field theory) which plays, with respect to homology theory for manifolds, the role of sheaves with respect to singular cohomology. We mention some applications to the study of mapping spaces and study several examples, including some over stratified spaces.Comment: 122 pages. A few examples adde

Machine learning‐based classification of Alzheimer's disease and its at‐risk states using personality traits, anxiety, and depression

Background Alzheimer's disease (AD) is often preceded by stages of cognitive impairment, namely subjective cognitive decline (SCD) and mild cognitive impairment (MCI). While cerebrospinal fluid (CSF) biomarkers are established predictors of AD, other non-invasive candidate predictors include personality traits, anxiety, and depression, among others. These predictors offer non-invasive assessment and exhibit changes during AD development and preclinical stages. Methods In a cross-sectional design, we comparatively evaluated the predictive value of personality traits (Big Five), geriatric anxiety and depression scores, resting-state functional magnetic resonance imaging activity of the default mode network, apoliprotein E (ApoE) genotype, and CSF biomarkers (tTau, pTau181, Aβ42/40 ratio) in a multi-class support vector machine classification. Participants included 189 healthy controls (HC), 338 individuals with SCD, 132 with amnestic MCI, and 74 with mild AD from the multicenter DZNE-Longitudinal Cognitive Impairment and Dementia Study (DELCODE). Results Mean predictive accuracy across all participant groups was highest when utilizing a combination of personality, depression, and anxiety scores. HC were best predicted by a feature set comprised of depression and anxiety scores and participants with AD were best predicted by a feature set containing CSF biomarkers. Classification of participants with SCD or aMCI was near chance level for all assessed feature sets. Conclusion Our results demonstrate predictive value of personality trait and state scores for AD. Importantly, CSF biomarkers, personality, depression, anxiety, and ApoE genotype show complementary value for classification of AD and its at-risk stages

Perivascular space enlargement accelerates in ageing and Alzheimer’s disease pathology: evidence from a three-year longitudinal multicentre study

Background Perivascular space (PVS) enlargement in ageing and Alzheimer’s disease (AD) and the drivers of such a structural change in humans require longitudinal investigation. Elucidating the effects of demographic factors, hypertension, cerebrovascular dysfunction, and AD pathology on PVS dynamics could inform the role of PVS in brain health function as well as the complex pathophysiology of AD. Methods We studied PVS in centrum semiovale (CSO) and basal ganglia (BG) computationally over three to four annual visits in 503 participants (255 females; meanage = 70.78 ± 5.78) of the ongoing observational multicentre “DZNE Longitudinal Cognitive Impairment and Dementia Study” (DELCODE) cohort. We analysed data from subjects who were cognitively unimpaired (n = 401), had amnestic mild cognitive impairment (n = 71), or had AD (n = 31). We used linear mixed-effects modelling to test for changes of PVS volumes in relation to cross-sectional and longitudinal age, as well as sex, years of education, hypertension, white matter hyperintensities, AD diagnosis, and cerebrospinal-fluid-derived amyloid (A) and tau (T) status (available for 46.71%; A-T-/A + T-/A + T + n = 143/48/39). Results PVS volumes increased significantly over follow-ups (CSO: B = 0.03 [0.02, 0.05], p  A-T-, pFDR = 0.004) or who were amyloid positive but tau negative (A + T +  > A + T-, pFDR = 0.07). CSO-PVS volumes increased at a faster rate with amyloid positivity as compared to amyloid negativity (A + T-/A + T +  > A-T-, pFDR = 0.021). Conclusion Our longitudinal evidence supports the relevance of PVS enlargement in presumably healthy ageing as well as in AD pathology. We further discuss the region-specific involvement of white matter hyperintensities and neurotoxic waste accumulation in PVS enlargement and the possibility of additional factors contributing to PVS progression. A comprehensive understanding of PVS dynamics could facilitate the understanding of pathological cascades and might inform targeted treatment strategies