More Like this

1. Cohomological Invariants in Positive Characteristic

   https://doi.org/10.1093/imrn/rnaa321  

   Totaro, Burt ( January 2021 , International Mathematics Research Notices)

   Abstract We determine the mod $p$ cohomological invariants for several affine group schemes $G$ in characteristic $p$. These are invariants of $G$-torsors with values in étale motivic cohomology, or equivalently in Kato’s version of Galois cohomology based on differential forms. In particular, we find the mod 2 cohomological invariants for the symmetric groups and the orthogonal groups in characteristic 2, which Serre computed in characteristic not 2. We also determine all operations on the mod $p$ étale motivic cohomology of fields, extending Vial’s computation of the operations on the mod $p$ Milnor K-theory of fields. 

   more » « less
2. Ultrafast learning of four-node hybridization cycles in phylogenetic networks using algebraic invariants

   https://doi.org/10.1093/bioadv/vbae014  

   Wu, Zhaoxing ; Solís-Lemus, Claudia ; Ouangraoua, ed., Aida ( February 2024 , Bioinformatics Advances)

   The abundance of gene flow in the Tree of Life challenges the notion that evolution can be represented with a fully bifurcating process which cannot capture important biological realities like hybridization, introgression, or horizontal gene transfer. Coalescent-based network methods are increasingly popular, yet not scalable for big data, because they need to perform a heuristic search in the space of networks as well as numerical optimization that can be NP-hard. Here, we introduce a novel method to reconstruct phylogenetic networks based on algebraic invariants. While there is a long tradition of using algebraic invariants in phylogenetics, our work is the first to define phylogenetic invariants on concordance factors (frequencies of four-taxon splits in the input gene trees) to identify level-1 phylogenetic networks under the multispecies coalescent model.

   Our novel hybrid detection methodology is optimization-free as it only requires the evaluation of polynomial equations, and as such, it bypasses the traversal of network space, yielding a computational speed at least 10 times faster than the fastest-to-date network methods. We illustrate our method’s performance on simulated and real data from the genus Canis.

   We present an open-source publicly available Julia package PhyloDiamond.jl available at https://github.com/solislemuslab/PhyloDiamond.jl with broad applicability within the evolutionary community.

    

   more » « less
3. New Methodologies for Grain Boundary Detection in EBSD Data of Microstructures

   https://doi.org/10.2514/6.2022-1424  

   Catania, Richard K. ; Senthilnathan, Arulmurugan ; Sions, John ; Snyder, Kyle ; Al-Ghaib, Huda ; Zimmerman, Ben ; Acar, Pinar ( December 2021 , New Methodologies for Grain Boundary Detection in EBSD Data of Microstructures)

   This work discusses new methodologies for identifying the grain boundaries in color images of metallic microstructures and the quantification of their grain topology. Grain boundaries have a large impact on the macro-scale material properties. Particularly, this work employs the experimental microstructure data of Titanium-Aluminum alloys, which can be used for various aerospace components owing to their outstanding mechanical performance in elevated temperatures. The grain topology of these metallic microstructures is quantified using the concept of shape moment invariants. In order to capture the grains using the shape moment invariants, it is necessary to identify the grain boundaries and separate them from their respective grains. We present two methodologies to detect the grain boundaries. The first method is the tolerance-based neighbor analysis. The second method focuses on creating three-dimensional space of pixel intensity values based on the three color channels and measuring the Euclidean distance to separate different grains. Additionally, since the grain boundaries may not possess the same material properties as the grain itself, this work investigates the effect of including the grain boundaries when determining the homogenized material properties of the given microstructure. To generate adequate statistical information, microstructures are reconstructed from the experimental data using the Markov Random Field (MRF) method. Upon separating the grains, we use the shape moment invariants to quantify the shapes of different grains. Using the shape moment invariants and the experimental material property values, three neural network functions are developed to investigate the effects of grain boundaries on material property predictions. 

   more » « less
4. Classifying Topology in Photonic Heterostructures with Gapless Environments

   https://doi.org/10.1103/PhysRevLett.131.213801  

   Dixon, Kahlil Y. ; Loring, Terry A. ; Cerjan, Alexander ( November 2023 , Physical Review Letters)

   Photonic topological insulators exhibit bulk-boundary correspondence, which requires that boundary-localized states appear at the interface formed between topologically distinct insulating materials. However, many topological photonic devices share a boundary with free space, which raises a subtle but critical problem as free space is gapless for photons above the light-line. Here, we use a local theory of topological materials to resolve bulk-boundary correspondence in heterostructures containing gapless materials and in radiative environments. In particular, we construct the heterostructure’s spectral localizer, a composite operator based on the system’s real-space description that provides a local marker for the system’s topology and a corresponding local measure of its topological protection; both quantities are independent of the material’s bulk band gap (or lack thereof). Moreover, we show that approximating radiative outcoupling as material absorption overestimates a heterostructure’s topological protection. As the spectral localizer is applicable to systems in any physical dimension and in any discrete symmetry class, our results show how to calculate topological invariants, quantify topological protection, and locate topological boundary-localized resonances in topological materials that interface with gapless media in general. 

   more » « less
5. Lifting Weak Supervision To Structured Prediction

   Vishwakarma, H ; Sala, F ( January 2022 , Advances in neural information processing systems)

   Weak supervision (WS) is a rich set of techniques that produce pseudolabels by aggregating easily obtained but potentially noisy label estimates from various sources. WS is theoretically well-understood for binary classification, where simple approaches enable consistent estimation of pseudolabel noise rates. Using this result, it has been shown that downstream models trained on the pseudolabels have generalization guarantees nearly identical to those trained on clean labels. While this is exciting, users often wish to use WS for\emph {structured prediction}, where the output space consists of more than a binary or multi-class label set: eg rankings, graphs, manifolds, and more. Do the favorable theoretical properties of WS for binary classification lift to this setting? We answer this question in the affirmative for a wide range of scenarios. For labels taking values in a finite metric space, we introduce techniques new to weak supervision based on pseudo-Euclidean embeddings and tensor decompositions, providing a nearly-consistent noise rate estimator. For labels in constant-curvature Riemannian manifolds, we introduce new invariants that also yield consistent noise rate estimation. In both cases, when using the resulting pseudolabels in concert with a flexible downstream model, we obtain generalization guarantees nearly identical to those for models trained on clean data. Several of our results, which can be viewed as robustness guarantees in structured prediction with noisy labels, may be of independent interest. 

   more » « less