arXiv subjects

David Lanners

Publications and source records attributed to David Lanners.

3 recordsLinked to original sources

arXiv2026

From Frames to Features: Scalable Zigzag Persistence for Binary Video

Zigzag persistence tracks topological features in spatio-temporal data through combinatorial invariants called barcodes. For binary videos, existing methods are bottlenecked by the construction of prohibitively large cubical complexes and performing Gaussian elimination on large boundary matrices, rendering high-resolution videos out of reach. We show that the $H_0$ and $H_1$ barcodes can be extracted directly from connected-component dynamics. By encoding these dynamics in a graph, we bypass cubical complexes entirely and are able to leverage the near-linear time barcode decomposition algorithm by Dey and Hou, leading to significant speedups. The total runtime of our pipeline is dominated by the construction of the underlying graph structures, which scales linearly with pixel count and is embarrassingly parallel across frames, ensuring excellent scalability. We demonstrate how this approach enables zigzag persistence on 4k video at real-time rates on consumer hardware.

math.AT↗

arXiv2026

Computing Diffusion Geometry

Calculus and geometry are ubiquitous in the theoretical modelling of scientific phenomena, but have historically been very challenging to apply directly to real data as statistics. Diffusion geometry is a new theory that reformulates classical calculus and geometry in terms of a diffusion process, allowing these theories to generalise beyond manifolds and be computed from data. This work introduces a new computational framework for diffusion geometry that substantially broadens its practical scope and improves its precision, robustness to noise, and computational complexity. We present a range of new computational methods, including all the standard objects from vector calculus and Riemannian geometry, and apply them to solve spatial PDEs and vector field flows, find geodesic (intrinsic) distances, curvature, and several new topological tools like de Rham cohomology, circular coordinates, and Morse theory. These methods are data-driven, scalable, and can exploit highly optimised numerical tools for linear algebra.

math.DG↗

arXiv2025

Emergent photons and mechanisms of confinement

We numerically study $\mathbb{Z}_N$ lattice gauge theories in 4D as prototypical models of systems with $\mathbb{Z}_N$ 1-$\textit{form symmetry}$. For $N \geq 3$, we provide evidence that such systems exhibit not only the expected phases with spontaneously broken/restored symmetry but also a third photon phase. When present, the 1-form symmetry provides a precise notion of confinement, and it is commonly believed that confinement ensues due to the proliferation of extended, string-like objects known as $\textit{center vortices}$, which carry a $\mathbb{Z}_N$ flux. However, this picture is challenged by the three-phase scenario investigated here. We show that both the confined and the photon phases are associated with the proliferation of center vortices and that the key difference between them lies in whether or not vortex-junctions - the $\textit{monopoles}$ - proliferate.

hep-lat↗