@phdthesis{18667, abstract = {Many chemical and physical properties of materials are determined by the material’s shape, for example the size of its pores and the width of its tunnels. This makes materials science a prime application area for geometrical and topological methods. Nevertheless many methods in topological data analysis have not been satisfyingly extended to the needs of materials science. This thesis provides new methods and new mathematical theorems targeted at those specific needs by answering four different research questions. While the motivation for each of the research questions arises from materials science, the methods are versatile and can be applied in different areas as well. The first research question is concerned with image data, for example a three-dimensional computed tomography (CT) scan of a material, like sand or stone. There are two commonly used topologies for digital images and depending on the application either of them might be required. However, software for computing the topological data analysis method persistence homology, usually supports only one of the two topologies. We answer the question how to compute persistent homology of an image with respect to one of the two topologies using software that is intended for the other topology. The second research question is concerned with image data as well, and asks how much of the topological information of an image is lost when the resolution is coarsened. As computer tomography scanners are more expensive the higher the resolution, it is an important question in materials science to know which resolution is enough to get satisfying persistent homology. We give theoretical bounds on the information loss based on different geometrical properties of the object to be scanned. In addition, we conduct experiments on sand and stone CT image data. The third research question is motivated by comparing crystalline materials efficiently. As the atoms within a crystal repeat periodically, crystalline materials are either modeled by unmanageable infinite periodic point sets, or by one of their fundamental domains, which is unstable under perturbation. Therefore a fingerprint of crystalline materials is needed, with appropriate properties such that comparing the crystals can be eased by comparing the fingerprints instead. We define the density fingerprint and prove the necessary properties. The fourth research question is motivated by studying the hole-structure or connectedness, i.e. persistent homology or merge trees, of crystalline materials. A common way to deal with periodicity is to take a fundamental domain and identify opposite boundaries to form a torus. However, computing persistent homology or merge trees on that torus loses some of the information materials scientists are interested in and is additionally not stable under certain noise. We therefore decorate the merge tree stemming from the torus with additional information describing the density and growth rate of the periodic copies of a component within a growing spherical window. We prove all desired properties, like stability and efficient computability.}, author = {Heiss, Teresa}, isbn = {978-3-99078-052-7}, issn = {2663-337X}, keywords = {persistent homology, topological data analysis, periodic, crystalline materials, images, fingerprint}, pages = {111}, publisher = {Institute of Science and Technology Austria}, title = {{New methods for applying topological data analysis to materials science}}, doi = {10.15479/at:ista:18667}, year = {2024}, } @phdthesis{15094, abstract = {Point sets, geometric networks, and arrangements of hyperplanes are fundamental objects in discrete geometry that have captivated mathematicians for centuries, if not millennia. This thesis seeks to cast new light on these structures by illustrating specific instances where a topological perspective, specifically through discrete Morse theory and persistent homology, provides valuable insights. At first glance, the topology of these geometric objects might seem uneventful: point sets essentially lack of topology, arrangements of hyperplanes are a decomposition of Rd, which is a contractible space, and the topology of a network primarily involves the enumeration of connected components and cycles within the network. However, beneath this apparent simplicity, there lies an array of intriguing structures, a small subset of which will be uncovered in this thesis. Focused on three case studies, each addressing one of the mentioned objects, this work will showcase connections that intertwine topology with diverse fields such as combinatorial geometry, algorithms and data structures, and emerging applications like spatial biology. }, author = {Cultrera di Montesano, Sebastiano}, issn = {2663-337X}, pages = {108}, publisher = {Institute of Science and Technology Austria}, title = {{Persistence and Morse theory for discrete geometric structures}}, doi = {10.15479/at:ista:15094}, year = {2024}, } @inproceedings{15093, abstract = {We present a dynamic data structure for maintaining the persistent homology of a time series of real numbers. The data structure supports local operations, including the insertion and deletion of an item and the cutting and concatenating of lists, each in time O(log n + k), in which n counts the critical items and k the changes in the augmented persistence diagram. To achieve this, we design a tailor-made tree structure with an unconventional representation, referred to as banana tree, which may be useful in its own right.}, author = {Cultrera di Montesano, Sebastiano and Edelsbrunner, Herbert and Henzinger, Monika H and Ost, Lara}, booktitle = {Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)}, editor = {Woodruff, David P.}, location = {Alexandria, VA, USA}, pages = {243 -- 295}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Dynamically maintaining the persistent homology of time series}}, doi = {10.1137/1.9781611977912.11}, year = {2024}, } @unpublished{18981, abstract = {We establish several results combining discrete Morse theory and microlocal sheaf theory in the setting of finite posets and simplicial complexes. Our primary tool is a computationally tractable description of the bounded derived category of sheaves on a poset with the Alexandrov topology. We prove that each bounded complex of sheaves on a finite poset admits a unique (up to isomorphism of complexes) minimal injective resolution, and we provide algorithms for computing minimal injective resolution of an injective complex, as well as several useful functors between derived categories of sheaves. For the constant sheaf on a simplicial complex, we give asymptotically tight bounds on the complexity of computing the minimal injective resolution using those algorithms. Our main result is a novel definition of the discrete microsupport of a bounded complex of sheaves on a finite poset. We detail several foundational properties of the discrete microsupport, as well as a microlocal generalization of the discrete homological Morse theorem and Morse inequalities.}, author = {Brown, Adam and Draganov, Ondrej}, booktitle = {arXiv}, title = {{Discrete microlocal Morse theory}}, doi = {10.48550/arXiv.2209.14993}, year = {2024}, } @unpublished{18673, abstract = {Motivated by applications to crystalline materials, we generalize the merge tree and the related barcode of a filtered complex to the periodic setting in Euclidean space. They are invariant under isometries, changing bases, and indeed changing lattices. In addition, we prove stability under perturbations and provide an algorithm that under mild geometric conditions typically satisfied by crystalline materials takes O((n+m)logn) time, in which n and m are the numbers of vertices and edges in the quotient complex, respectively. }, author = {Edelsbrunner, Herbert and Heiss, Teresa}, booktitle = {arXiv}, title = {{Merge trees of periodic filtrations}}, doi = {10.48550/arXiv.2408.16575}, year = {2024}, } @unpublished{15091, abstract = {Motivated by applications in the medical sciences, we study finite chromatic sets in Euclidean space from a topological perspective. Based on the persistent homology for images, kernels and cokernels, we design provably stable homological quantifiers that describe the geometric micro- and macro-structure of how the color classes mingle. These can be efficiently computed using chromatic variants of Delaunay and alpha complexes, and code that does these computations is provided.}, author = {Cultrera di Montesano, Sebastiano and Draganov, Ondrej and Edelsbrunner, Herbert and Saghafian, Morteza}, booktitle = {arXiv}, title = {{Chromatic alpha complexes}}, doi = {10.48550/arXiv.2212.03128}, year = {2024}, } @article{18564, abstract = {The recently discovered Pa 30 nebula, the putative type Iax supernova remnant associated with the historical supernova of 1181 AD, shows puzzling characteristics that make it unique among known supernova remnants. In particular, Pa 30 exhibits a complex morphology, with a unique radial and filamentary structure, and it hosts a hot stellar remnant at its center, which displays oxygen-dominated, ultrafast winds. Because of the surviving stellar remnant and the lack of hydrogen and helium in its filaments, it has been suggested that Pa 30 is the product of a failed thermonuclear explosion in a near- or super-Chandrasekhar white dwarf, which created a subluminous transient, a rare subtype of the Ia class of supernovae called type Iax. We present here a detailed study of the 3D structure and velocities of a full radial section of the remnant. The Integral Field Unit observations, obtained with the new red channel of the Keck Cosmic Web Imager spectrograph, reveal that the ejecta are consistent with being ballistic, with velocities close to the free-expansion velocity. Additionally, we detect a large cavity inside the supernova remnant and a sharp inner edge to the filamentary structure, which coincides with the outer edge of a bright ring detected in infrared images. Finally, we detect a strong asymmetry in the amount of ejecta along the line of sight, which might hint at an asymmetric explosion. Our analysis provides strong confirmation that the explosion originated from SN 1181.}, author = {Cunningham, Tim and Caiazzo, Ilaria and Prusinski, Nikolaus Z. and Fuller, James and Raymond, John C. and Kulkarni, S. R. and Neill, James D. and Duffell, Paul and Martin, Chris and Toloza, Odette and Charbonneau, David and Kenyon, Scott J. and Lin, Zeren and Matuszewski, Mateusz and McGurk, Rosalie and Polin, Abigail and Yao, Philippe Z.}, issn = {2041-8213}, journal = {The Astrophysical Journal Letters}, number = {1}, publisher = {IOP Publishing}, title = {{Expansion properties of the young supernova type Iax remnant Pa 30 revealed}}, doi = {10.3847/2041-8213/ad713b}, volume = {975}, year = {2024}, } @article{18709, abstract = {We measure the mass distribution of main-sequence (MS) companions to hot subdwarf B stars (sdBs) in post-common envelope binaries (PCEBs). We carried out a spectroscopic survey of 14 eclipsing systems ("HW Vir binaries") with orbital periods of 3.8 < Porb < 12 hr, resulting in a well-understood selection function and a near-complete sample of HW Vir binaries with G < 16. We constrain companion masses from the radial velocity curves of the sdB stars. The companion mass distribution peaks at MMS ≈ 0.15 M⊙ and drops off at MMS > 0.2 M⊙, with only two systems hosting companions above the fully convective limit. There is no correlation between Porb and MMS within the sample. A similar drop-off in the companion mass distribution of white dwarf (WD) + MS PCEBs has been attributed to disrupted magnetic braking (MB) below the fully convective limit. We compare the sdB companion mass distribution to predictions of binary evolution simulations with a range of MB laws. Because sdBs have short lifetimes compared to WDs, explaining the lack of higher-mass MS companions to sdBs with disrupted MB requires MB to be boosted by a factor of 20–100 relative to MB laws inferred from the rotation evolution of single stars. We speculate that such boosting may be a result of irradiation-driven enhancement of the MS stars' winds. An alternative possibility is that common envelope evolution favors low-mass companions in short-period orbits, but the existence of massive WD companions to sdBs with similar periods disfavors this scenario.}, author = {Blomberg, Lisa and El-Badry, Kareem and Breivik, Katelyn and Caiazzo, Ilaria and Nagarajan, Pranav and Rodriguez, Antonio and Van Roestel, Jan and Vanderbosch, Zachary P. and Yamaguchi, Natsuko}, issn = {0004-6280}, journal = {Publications of the Astronomical Society of the Pacific}, number = {12}, publisher = {IOP Publishing}, title = {{The companion mass distribution of post common envelope hot subdwarf binaries: Evidence for boosted and disrupted magnetic braking?}}, doi = {10.1088/1538-3873/ad94a2}, volume = {136}, year = {2024}, } @article{15372, abstract = {CRISPR-Cas9 is a powerful tool for genome editing, but the strict requirement for an NGG protospacer-adjacent motif (PAM) sequence immediately next to the DNA target limits the number of editable genes. Recently developed Cas9 variants have been engineered with relaxed PAM requirements, including SpG-Cas9 (SpG) and the nearly PAM-less SpRY-Cas9 (SpRY). However, the molecular mechanisms of how SpRY recognizes all potential PAM sequences remains unclear. Here, we combine structural and biochemical approaches to determine how SpRY interrogates DNA and recognizes target sites. Divergent PAM sequences can be accommodated through conformational flexibility within the PAM-interacting region, which facilitates tight binding to off-target DNA sequences. Nuclease activation occurs ~1000-fold slower than for Streptococcus pyogenes Cas9, enabling us to directly visualize multiple on-pathway intermediate states. Experiments with SpG position it as an intermediate enzyme between Cas9 and SpRY. Our findings shed light on the molecular mechanisms of PAMless genome editing.}, author = {Hibshman, Grace N. and Bravo, Jack Peter Kelly and Hooper, Matthew M. and Dangerfield, Tyler L. and Zhang, Hongshan and Finkelstein, Ilya J. and Johnson, Kenneth A. and Taylor, David W.}, issn = {2041-1723}, journal = {Nature Communications}, publisher = {Springer Nature}, title = {{Unraveling the mechanisms of PAMless DNA interrogation by SpRY-Cas9}}, doi = {10.1038/s41467-024-47830-3}, volume = {15}, year = {2024}, } @article{17442, abstract = {Although eukaryotic Argonautes have a pivotal role in post-transcriptional gene regulation through nucleic acid cleavage, some short prokaryotic Argonaute variants (pAgos) rely on auxiliary nuclease factors for efficient foreign DNA degradation1. Here we reveal the activation pathway of the DNA defence module DdmDE system, which rapidly eliminates small, multicopy plasmids from the Vibrio cholerae seventh pandemic strain (7PET)2. Through a combination of cryo-electron microscopy, biochemistry and in vivo plasmid clearance assays, we demonstrate that DdmE is a catalytically inactive, DNA-guided, DNA-targeting pAgo with a distinctive insertion domain. We observe that the helicase-nuclease DdmD transitions from an autoinhibited, dimeric complex to a monomeric state upon loading of single-stranded DNA targets. Furthermore, the complete structure of the DdmDE–guide–target handover complex provides a comprehensive view into how DNA recognition triggers processive plasmid destruction. Our work establishes a mechanistic foundation for how pAgos utilize ancillary factors to achieve plasmid clearance, and provides insights into anti-plasmid immunity in bacteria. }, author = {Bravo, Jack Peter Kelly and Ramos, Delisa A. and Fregoso Ocampo, Rodrigo and Ingram, Caiden and Taylor, David W.}, issn = {1476-4687}, journal = {Nature}, number = {8018}, pages = {961--967}, publisher = {Springer Nature}, title = {{Plasmid targeting and destruction by the DdmDE bacterial defence system}}, doi = {10.1038/s41586-024-07515-9}, volume = {630}, year = {2024}, } @article{17494, author = {Bravo, Jack Peter Kelly}, issn = {1746-0921}, journal = {Future Microbiology}, number = {15}, pages = {1269--1272}, publisher = {Taylor & Francis}, title = {{Anti-plasmid immunity: A key to pathogen success?}}, doi = {10.1080/17460913.2024.2389720}, volume = {19}, year = {2024}, } @article{13318, abstract = {Bohnenblust–Hille inequalities for Boolean cubes have been proven with dimension-free constants that grow subexponentially in the degree (Defant et al. in Math Ann 374(1):653–680, 2019). Such inequalities have found great applications in learning low-degree Boolean functions (Eskenazis and Ivanisvili in Proceedings of the 54th annual ACM SIGACT symposium on theory of computing, pp 203–207, 2022). Motivated by learning quantum observables, a qubit analogue of Bohnenblust–Hille inequality for Boolean cubes was recently conjectured in Rouzé et al. (Quantum Talagrand, KKL and Friedgut’s theorems and the learnability of quantum Boolean functions, 2022. arXiv preprint arXiv:2209.07279). The conjecture was resolved in Huang et al. (Learning to predict arbitrary quantum processes, 2022. arXiv preprint arXiv:2210.14894). In this paper, we give a new proof of these Bohnenblust–Hille inequalities for qubit system with constants that are dimension-free and of exponential growth in the degree. As a consequence, we obtain a junta theorem for low-degree polynomials. Using similar ideas, we also study learning problems of low degree quantum observables and Bohr’s radius phenomenon on quantum Boolean cubes.}, author = {Volberg, Alexander and Zhang, Haonan}, issn = {1432-1807}, journal = {Mathematische Annalen}, pages = {1657--1676}, publisher = {Springer Nature}, title = {{Noncommutative Bohnenblust–Hille inequalities}}, doi = {10.1007/s00208-023-02680-0}, volume = {389}, year = {2024}, } @article{17282, abstract = {Let X be a vector field and Y be a co-vector field on a smooth manifold M. Does there exist a smooth Riemannian metric gαβ on M such that Yβ=gαβXα ? The main result of this note gives necessary and sufficient conditions for this to be true. As an application of this result we show that a finite-dimensional ergodic Lindblad equation admits a gradient flow structure for the von Neumann relative entropy if and only if the condition of BKM-detailed balance holds.}, author = {Brooks, Morris and Maas, Jan}, issn = {1432-0835}, journal = {Calculus of Variations and Partial Differential Equations}, number = {6}, publisher = {Springer Nature}, title = {{Characterisation of gradient flows for a given functional}}, doi = {10.1007/s00526-024-02755-z}, volume = {63}, year = {2024}, } @article{17143, abstract = {This paper deals with local criteria for the convergence to a global minimiser for gradient flow trajectories and their discretisations. To obtain quantitative estimates on the speed of convergence, we consider variations on the classical Kurdyka–Łojasiewicz inequality for a large class of parameter functions. Our assumptions are given in terms of the initial data, without any reference to an equilibrium point. The main results are convergence statements for gradient flow curves and proximal point sequences to a global minimiser, together with sharp quantitative estimates on the speed of convergence. These convergence results apply in the general setting of lower semicontinuous functionals on complete metric spaces, generalising recent results for smooth functionals on Rn. While the non-smooth setting covers very general spaces, it is also useful for (non)-smooth functionals on Rn. .}, author = {Dello Schiavo, Lorenzo and Maas, Jan and Pedrotti, Francesco}, issn = {1088-6850}, journal = {Transactions of the American Mathematical Society}, number = {6}, pages = {3779--3804}, publisher = {American Mathematical Society}, title = {{Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces}}, doi = {10.1090/tran/9156}, volume = {377}, year = {2024}, } @article{13271, abstract = {In this paper, we prove the convexity of trace functionals (A,B,C)↦Tr|BpACq|s, for parameters (p, q, s) that are best possible, where B and C are any n-by-n positive-definite matrices, and A is any n-by-n matrix. We also obtain the monotonicity versions of trace functionals of this type. As applications, we extend some results in Carlen et al. (Linear Algebra Appl 490:174–185, 2016), Hiai and Petz (Publ Res Inst Math Sci 48(3):525-542, 2012) and resolve a conjecture in Al-Rashed and Zegarliński (Infin Dimens Anal Quantum Probab Relat Top 17(4):1450029, 2014) in the matrix setting. Other conjectures in Al-Rashed and Zegarliński (Infin Dimens Anal Quantum Probab Relat Top 17(4):1450029, 2014) will also be discussed. We also show that some related trace functionals are not concave in general. Such concavity results were expected to hold in different problems.}, author = {Zhang, Haonan}, issn = {1424-0637}, journal = {Annales Henri Poincare}, pages = {2087--2106}, publisher = {Springer Nature}, title = {{Some convexity and monotonicity results of trace functionals}}, doi = {10.1007/s00023-023-01345-7}, volume = {25}, year = {2024}, } @article{15252, abstract = {A measurable map between measure spaces is shown to have bounded compression if and only if its image via the measure-algebra functor is Lipschitz-continuous w.r.t. the measure-algebra distances. This provides a natural interpretation of maps of bounded compression/deformation by means of the measure-algebra functor and corrobo-rates the assertion that maps of bounded deformation are a natural class of morphisms for the category of complete and separable metric measure spaces.}, author = {Dello Schiavo, Lorenzo}, issn = {1848-8013}, journal = {Mathematical Communications}, number = {1}, pages = {137--142}, publisher = {Udruga Matematicara Osijek}, title = {{A characterization of maps of bounded compression}}, volume = {29}, year = {2024}, } @article{18490, abstract = {For large classes of even-dimensional Riemannian manifolds (Formula presented.), we construct and analyze conformally invariant random fields. These centered Gaussian fields (Formula presented.), called co-polyharmonic Gaussian fields, are characterized by their covariance kernels k which exhibit a precise logarithmic divergence: (Formula presented.). They share a fundamental quasi-invariance property under conformal transformations. In terms of the co-polyharmonic Gaussian field (Formula presented.), we define the Liouville Quantum Gravity measure, a random measure on (Formula presented.), heuristically given as (Formula presented.) and rigorously obtained as almost sure weak limit of the right-hand side with (Formula presented.) replaced by suitable regular approximations (Formula presented.). In terms on the Liouville Quantum Gravity measure, we define the Liouville Brownian motion on (Formula presented.) and the random GJMS operators. Finally, we present an approach to a conformal field theory in arbitrary even dimension with an ansatz based on Branson's (Formula presented.) -curvature: we give a rigorous meaning to the Polyakov–Liouville measure (Formula presented.) and we derive the corresponding conformal anomaly. The set of admissible manifolds is conformally invariant. It includes all compact 2-dimensional Riemannian manifolds, all compact non-negatively curved Einstein manifolds of even dimension, and large classes of compact hyperbolic manifolds of even dimension. However, not every compact even-dimensional Riemannian manifold is admissible. Our results concerning the logarithmic divergence of the kernel (Formula presented.) rely on new sharp estimates for heat kernels and higher order Green kernels on arbitrary closed manifolds. }, author = {Dello Schiavo, Lorenzo and Herry, Ronan and Kopfer, Eva and Sturm, Karl Theodor}, issn = {1469-7750}, journal = {Journal of the London Mathematical Society}, number = {5}, publisher = {London Mathematical Society}, title = {{Conformally invariant random fields, Liouville quantum gravity measures, and random Paneitz operators on Riemannian manifolds of even dimension}}, doi = {10.1112/jlms.70003}, volume = {110}, year = {2024}, } @article{15350, abstract = {We extend three related results from the analysis of influences of Boolean functions to the quantum setting, namely the KKL theorem, Friedgut’s Junta theorem and Talagrand’s variance inequality for geometric influences. Our results are derived by a joint use of recently studied hypercontractivity and gradient estimates. These generic tools also allow us to derive generalizations of these results in a general von Neumann algebraic setting beyond the case of the quantum hypercube, including examples in infinite dimensions relevant to quantum information theory such as continuous variables quantum systems. Finally, we comment on the implications of our results as regards to noncommutative extensions of isoperimetric type inequalities, quantum circuit complexity lower bounds and the learnability of quantum observables.}, author = {Rouzé, Cambyse and Wirth, Melchior and Zhang, Haonan}, issn = {1432-0916}, journal = {Communications in Mathematical Physics}, number = {4}, publisher = {Springer Nature}, title = {{Quantum Talagrand, KKL and Friedgut’s theorems and the learnability of quantum boolean functions}}, doi = {10.1007/s00220-024-04981-0}, volume = {405}, year = {2024}, } @article{18158, abstract = {We study the geometry of Poisson point processes from the point of view of optimal transport and Ricci lower bounds. We construct a Riemannian structure on the space of point processes and the associated distance W that corresponds to the Benamou–Brenier variational formula. Our main tool is a non-local continuity equation formulated with the difference operator. The closure of the domain of the relative entropy is a complete geodesic space, when endowed with W. The geometry of this non-local infinite-dimensional space is analogous to that of spaces with positive Ricci curvature. Among others: (a) the Ornstein–Uhlenbeck semi-group is the gradient flow of the relative entropy; (b) the Poisson space has an entropic Ricci curvature bounded from below by 1; (c) W satisfies an HWI inequality.}, author = {Dello Schiavo, Lorenzo and Herry, Ronan and Suzuki, Kohei}, issn = {2270-518X}, journal = {Journal de l'Ecole Polytechnique - Mathematiques}, pages = {957--1010}, publisher = {Ecole Polytechnique}, title = {{Wasserstein geometry and Ricci curvature bounds for Poisson spaces}}, doi = {10.5802/jep.270}, volume = {11}, year = {2024}, } @article{14934, abstract = {We study random perturbations of a Riemannian manifold (M, g) by means of so-called Fractional Gaussian Fields, which are defined intrinsically by the given manifold. The fields h• : ω → hω will act on the manifold via the conformal transformation g → gω := e2hω g. Our focus will be on the regular case with Hurst parameter H > 0, the critical case H = 0 being the celebrated Liouville geometry in two dimensions. We want to understand how basic geometric and functional-analytic quantities like diameter, volume, heat kernel, Brownian motion, spectral bound, or spectral gap change under the influence of the noise. And if so, is it possible to quantify these dependencies in terms of key parameters of the noise? Another goal is to define and analyze in detail the Fractional Gaussian Fields on a general Riemannian manifold, a fascinating object of independent interest.}, author = {Dello Schiavo, Lorenzo and Kopfer, Eva and Sturm, Karl Theodor}, issn = {1572-929X}, journal = {Potential Analysis}, pages = {501--553}, publisher = {Springer Nature}, title = {{A discovery tour in random Riemannian geometry}}, doi = {10.1007/s11118-023-10118-0}, volume = {61}, year = {2024}, }