Tomás S. R. Silva

Daniel Fadel, Henrique N. Sá Earp, Tomás S. R. Silva

This survey revisits classical results in vector calculus and analysis by exploring a generalised perspective on the exterior derivative, interpreting it as a measure of "infinitesimal flux". This viewpoint leads to a higher-dimensional analogue of the Mean Value Theorem, valid for differential $k$-forms, and provides a natural formulation of Stokes' theorem that mirrors the exact hypotheses of the Fundamental Theorem of Calculus -- without requiring full $C^1$ smoothness of the differential form. As a numerical application, we propose an algorithm for exterior differentiation in $\mathbb{R}^n$ that relies solely on black-box access to the differential form, offering a practical tool for computation without the need for mesh discretization or explicit symbolic expressions.

Edward Hirst, Henrique N. Sá Earp, Tomás S. R. Silva

The neural Willmore flow of a closed oriented $2$-surface in $\mathbb{R}^3$ is introduced as a natural evolution process to minimise the Willmore energy, which is the squared $L^2$-norm of mean curvature. Neural architectures are used to model maps from topological $2d$ domains to $3d$ Euclidean space, where the learning process minimises a PINN-style loss for the Willmore energy as a functional on the embedding. Training reproduces the expected round sphere for genus $0$ surfaces, and the Clifford torus for genus $1$ surfaces, respectively. Furthermore, the experiment in the genus $2$ case provides a novel approach to search for minimal Willmore surfaces in this open problem.

Tomás S. R. Silva

We introduce a nonnegative functional on the space of line arrangements in $\mathbb{P}^2$ that vanishes precisely on free arrangements, obtained as a semicontinuous relaxation of Saito's criterion for freeness. Given an arrangement $\mathcal{A}$ of $n$ lines with candidate exponents $(d_1, d_2)$, we parameterize the spaces of logarithmic derivations of degrees $d_1$ and $d_2$ via the null spaces of the associated derivation matrices and express the Saito determinant as a bilinear map into the space of degree $n$ polynomials. The functional then admits a natural geometric interpretation: it measures the squared sine of the angle between the image of this bilinear map and the direction of the defining polynomial $Q(\mathcal{A})$ in coefficient space, and equals zero if and only if its image contains the line spanned by $Q(\mathcal{A})$. This provides a computable measure of how far a given arrangement is from admitting a free basis of logarithmic derivations of the expected degrees. Using this functional as a reward signal, we develop a sequential construction procedure in which lines are added one at a time so as to minimize the angular distance to freeness, implemented via reinforcement learning with an adaptive curriculum over arrangement sizes and exponent types. Our results suggest that semicontinuous relaxation techniques, grounded in the geometry of polynomial coefficient spaces, offer a viable approach to the computational exploration of freeness in the theory of line arrangements.