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Evaluating spatial patterns in data is an integral task across various domains, including geostatistics, astronomy, and spatial tissue biology. The analysis of transcriptomics data in particular relies on methods for detecting spatially-dependent features that exhibit significant spatial patterns for both explanatory analysis and feature selection. However, given the complex and high-dimensional nature of these data, there is a need for robust, stable, and reliable descriptors of spatial dependence. We leverage the stability and multiscale properties of persistent homology to address this task. To this end, we introduce a novel framework using functional topological summaries, such as Betti curves and persistence landscapes, for identifying and describing non-random patterns in spatial data. In particular, we propose a non-parametric one-sample permutation test for spatial dependence and investigate its utility across both simulated and real spatial omics data. Our vectorised approach outperforms baseline methods at accurately detecting spatial dependence. Further, we find that our method is more robust to outliers than alternative tests using Moran's I.

We present the first formally-verified Internet router, which is part of the SCION Internet architecture. SCION routers run a cryptographic protocol for secure packet forwarding in an adversarial environment. We verify both the protocol's network-wide security properties and low-level properties of its implementation. More precisely, we develop a series of protocol models by refinement in Isabelle/HOL and we use an automated program verifier to prove that the router's Go code satisfies memory safety, crash freedom, freedom from data races, and adheres to the protocol model. Both verification efforts are soundly linked together. Our work demonstrates the feasibility of coherently verifying a critical network component from high-level protocol models down to performance-optimized production code, developed by an independent team. In the process, we uncovered critical bugs in both the protocol and its implementation, which were confirmed by the code developers, and we strengthened the protocol's security properties. This paper explains our approach, summarizes the main results, and distills lessons for the design and implementation of verifiable systems, for the handling of continuous changes, and for the verification techniques and tools employed.

The magnitude of a metric space is a novel invariant that provides a measure of the 'effective size' of a space across multiple scales, while also capturing numerous geometrical properties, such as curvature, density, or entropy. We develop a family of magnitude-based measures of the intrinsic diversity of latent representations, formalising a novel notion of dissimilarity between magnitude functions of finite metric spaces. Our measures are provably stable under perturbations of the data, can be efficiently calculated, and enable a rigorous multi-scale characterisation and comparison of latent representations. We show their utility and superior performance across different domains and tasks, including (i) the automated estimation of diversity, (ii) the detection of mode collapse, and (iii) the evaluation of generative models for text, image, and graph data.

Deep learning models have seen significant successes in numerous applications, but their inner workings remain elusive. The purpose of this work is to quantify the learning process of deep neural networks through the lens of a novel topological invariant called magnitude. Magnitude is an isometry invariant; its properties are an active area of research as it encodes many known invariants of a metric space. We use magnitude to study the internal representations of neural networks and propose a new method for determining their generalisation capabilities. Moreover, we theoretically connect magnitude dimension and the generalisation error, and demonstrate experimentally that the proposed framework can be a good indicator of the latter.

Vanadium dioxide undergoes a metal-insulator transition (MIT) from an insulating (monoclinic) to a metallic (tetragonal) phase close to room temperature, which makes it a promising functional material for many applications, e.g. as chemical sensors. Not much is known about its surface and interface properties, although these are critical in many of its applications. This work presents an atomic-scale investigation of the tetragonal rutile VO$_2$(110)$_\text{T}$ single-crystal surface and reports results obtained with scanning tunneling microscopy (STM), low-energy electron diffraction (LEED) and X-ray photoelectron spectroscopy (XPS), supported by density-functional theory-based (DFT) calculations. The surface reconstructs into an oxygen-rich (2$\times$2) superstructure that coexists with small patches of the underlying, unreconstructed (110)-(1$\times$1) surface. The best structural model for the (2$\times$2) surface termination, conceptually derived from a vanadium pentoxide (001) monolayer, consists of rings of corner-shared tetrahedra. Over a wide range of oxygen chemical potentials this reconstruction is more stable than the unreconstructed (110) surface as well as models proposed in the literature.

Strange metal behavior is ubiquitous to correlated materials ranging from cuprate superconductors to bilayer graphene. There is increasing recognition that it arises from physics beyond the quantum fluctuations of a Landau order parameter which, in quantum critical heavy fermion antiferromagnets, may be realized as critical Kondo entanglement of spin and charge. The dynamics of the associated electronic delocalization transition could be ideally probed by optical conductivity, but experiments in the corresponding frequency and temperature ranges have remained elusive. We present terahertz time-domain transmission spectroscopy on molecular beam epitaxy-grown thin films of YbRh$_2$Si$_2$, a model strange metal compound. We observe frequency over temperature scaling of the optical conductivity as a hallmark of beyond-Landau quantum criticality. Our discovery implicates critical charge fluctuations as playing a central role in the strange metal behavior, thereby elucidating one of the longstanding mysteries of correlated quantum matter., Comment: preprint with 15 pages, 4 figures

The (001) surfaces of cleaved Sr$_3$Ru$_2$O$_7$ and Sr$_2$RuO$_4$ samples were investigated using low-temperature scanning tunneling microscopy and density functional theory calculations. Intrinsic defects are not created during cleaving. This experimental observation is consistent with calculations, where the formation energy for a Sr and O vacancy, 4.19 eV and 3.81 eV, respectively, is significantly larger than that required to cleave the crystal, 1.11 eV/(1 $\times$ 1) unit cell. Surface oxygen vacancies can be created through electron bombardment, however, and their appearance is shown to vary strongly with the imaging conditions. Point defects observed on as-cleaved surfaces result from bulk impurities and adsorption from the residual gas.

A comprehensive bulk and surface investigation of high-quality In$_2$O$_3$(001) single crystals is reported. The transparent-yellow, cube-shaped single crystals were grown using the flux method. Inductively coupled plasma mass spectrometry (ICP-MS) reveals small residues of Pb, Mg, and Pt in the crystals. Four-point-probe measurements show a resistivity of 2.0 $\pm$ 0.5 $\times$ 10$^5$ {\Omega} cm, which translates into a carrier concentration of $\approx$10$^{12}$ cm$^{-3}$. The results from x-ray diffraction (XRD) measurements revise the lattice constant to 10.1150(5) {\AA} from the previously accepted value of 10.117 {\AA}. Scanning tunneling microscopy (STM) images of a reduced (sputtered/annealed) and oxidized (exposure to atomic oxygen at 300 {\deg}C) surface show a step height of 5 {\AA}, which indicates a preference for one type of surface termination. The surfaces stay flat without any evidence for macroscopic faceting under any of these preparation conditions. A combination of low-energy ion scattering (LEIS) and atomically resolved STM indicates an indium-terminated surface with small islands of 2.5 {\AA} height under reducing conditions, with a surface structure corresponding to a strongly distorted indium lattice. Scanning tunneling spectroscopy (STS) reveals a pronounced surface state at the Fermi level ($E_F$). Photoelectron spectroscopy (PES) shows additional, deep-lying band gap states, which can be removed by exposure of the surface to atomic oxygen. Oxidation also results in a shoulder at the O 1s core level at a higher binding energy, possibly indicative of a surface peroxide species. A downward band bending of 0.4 eV is observed for the reduced surface, while the band bending of the oxidized surface is of the order of 0.1 eV or less.