In this work, the classical Nelson -- Hadwiger problem is studied which lies on the edge of combinatorial geometry and graph theory. It concerns colorings of distance graphs in $ {\mathbb R}^n $, i.e., graphs such that their vertices are vectors and...
As a standard example in toric geometry, the Atiyah flop of a $(-1,-1)$-curve in a smooth 3-fold can be described combinatorially in terms of the two possible triangulations of a square cone. The flop of $(-2,0)$-curve cannot be realised in terms of...
equilateral-triangle faces. This is an example of Goldberg polyhedron GPIII(2,0) or {3+,3}2,0, containing triangular and hexagonal faces. Its dual is the alternate-triakis
In this study, we report the first Cu-filled through silicon via (TSV) integrated ion trap. TSVs are placed directly underneath electrodes as vertical interconnections between ion trap and a glass interposer, facilitating the arbitrary geometry desig...
In this contribution, we study systems with a finite number of degrees of freedom as in robotics. A key idea is to consider the mass tensor associated to the kinetic energy as a metric in a Riemannian configuration space. We apply Pontryagin's framew...
Shape completion, i.e., predicting the complete geometry of an object from a partial observation, is highly relevant for several downstream tasks, most notably robotic manipulation. When basing planning or prediction of real grasps on object shape re...
Special geometry is most known from 4-dimensional N=2 supergravity, though it contains also quaternionic and real geometries. In this review, we first repeat the connections between the various special geometries. Then the constructions are given s...
Those are the notes of the two talks I gave in april 2013 in St-John (US Virgin Islands) during the Simons Symposium on non-Archimedean and tropical geometry. They essentially consist of a survey of Hrushovski and Loeser's work on the homotopy type o...
We theoretically study channel plasmon-polaritons (CPPs) with a geometry similar to that in recent experiments at telecom wavelengths (Bozhevolnyi et al., Nature 440, 508 (2006)). The CPP modal shape, dispersion relation, and losses are simulated u...
A parametric aircraft geometry tool (⭐ 724)
Silver nanocrystals have been prepared by reacting silver nitrate with ascorbic acid in aqueous solution containing a low concentration of a commercial polynaphtalene sulphonate polymer (Daxad 19). Various crystalline morphologies have been obtained...
We study open book foliations on surfaces in 3-manifolds, and give applications to contact geometry of dimension 3. We prove a braid-theoretic formula of the self-linking number of transverse links, which reveals an unexpected link to the Johnson-Mor...
Continuous image morphing is a classical task in image processing. The metamorphosis model proposed by Trouvé, Younes and coworkers casts this problem in the frame of Riemannian geometry and geodesic paths between images. The associated metric in th...
Recent work studying the generalization of diffusion models with UNet-based denoisers reveals inductive biases that can be expressed via geometry-adaptive harmonic bases. However, in practice, more recent denoising networks are often based on transfo...
With special relativity, we seem to be facing a conundrum. It is a very well-tested theory; in this way, the Minkowski spacetime must be "capturing" essential features of space and time. However, its geometry seems to be incompatible with any sort of...
An analogy with real Clifford algebras on even-dimensional vector spaces suggests to assign a couple of space and time dimensions modulo 8 to any algebra (represented over a complex Hilbert space) containing two self-adjoint involutions and an anti-u...
The Standard Model of particle physics can be deduced from a small number of axioms within Connes' noncommutative geometry (NCG). Boyle and Farnsworth [New J. Phys. 16 (2014) 123027] proposed to interpret Connes' approach as an algebra extension in t...
In this two-part paper we propose an extension of Connes' notion of even spectral triple to the Lorentzian setting. This extension, which we call a spectral spacetime, is discussed in part II where several natural examples are given which are not cov...
The Deutsch multiverse is collection of parallel universes. In this article a formal theory and a topos-theoretic model of the Deutsch multiverse are given. For this the Lawvere-Kock Synthetic Differential Geometry and topos models for smooth infin...
We report on the experimental observation of solitons propagating along a torus of fluid. We show that such a periodic system leads to significant differences compared to the classical plane geometry. In particular, we highlight the observation of su...