arXiv subjects

Andreas Ott

Publications and source records attributed to Andreas Ott.

10 recordsLinked to original sources

arXiv2022

MuRiT: Efficient Computation of Pathwise Persistence Barcodes in Multi-Filtered Flag Complexes via Vietoris-Rips Transformations

Multi-parameter persistent homology naturally arises in applications of persistent topology to data that come with extra information depending on additional parameters, like for example time series data. We introduce the concept of a Vietoris-Rips transformation, a method that reduces the computation of the one-parameter persistent homology of pathwise subcomplexes in multi-filtered flag complexes to the computation of the Vietoris-Rips persistent homology of certain semimetric spaces. The corresponding pathwise persistence barcodes track persistence features of the ambient multi-filtered complex and can in particular be used to recover the rank invariant in multi-parameter persistent homology. We present MuRiT, a scalable algorithm that computes the pathwise persistence barcodes of multi-filtered flag complexes by means of Vietoris-Rips transformations. Moreover, we provide an efficient software implementation of the MuRiT algorithm which resorts to Ripser for the actual computation of Vietoris-Rips persistence barcodes. To demonstrate the applicability of MuRiT to real-world datasets, we establish MuRiT as part of our CoVtRec pipeline for the surveillance of the convergent evolution of the coronavirus SARS-CoV-2 in the current COVID-19 pandemic.

math.AT↗

arXiv2021

Ultrafast topological data analysis reveals pandemic-scale dynamics of convergent evolution

Genome variants which re-occur independently across evolutionary lineages are key molecular signatures of adaptation. Inferring the dynamics of such genetic changes from pandemic-scale genomic datasets is now possible, which opens up unprecedented insight into evolutionary processes. However, existing approaches depend on the construction of accurate phylogenetic trees, which remains challenging at scale. Here we present EVOtRec, an organism-agnostic, fast and scalable Topological Data Analysis approach that enables the inference of convergently evolving genomic variants over time directly from topological patterns in the dataset, without requiring the construction of a phylogenetic tree. Using data from both simulations and published experiments, we show that EVOtRec can robustly identify variants under positive selection and performs orders of magnitude faster than state-of-the-art phylogeny-based approaches, with comparable results. We apply EVOtRec to three large viral genome datasets: SARS-CoV-2, influenza virus A subtype H5N1 and HIV-1. We identify key convergent genome variants and demonstrate how EVOtRec facilitates the real-time tracking of high fitness variants in large datasets with millions of genomes, including effects modulated by varying genomic backgrounds. We envision our Topological Data Analysis approach as a new framework for efficient comparative genomics.

q-bio.PE↗

arXiv2020

Bounded cohomology, Higgs bundles, and Milnor-Wood inequalities

We explain how the generalized Milnor-Wood inequality for reductive representations of a cocompact complex-hyperbolic lattice into a Hermitian Lie group translates, under the non-abelian Hodge correspondence, into various kinds of Milnor-Wood inequalities for Higgs bundles. This clarifies the relation between the representation theoretic generalized Milnor-Wood inequality and the various different versions of Milnor-Wood inequalities for Higgs bundles that are known in the literature.

math.DG↗

arXiv2020

Higgs bundles, harmonic maps, and pleated surfaces

This paper unites the gauge-theoretic and hyperbolic-geometric perspectives on the asymptotic geometry of the character variety of SL(2,C) representations of a surface group. Specifically, we find an asymptotic correspondence between the analytically defined limiting configuration of a sequence of solutions to the SU(2) self-duality equations on a closed Riemann surface constructed by Mazzeo-Swoboda-Weiss-Witt, and the geometric topological shear-bend parameters of equivariant pleated surfaces in hyperbolic three-space due to Bonahon and Thurston. The geometric link comes from the nonabelian Hodge correspondence and a study of high energy degenerations of harmonic maps. Our result has several applications. We prove: (1) the local invariance of the partial compactification of the moduli space of solutions to the self-duality equations by limiting configurations; (2) a refinement of the harmonic maps characterization of the Morgan-Shalen compactification of the character variety; and (3) a comparison between the family of complex projective structures defined by a quadratic differential and the realizations of the corresponding flat connections as Higgs bundles, as well as a determination of the asymptotic shear-bend cocycle of Thurston's pleated surface.

math.DG↗

arXiv2018

Transgression in bounded cohomology and a conjecture of Monod

We develop an algebro-analytic framework for the systematic study of the continuous bounded cohomology of Lie groups in large degree. As an application, we examine the continuous bounded cohomology of PSL(2,R) with trivial real coefficients in all degrees greater than two. We prove a vanishing result for strongly reducible classes, thus providing further evidence for a conjecture of Monod. On the cochain level, our method yields explicit formulas for cohomological primitives of arbitrary bounded cocycles.

math.GR↗

arXiv2016

Thermosensitive Cu2O-PNIPAM core-shell nanoreactors with tunable photocatalytic activity

We report a facile and novel method for the fabrication of Cu2O@PNIPAM core-shell nanoreactors using Cu2O nanocubes as the core. The PNIPAM shell not only effectively protects the Cu2O nanocubes from oxidation, but also improves the colloidal stability of the system. The Cu2O@PNIPAM core-shell microgels can work efficiently as photocatalyst for the decomposition of methyl orange under visible light. A significant enhancement in the catalytic activity has been observed for the core-shell microgels compared with the pure Cu2O nanocubes. Most importantly, the photocatalytic activity of the Cu2O nanocubes can be further tuned by the thermosensitive PNIPAM shell, as rationalized by our recent theory.

physics.chem-ph↗

arXiv2016

Perturbations of the Spence-Abel equation and deformations of the dilogarithm function

We analyze existence, uniqueness and regularity of solutions for perturbations of the Spence-Abel equation for the Rogers' dilogarithm. As an application we deduce a version of Hyers-Ulam stability for the Spence-Abel equation. Our analysis makes use of a well-known cohomological interpretation of the Spence-Abel equation and is based on our recent results on continuous bounded cohomology of SL(2,R).

math.NT↗

arXiv2013

Bounded cohomology via partial differential equations, I

We present a new technique that employs partial differential equations in order to explicitly construct primitives in the continuous bounded cohomology of Lie groups. As an application, we prove a vanishing theorem for the continuous bounded cohomology of SL(2, R) in degree four, establishing a special case of a conjecture of Monod.

math.GR↗

arXiv2013

Removal of singularities and Gromov compactness for symplectic vortices

We prove that the moduli space of gauge equivalence classes of symplectic vortices with uniformly bounded energy in a compact Hamiltonian manifold admits a Gromov compactification by polystable vortices. This extends results of Mundet i Riera and Tian for circle actions to the case of arbitrary compact Lie groups. Our argument relies on an a priori estimate for vortices that allows us to apply techniques used by McDuff and Salamon in their proof of Gromov compactness for pseudoholomorphic curves. As an intermediate result we prove a removable singularity theorem for vortices.

math.SG↗

arXiv2009

Surjectivity of the comparison map in bounded cohomology for Hermitian Lie groups

We prove surjectivity of the comparison map from continuous bounded cohomology to continuous cohomology for Hermitian Lie groups with finite center. For general semisimple Lie groups with finite center, the same argument shows that the image of the comparison map contains all the even generators. Our proof uses a Hirzebruch type proportionality principle in combination with Gromov's results on boundedness of primary characteristic classes and classical results of Cartan and Borel on the cohomology of compact homogeneous spaces.

math.AT↗