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We compute and compare the (intersection) cohomology of various natural geometric compactifications of the moduli space of cubic threefolds: the GIT compactification and its Kirwan blowup, as well as the Baily–Borel and toroidal compactifications of the ball quotient model, due to Allcock–Carlson–Toledo. Our starting point is Kirwan’s method. We then follow by investigating the behavior of the cohomology under the birational maps relating the various models, using the decomposition theorem in different ways, and via a detailed study of the boundary of the ball quotient model. As an easy illustration of our methods, the simpler case of the moduli space of cubic surfaces is discussed in an appendix.

In this paper, we prove compatibilities of various definitions of relatively unipotent log de Rham fundamental groups for certain proper log smooth integral morphisms of fine log schemes of characteristic zero. Our proofs are purely algebraic. As an application, we give a purely algebraic calculation of the monodromy action on the unipotent log de Rham fundamental group of a stable log curve. As a corollary we give a purely algebraic proof to the transcendental part of Andreatta–Iovita–Kim’s article: obtaining in this way a complete algebraic criterion for good reduction for curves.

For a proper, smooth scheme $X$ over a $p$ -adic field $K$ , we show that any proper, flat, semistable ${\\mathcal{O}}_{K}$ -model ${\\mathcal{X}}$ of $X$ whose logarithmic de Rham cohomology is torsion free determines the same ${\\mathcal{O}}_{K}$ -lattice inside $H_{\\text{dR}}^{i}(X/K)$ and, moreover, that this lattice is functorial in $X$ . For this, we extend the results of Bhatt–Morrow–Scholze on the construction and the analysis of an $A_{\\text{inf}}$ -valued cohomology theory of $p$ -adic formal, proper, smooth ${\\mathcal{O}}_{\\overline{K}}$ -schemes $\\mathfrak{X}$ to the semistable case. The relation of the $A_{\\text{inf}}$ -cohomology to the $p$ -adic étale and the logarithmic crystalline cohomologies allows us to reprove the semistable conjecture of Fontaine–Jannsen.

We construct combinatorial bases of the$T$-equivariant cohomology$H_{T}^{\\bullet }(\\unicode[STIX]{x1D6F4},k)$of the Bott–Samelson variety$\\unicode[STIX]{x1D6F4}$under some mild restrictions on the field of coefficients$k$. These bases allow us to prove the surjectivity of the restrictions$H_{T}^{\\bullet }(\\unicode[STIX]{x1D6F4},k)\\rightarrow H_{T}^{\\bullet }(\\unicode[STIX]{x1D70B}^{-1}(x),k)$and$H_{T}^{\\bullet }(\\unicode[STIX]{x1D6F4},k)\\rightarrow H_{T}^{\\bullet }(\\unicode[STIX]{x1D6F4}\\setminus \\unicode[STIX]{x1D70B}^{-1}(x),k)$, where$\\unicode[STIX]{x1D70B}:\\unicode[STIX]{x1D6F4}\\rightarrow G/B$is the canonical resolution. In fact, we also construct bases of the targets of these restrictions by picking up certain subsets of certain bases of$H_{T}^{\\bullet }(\\unicode[STIX]{x1D6F4},k)$and restricting them to$\\unicode[STIX]{x1D70B}^{-1}(x)$or$\\unicode[STIX]{x1D6F4}\\setminus \\unicode[STIX]{x1D70B}^{-1}(x)$respectively. As an application, we calculate the cohomology of the costalk-to-stalk embedding for the direct image$\\unicode[STIX]{x1D70B}_{\\ast }\\text{}\\underline{k}_{_{\\unicode[STIX]{x1D6F4}}}$. This algorithm avoids division by 2, which allows us to re-establish 2-torsion for parity sheaves in Braden’s example, Braden and Williamson [‘Modular intersection cohomology complexes on flag varieties’, Math. Z. 272 (3–4) (2012), 697–727].

The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in his pioneering paper The first aim of this paper is to lay the foundation of the theory of multicomplexes. After setting the main definitions, we construct the singular multicomplex In the second part of this work we apply the theory of multicomplexes to the study of the bounded cohomology of topological spaces. Our constructions and arguments culminate in the complete proofs of Gromov’s Mapping Theorem (which implies in particular that the bounded cohomology of a space only depends on its fundamental group) and of Gromov’s Vanishing Theorem, which ensures the vanishing of the simplicial volume of closed manifolds admitting an amenable cover of small multiplicity. The third and last part of the paper is devoted to the study of locally finite chains on non-compact spaces, hence to the simplicial volume of open manifolds. We expand some ideas of Gromov to provide detailed proofs of a criterion for the vanishing and a criterion for the finiteness of the simplicial volume of open manifolds. As a by-product of these results, we prove a criterion for the

Background Genome annotation is of key importance in many research questions. The identification of protein-coding genes is often based on transcriptome sequencing data, ab-initio or homology-based prediction. Recently, it was demonstrated that intron position conservation improves homology-based gene prediction, and that experimental data improves ab-initio gene prediction. Results Here, we present an extension of the gene prediction program GeMoMa that utilizes amino acid sequence conservation, intron position conservation and optionally RNA-seq data for homology-based gene prediction. We show on published benchmark data for plants, animals and fungi that GeMoMa performs better than the gene prediction programs BRAKER1, MAKER2, and CodingQuarry, and purely RNA-seq-based pipelines for transcript identification. In addition, we demonstrate that using multiple reference organisms may help to further improve the performance of GeMoMa. Finally, we apply GeMoMa to four nematode species and to the recently published barley reference genome indicating that current annotations of protein-coding genes may be refined using GeMoMa predictions. Conclusions GeMoMa might be of great utility for annotating newly sequenced genomes but also for finding homologs of a specific gene or gene family. GeMoMa has been published under GNU GPL3 and is freely available at http://www.jstacs.de/index.php/GeMoMa .

Not applicable.

Reflexive homology is the homology theory associated to the reflexive crossed simplicial group; one of the fundamental crossed simplicial groups. It is the most general way to extend Hochschild homology to detect an order-reversing involution. In this paper we study the relationship between reflexive homology and the $C_2$-equivariant homology of free loop spaces. We define reflexive homology in terms of functor homology. We give a bicomplex for computing reflexive homology together with some calculations, including the reflexive homology of a tensor algebra. We prove that the reflexive homology of a group algebra is isomorphic to the homology of the $C_2$-equivariant Borel construction on the free loop space of the classifying space. We give a direct sum decomposition of the reflexive homology of a group algebra indexed by conjugacy classes of group elements, where the summands are defined in terms of a reflexive analogue of group homology. We define a hyperhomology version of reflexive homology and use it to study the $C_2$-equivariant homology of certain free loop and free loop-suspension spaces. We show that reflexive homology satisfies Morita invariance. We prove that under nice conditions the involutive Hochschild homology studied by Braun and by Fernàndez-València and Giansiracusa coincides with reflexive homology.

Background HH-suite is a widely used open source software suite for sensitive sequence similarity searches and protein fold recognition. It is based on pairwise alignment of profile Hidden Markov models (HMMs), which represent multiple sequence alignments of homologous proteins. Results We developed a single-instruction multiple-data (SIMD) vectorized implementation of the Viterbi algorithm for profile HMM alignment and introduced various other speed-ups. These accelerated the search methods HHsearch by a factor 4 and HHblits by a factor 2 over the previous version 2.0.16. HHblits3 is ∼10× faster than PSI-BLAST and ∼20× faster than HMMER3. Jobs to perform HHsearch and HHblits searches with many query profile HMMs can be parallelized over cores and over cluster servers using OpenMP and message passing interface (MPI). The free, open-source, GPLv3-licensed software is available at https://github.com/soedinglab/hh-suite . Conclusion The added functionalities and increased speed of HHsearch and HHblits should facilitate their use in large-scale protein structure and function prediction, e.g. in metagenomics and genomics projects.