arXiv subjects

Roland Denis

Publications and source records attributed to Roland Denis.

4 recordsLinked to original sources

arXiv2025

An Algorithm to compute the Kronecker cone and other moment cones

We describe a new algorithm that computes the minimal list of inequalities for the moment cone of any representation of a complex reductive group, with implementation details for two fundamental cases: the Kronecker cone (governing the asymptotic support of Kronecker coefficients) and the fermionic cone. These correspond to the actions of ${\mathrm GL}\_{d\_1}({\mathbb C})\times\cdots\times {\mathrm GL}\_{d\_s}({\mathbb C})$ on ${\mathbb C}^{d\_1}\otimes\cdots\otimes {\mathbb C}^{d\_s}$ and ${\mathrm GL}\_d({\mathbb C})$ on $\bigwedge^r{\mathbb C}^d$, respectively. An implementation for these two cases in Python-Sage is available at https://ea-icj.github.io/. Our work overcomes the fundamental limitations that previously restricted such computations to cases like ${\mathbb C}^4\otimes{\mathbb C}^4\otimes{\mathbb C}^4$. The state-of-the-art method by Vergne-Walter faced two major bottlenecks: one from combinatorial geometry in finite-dimensional vector spaces, and another from deciding whether certain dominant morphisms are birational - a problem in effective algebraic geometry that lacked a direct algorithmic solution. We surmount these obstacles by: a novel use of Weyl group actions to master combinatorial complexity, and an original algorithm for deciding birationality that replaces previous workarounds relying on convex geometry. Our approach allow us to tackle problems at a new scale. We compute the minimal list of 5,333 (up to $\mathfrak S\_3$) inequalities for the Kronecker cone ${\mathbb C}^6\otimes{\mathbb C}^6\otimes{\mathbb C}^6$ in 2 hours. Furthermore, a parallel implementation computes the 64,792 (up to $\mathfrak S\_3$) inequalities for ${\mathbb C}^7\otimes{\mathbb C}^7\otimes{\mathbb C}^7$ in 188 hours.

math.AG↗

arXiv2023

The Hyperbolic Plane in $\mathbb{E}^3$

We build an explicit $C^1$ isometric embedding $f_{\infty}:\mathbb{H}^2\to\mathbb{E}^3$ of the hyperbolic plane whose image is relatively compact. Its limit set is a closed curve of Hausdorff dimension 1. Given an initial embedding $f_0$, our construction generates iteratively a sequence of maps by adding at each step $k$ a layer of $N_{k}$ corrugations. To understand the behavior of $df_\infty$ we introduce a $formal$ $corrugation$ $process$ leading to a $formal$ $analogue$ $Φ_{\infty}:\mathbb{H}^2\to \mathcal{L}(\mathbb{R}^2,\mathbb{R}^3)$. We show a self-similarity structure for $Φ_{\infty}$. We next prove that $df_\infty$ is close to $Φ_{\infty}$ up to a precision that depends on the sequence $N_*:= (N_{k})_k$. We then introduce the $pattern$ $maps$ $\boldsymbolν_{\infty}^Φ$ and $\boldsymbolν_{\infty}$, of respectively $Φ_{\infty}$ and $df_\infty$, that together with $df_0$ entirely describe the geometry of the Gauss maps associated to $Φ_{\infty}$ and $df_\infty$. For well chosen sequences of corrugation numbers, we finally show an asymptotic convergence of $\boldsymbolν_{\infty}$ towards $\boldsymbolν_{\infty}^Φ$ over circles of rational radii.

math.DG↗

arXiv2023

A multiphase Cahn-Hilliard system with mobilities and the numerical simulation of dewetting

We propose in this paper a new multiphase Cahn-Hilliard model with doubly degenerate mobilities. We prove by a formal asymptotic analysis that it approximates with second order accuracy the multiphase surface diffusion flow with mobility coefficients and surface tensions. To illustrate that it lends itself well to numerical approximation, we propose a simple and effective numerical scheme together with a very compact Matlab implementation. We provide the results of various numerical experiments to show the influence of mobility and surface tension coefficients. Thanks to its second order accuracy and its good suitability for numerical implementation, our model is very handy for tackling notably difficult surface diffusion problems. In particular, we show that it can be used very effectively to simulate numerically the dewetting of thin liquid tubes on arbitrary solid supports without requiring nonlinear boundary conditions.

math.NA↗

arXiv2022

Learning phase field mean curvature flows with neural networks

We introduce in this paper new and very effective numerical methods based on neural networks for the approximation of the mean curvature flow of either oriented or non-orientable surfaces. To learn the correct interface evolution law, our neural networks are trained on phase field representations of exact evolving interfaces. The structures of the networks draw inspiration from splitting schemes used for the discretization of the Allen-Cahn equation. But when the latter approximate the mean curvature motion of oriented interfaces only, the approach we propose extends very naturally to the non-orientable case. Through a variety of examples, we show that our networks, trained only on flows of smooth and simplistic interfaces, generalize very well to more complex interfaces, either oriented or non-orientable, and possibly with singularities. Furthermore, they can be coupled easily with additional constraints which opens the way to various applications illustrating the flexibility and effectiveness of our approach: mean curvature flows with volume constraint, multiphase mean curvature flows, numerical approximation of Steiner trees or minimal surfaces.

math.NA↗