arXiv subjects

Konrad Mundinger

Publications and source records attributed to Konrad Mundinger.

5 recordsLinked to original sources

arXiv2026

Fast Isotopy Computation for T-Curves

A T-curve of degree $d$ is given by a regular unimodular triangulation of $d \cdot \Delta_2$ together with a sign distribution on its lattice points. By Viro's Patchworking Theorem, this determines the ambient isotopy type (a.k.a. real scheme) of a smooth real plane projective algebraic curve of the same degree. We present a near-quadratic time algorithm for extracting that isotopy type from the triangulation and the signs. Through a GPU-accelerated implementation, this allows one to compute billions of real schemes per second, enabling exhaustive enumeration at scale. This algorithm was essential for our recent construction of all 121 real schemes of degree seven by T-curves.

math.AG↗

arXiv2026

Limits of combinatorial patchworking

It is shown that there are real plane algebraic curves of degree eight that cannot be realized as T-curves, i.e., via combinatorial patchworking. In fact, this holds for several real schemes (i.e., ambient isotopy types) with the maximal number of real components, called $M$-curves. On the other hand, each nonempty real scheme of lower degree, maximal or not, arises as a T-curve. By constructing one patchwork of the dilated triangle $d\cdot\Delta_2$ for each nonempty real scheme of degree $d\leq 7$, we provide an explicit method for constructing polynomials realizing these real schemes. This resolves a question of Itenberg and Viro (1996).

math.AG↗

arXiv2025

Neural Discovery in Mathematics: Do Machines Dream of Colored Planes?

We demonstrate how neural networks can drive mathematical discovery through a case study of the Hadwiger-Nelson problem, a long-standing open problem at the intersection of discrete geometry and extremal combinatorics that is concerned with coloring the plane while avoiding monochromatic unit-distance pairs. Using neural networks as approximators, we reformulate this mixed discrete-continuous geometric coloring problem with hard constraints as an optimization task with a probabilistic, differentiable loss function. This enables gradient-based exploration of admissible configurations that most significantly led to the discovery of two novel six-colorings, providing the first improvement in thirty years to the off-diagonal variant of the original problem. Here, we establish the underlying machine learning approach used to obtain these results and demonstrate its broader applicability through additional numerical insights.

cs.LG↗

arXiv2024

Extending the Continuum of Six-Colorings

We present two novel six-colorings of the Euclidean plane that avoid monochromatic pairs of points at unit distance in five colors and monochromatic pairs at another specified distance $d$ in the sixth color. Such colorings have previously been known to exist for $0.41 < \sqrt{2} - 1 \le d \le 1 / \sqrt{5} < 0.45$. Our results significantly expand that range to $0.354 \le d \le 0.657$, the first improvement in 30 years. Notably, the constructions underlying this were derived by formalizing colorings suggested by a custom machine learning approach.

math.CO↗

arXiv2024

Neural Parameter Regression for Explicit Representations of PDE Solution Operators

We introduce Neural Parameter Regression (NPR), a novel framework specifically developed for learning solution operators in Partial Differential Equations (PDEs). Tailored for operator learning, this approach surpasses traditional DeepONets (Lu et al., 2021) by employing Physics-Informed Neural Network (PINN, Raissi et al., 2019) techniques to regress Neural Network (NN) parameters. By parametrizing each solution based on specific initial conditions, it effectively approximates a mapping between function spaces. Our method enhances parameter efficiency by incorporating low-rank matrices, thereby boosting computational efficiency and scalability. The framework shows remarkable adaptability to new initial and boundary conditions, allowing for rapid fine-tuning and inference, even in cases of out-of-distribution examples.

cs.LG↗