Objectives We aimed to assess how the 2018 and 1999 classifications of periodontal diseases reflect (a) patients’ characteristics, (b) disease severity/extent/progression and (c) tooth loss (TL) during observation period. Methods A total of 251 patients were followed over 21.8 ± 6.2 years. For the 1999 classification, using clinical attachment level (CAL), patients were classified as localized/generalized, mild/moderate/severe and aggressive/chronic periodontitis. For the 2018 classification, patients were staged according to their CAL or bone loss (BL) and the number of lost teeth (stages I–IV). Further factors like probing pocket depths (PPD) or furcation involvement modified the stage. The extent was sub‐classified as generalized/localized. Patients were graded according to the BL/age index, smoking and/or diabetes. Results According to the 1999 classification, most patients suffered from generalized severe chronic periodontitis (203/251) or generalized aggressive periodontitis (45/251). Patients with aggressive periodontitis were younger and less often female or smokers. They showed similar TL (0.25 ± 0.22 teeth/patient*year) as generalized severe chronic periodontitis patients (0.23 ± 0.25 teeth/patient*year). According to the 2018 classification, most patients were classified as generalized III‐C (140/251), III‐B (31/251) or IV‐C (64/251). Patients’ age, smoking status, CAL, PPD and BL were well reflected. TL differed between IV‐C (0.36 ± 0.47), generalized III‐C (0.21 ± 0.24) and localized forms (0.10–0.15). Conclusions Patients’ characteristics, disease severity/extent/progression and TL were well reflected by the 2018 classification.

We use Scholze's framework of diamonds to gain new insights in correspondences between p-adic vector bundles and local systems. Such correspondences arise in the context of p-adic Simpson theory in the case of vanishing Higgs fields. In the present paper we provide a detailed analysis of local systems on diamonds for the étale, proétale, and the v-topology, and study the structure sheaves for all three topologies in question. Applied to proper adic spaces of finite type over C p this enables us to prove a category equivalence between C p -local systems with integral models, and modules under the v-structure sheaf which modulo each p n can be trivialized on a proper cover. The flexibility of the v-topology together with a descent result on integral models of local systems allows us to prove that the trivializability condition in the module category may be checked on any normal proper cover. This result leads to an extension of the parallel transport theory by Deninger and the second author to vector bundles with numerically flat reduction on a proper normal cover. 2020 MSC: 14G45, 14G22, 11G25. for suggesting to look into the v-topology. Moreover, we would like to thank Christopher Deninger for his comments, Torsten Wedhorn for useful hints on adic spaces and Matti Würthen for many helpful conversations.

For an abeloid variety A over a complete algebraically closed field extension K of $$\mathbb {Q}_p$$ Q p , we construct a p-adic Corlette–Simpson correspondence, namely an equivalence between finite-dimensional continuous K-linear representations of the Tate module and a certain subcategory of the Higgs bundles on A. To do so, our central object of study is the category of vector bundles for the v-topology on the diamond associated to A. We prove that any pro-finite-étale v-vector bundle can be built from pro-finite-étale v-line bundles and unipotent v-bundles. To describe the latter, we extend the theory of universal vector extensions to the v-topology and use this to generalise a result of Brion by relating unipotent v-bundles on abeloids to representations of vector groups.

We develop a full 6-functor formalism for p-torsion étale sheaves in rigid-analytic geometry. More concretely, we use the recently developed condensed mathematics by Clausen-Scholze to associate to every small v-stack (e.g. rigid-analytic variety) X with pseudouniformizer π an ∞-category D a (O + X /π) of "derived quasicoherent complete topological O + X /π-modules" on X. We then construct the six functors ⊗, Hom, f * , f * , f ! and f ! in this setting and show that they satisfy all the expected compatibilities, similar to the ℓ-adic case. By introducing ϕ-module structures and proving a version of the p-torsion Riemann-Hilbert correspondence we relate O + X /πsheaves to F p -sheaves. As a special case of this formalism we prove Poincaré duality for F p -cohomology on rigid-analytic varieties. In the process of constructing D a (O + X /π) we also develop a general descent formalism for condensed modules over condensed rings.

We use Scholze’s framework of diamonds to gain new insights in correspondences between $p$-adic vector bundles and local systems. Such correspondences arise in the context of $p$-adic Simpson theory in the case of vanishing Higgs fields. In the present paper, we provide a detailed analysis of local systems on diamonds for the étale, pro-étale, and the $v$-topology and study the structure sheaves for all three topologies in question. Applied to proper adic spaces of finite type over $\mathbb {C}_p$, this enables us to prove a category equivalence between $\mathbb {C}_p$-local systems with integral models, and modules under the $v$-structure sheaf that modulo each $p^n$ can be trivialized on a proper cover. The flexibility of the $v$-topology together with a descent result on integral models of local systems allows us to prove that the trivializability condition in the module category may be checked on any normal proper cover. This result leads to an extension of the parallel transport theory by Deninger and the second author to vector bundles with numerically flat reduction on a proper normal cover. 2020 MSC: 14G45, 14G22, 11G25.