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Christian Lange

Publications and source records attributed to Christian Lange.

At least 19 recordsLinked to original sources
arXiv2026

Hausdorff-type metric geometry of the space of Cauchy hypersurfaces

We equip the space of Cauchy hypersurfaces in a globally hyperbolic spacetime with a natural Hausdorff-type metric. For a timelike Cauchy complete, smooth Lorentzian manifold with a compact Cauchy hypersurface, we show that the resulting metric space is geodesic and proper. We further discuss extensions to more general synthetic Lorentzian settings. For this purpose, we generalize results on completeness properties of spacetimes due to Beem and Takahashi.

math.DG↗
arXiv2025

Quantitative Lorentzian isoperimetric inequalities in conical Minkowski spacetimes

We establish optimal stability estimates in terms of the Fraenkel asymmetry with universal dimensional constants for a Lorentzian isoperimetric inequality due to Bahn and Ehrlich and, as a consequence, for a special version of a Lorentzian isoperimetric inequality due to Cavalletti and Mondino. For the Bahn--Ehrlich inequality the Fraenkel asymmetry enters the stability result quadratically like in the Euclidean case while for the Cavalletti--Mondino inequality the Fraenkel asymmetry enters linearly. As it turns out, refining the latter inequality through an additional geometric term allows us to recover the more common quadratic stability behavior. Along the way, we provide simple, self-contained proofs for the above isoperimetric-type inequalities. Moreover, in a fixed conical Minkowski spacetime, we use a Lipschitz bound, naturally provided by the causal structure, to upgrade our quantitative control to a Hausdorff stability estimate. This estimate is formulated in terms of a distance defined by Bahn and Ehrlich, which restricts to a natural Hausdorff-type metric on the space of Cauchy hypersurfaces.

math.DG↗
arXiv2025

Investigating Different Geo Priors for Image Classification

Species distribution models encode spatial patterns of species occurrence making them effective priors for vision-based species classification when location information is available. In this study, we evaluate various SINR (Spatial Implicit Neural Representations) models as a geographical prior for visual classification of species from iNaturalist observations. We explore the impact of different model configurations and adjust how we handle predictions for species not included in Geo Prior training. Our analysis reveals factors that contribute to the effectiveness of these models as Geo Priors, factors that may differ from making accurate range maps.

cs.CV↗
arXiv2025

Feedforward Few-shot Species Range Estimation

Knowing where a particular species can or cannot be found on Earth is crucial for ecological research and conservation efforts. By mapping the spatial ranges of all species, we would obtain deeper insights into how global biodiversity is affected by climate change and habitat loss. However, accurate range estimates are only available for a relatively small proportion of all known species. For the majority of the remaining species, we typically only have a small number of records denoting the spatial locations where they have previously been observed. We outline a new approach for few-shot species range estimation to address the challenge of accurately estimating the range of a species from limited data. During inference, our model takes a set of spatial locations as input, along with optional metadata such as text or an image, and outputs a species encoding that can be used to predict the range of a previously unseen species in a feedforward manner. We evaluate our approach on two challenging benchmarks, where we obtain state-of-the-art range estimation performance, in a fraction of the compute time, compared to recent alternative approaches.

cs.CV↗
arXiv2024

Combining Observational Data and Language for Species Range Estimation

Species range maps (SRMs) are essential tools for research and policy-making in ecology, conservation, and environmental management. However, traditional SRMs rely on the availability of environmental covariates and high-quality species location observation data, both of which can be challenging to obtain due to geographic inaccessibility and resource constraints. We propose a novel approach combining millions of citizen science species observations with textual descriptions from Wikipedia, covering habitat preferences and range descriptions for tens of thousands of species. Our framework maps locations, species, and text descriptions into a common space, facilitating the learning of rich spatial covariates at a global scale and enabling zero-shot range estimation from textual descriptions. Evaluated on held-out species, our zero-shot SRMs significantly outperform baselines and match the performance of SRMs obtained using tens of observations. Our approach also acts as a strong prior when combined with observational data, resulting in more accurate range estimation with less data. We present extensive quantitative and qualitative analyses of the learned representations in the context of range estimation and other spatial tasks, demonstrating the effectiveness of our approach.

cs.DB↗
arXiv2024

Generating Binary Species Range Maps

Accurately predicting the geographic ranges of species is crucial for assisting conservation efforts. Traditionally, range maps were manually created by experts. However, species distribution models (SDMs) and, more recently, deep learning-based variants offer a potential automated alternative. Deep learning-based SDMs generate a continuous probability representing the predicted presence of a species at a given location, which must be binarized by setting per-species thresholds to obtain binary range maps. However, selecting appropriate per-species thresholds to binarize these predictions is non-trivial as different species can require distinct thresholds. In this work, we evaluate different approaches for automatically identifying the best thresholds for binarizing range maps using presence-only data. This includes approaches that require the generation of additional pseudo-absence data, along with ones that only require presence data. We also propose an extension of an existing presence-only technique that is more robust to outliers. We perform a detailed evaluation of different thresholding techniques on the tasks of binary range estimation and large-scale fine-grained visual classification, and we demonstrate improved performance over existing pseudo-absence free approaches using our method.

q-bio.QM↗
arXiv2023

Active Learning-Based Species Range Estimation

We propose a new active learning approach for efficiently estimating the geographic range of a species from a limited number of on the ground observations. We model the range of an unmapped species of interest as the weighted combination of estimated ranges obtained from a set of different species. We show that it is possible to generate this candidate set of ranges by using models that have been trained on large weakly supervised community collected observation data. From this, we develop a new active querying approach that sequentially selects geographic locations to visit that best reduce our uncertainty over an unmapped species' range. We conduct a detailed evaluation of our approach and compare it to existing active learning methods using an evaluation dataset containing expert-derived ranges for one thousand species. Our results demonstrate that our method outperforms alternative active learning methods and approaches the performance of end-to-end trained models, even when only using a fraction of the data. This highlights the utility of active learning via transfer learned spatial representations for species range estimation. It also emphasizes the value of leveraging emerging large-scale crowdsourced datasets, not only for modeling a species' range, but also for actively discovering them.

cs.LG↗
arXiv2023

Superradiant and subradiant states in lifetime-limited organic molecules through laser-induced tuning

An array of radiatively coupled emitters is an exciting new platform for generating, storing, and manipulating quantum light. However, the simultaneous positioning and tuning of multiple lifetime-limited emitters into resonance remains a significant challenge. Here we report the creation of superradiant and subradiant entangled states in pairs of lifetime-limited and sub-wavelength spaced organic molecules by permanently shifting them into resonance with laser-induced tuning. The molecules are embedded as defects in an organic nanocrystal. The pump light redistributes charges in the nanocrystal and dramatically increases the likelihood of resonant molecules. The frequency spectra, lifetimes, and second-order correlation agree with a simple quantum model. This scalable tuning approach with organic molecules provides a pathway for observing collective quantum phenomena in sub-wavelength arrays of quantum emitters.

quant-ph↗
arXiv2023

Spatial Implicit Neural Representations for Global-Scale Species Mapping

Estimating the geographical range of a species from sparse observations is a challenging and important geospatial prediction problem. Given a set of locations where a species has been observed, the goal is to build a model to predict whether the species is present or absent at any location. This problem has a long history in ecology, but traditional methods struggle to take advantage of emerging large-scale crowdsourced datasets which can include tens of millions of records for hundreds of thousands of species. In this work, we use Spatial Implicit Neural Representations (SINRs) to jointly estimate the geographical range of 47k species simultaneously. We find that our approach scales gracefully, making increasingly better predictions as we increase the number of species and the amount of data per species when training. To make this problem accessible to machine learning researchers, we provide four new benchmarks that measure different aspects of species range estimation and spatial representation learning. Using these benchmarks, we demonstrate that noisy and biased crowdsourced data can be combined with implicit neural representations to approximate expert-developed range maps for many species.

cs.LG↗
arXiv2023

On the local maximizers of higher capacity ratios

We prove an analogue of the 4-dimensional local Viterbo conjecture for the higher Ekeland-Hofer capacities: on the space of 4-dimensional smooth star-shaped domains of unitary volume, endowed with the $C^3$ topology, the local maximizers of the $k$-th Ekeland-Hofer capacities are those domains symplectomorphic to suitable rational ellipsoids.

math.SG↗
arXiv2023

Orbifolds and manifold quotients with upper curvature bounds

We characterize Riemannian orbifolds with an upper curvature bound in the Alexandrov sense as reflectofolds, i.e. Riemannian orbifolds all of whose local groups are generated by reflections, with the same upper bound on the sectional curvature. Combined with a result by Lytchak--Thorbergsson this implies that a quotient of a Riemannian manifold by a closed group of isometries has locally bounded curvature (from above) in the Alexandrov sense if and only if it is a reflectofold.

math.DG↗
arXiv2023

On continuous billiard and quasigeodesic flows characterizing alcoves and isosceles tetrahedra

We characterize fundamental domains of affine reflection groups as those polyhedral convex bodies which support a continuous billiard dynamics. We interpret this characterization in the broader context of Alexandrov geometry and prove an analogous characterization for isosceles tetrahedra in terms of continuous quasigeodesic flows. Moreover, we show an optimal regularity result for convex bodies: the billiard dynamics is continuous if the boundary is of class $\mathcal{C}^{2,1}$. In particular, billiard trajectories converge to geodesics on the boundary in this case. Our proof of the latter continuity statement is based on Alexandrov geometry methods that we discuss resp. establish first.

math.DG↗
arXiv2022

Higher systolic inequalities for 3-dimensional contact manifolds

A contact form is called Besse when the associated Reeb flow is periodic. We prove that Besse contact forms on closed connected 3-manifolds are the local maximizers of suitable higher systolic ratios. Our result extends earlier ones for Zoll contact forms, that is, contact forms whose Reeb flow defines a free circle action.

math.SG↗
arXiv2021

How highly connected can an orbifold be?

On the one hand, we provide the first examples of arbitrarily highly connected (compact) bad orbifolds. On the other hand, we show that n-connected n-orbifolds are manifolds. The latter improves the best previously known bound of Lytchak by roughly a factor of 2. For compact orbifolds and in most dimensions we prove slightly better bounds. We obtain sharp results up to dimension 5.

math.DG↗
arXiv2021

Sharp systolic inequalities for rotationally symmetric 2-orbifolds

We show that suitably defined systolic ratios are globally bounded from above on the space of rotationally symmetric spindle orbifolds and that the upper bound is attained precisely at so-called Besse metrics, i.e. Riemannian orbifold metrics all of whose geodesics are closed.

math.DG↗
arXiv2021

A diameter gap for quotients of the unit sphere

We prove that for any isometric action of a group on a unit sphere of dimension larger than one, the quotient space has diameter zero or larger than a universal dimension-independent positive constant.

math.MG↗