arXiv subjects

Daniel Platt

Publications and source records attributed to Daniel Platt.

17 recordsLinked to original sources

arXiv2026

An isoperimetric problem for Fourier zeros of centrally symmetric convex bodies

Given a centrally symmetric convex body, consider the zero set of the Fourier transform of its characteristic function. We study how large the distance from this set to the origin can be when the volume of the body is fixed. A 2009 conjecture of Benguria, Levitin, and Parnovski asserts that the maximum is attained by a Euclidean ball. We disprove it in all dimensions greater than one. In the plane, every regular centrally symmetric polygon with at least twelve sides outperforms the disk, albeit by a tiny margin, and the regular dodecagon is best among them. In dimensions three and higher, no maximiser exists.

math.MG↗

arXiv2026

Nowhere-vanishing harmonic 1-forms on real loci of K3-fibred Calabi-Yau 3-folds

We develop an analytic construction of nowhere-vanishing harmonic $1$-forms on real loci of K3-fibred Calabi-Yau $3$-folds with collapsing Ricci-flat K\"ahler metrics. We apply our construction to examples whose real loci have connected components diffeomorphic to $S^1\times S^2$ and to both trivial and nontrivial mapping tori. As an application, we produce examples of compact $7$-manifold with holonomy $G_2$ via the Joyce-Karigiannis construction.

math.DG↗

arXiv2026

Non-uniqueness and symmetries for the Nirenberg problem using computer assistance

We apply verified numerics to the Nirenberg problem, proving that a genuine solution exists near two given computer-generated approximate solutions. This proves existence of a solution for a particular prescribed curvature that was previously predicted, but not proved, to exist. We are also able to determine the symmetry groups of the genuine solutions exactly, which in one case is different from the symmetry of the prescribed curvature. We expect the computer code for this proof can be reused to study other aspects of the Nirenberg problem.

math.DG↗

arXiv2025

IMProofBench: Benchmarking AI on Research-Level Mathematical Proof Generation

As the mathematical capabilities of large language models (LLMs) improve, it becomes increasingly important to evaluate their performance on research-level tasks at the frontier of mathematical knowledge. However, existing benchmarks are limited, as they focus solely on final-answer questions or high-school competition problems. To address this gap, we introduce IMProofBench, a private benchmark consisting of 77 peer-reviewed problems developed by expert mathematicians. Each problem requires a detailed proof and is paired with subproblems that have final answers, supporting both an evaluation by human experts and a large-scale quantitative analysis through automated grading. Furthermore, unlike prior benchmarks, the evaluation setup simulates a realistic research environment: models operate in an agentic framework with tools like web search for literature review and mathematical software such as SageMath. Our results show that current LLMs can already solve a significant percentage of research-level questions. IMProofBench will continue to evolve as a dynamic benchmark in collaboration with the mathematical community, ensuring its relevance for evaluating the next generation of LLMs.

cs.CL↗

arXiv2025

Lower bounds for the reach and applications

The reach of a submanifold of $\mathbb{R}^N$ is defined as the largest radius of a tubular neighbourhood around the submanifold that avoids self-intersections. While essential in geometric and topological applications, computing the reach explicitly is notoriously difficult. In this paper, we introduce a rigorous and practical method to compute a guaranteed lower bound for the reach of a submanifold described as the common zero-set of finitely many smooth functions, not necessarily polynomials. Our algorithm uses techniques from numerically verified proofs and is particularly suitable for high-performance parallel implementations. We illustrate the utility of this method through several applications. Of special note is a novel algorithm for computing the homology groups of planar curves, achieved by constructing a cubical complex that deformation retracts onto the curve--an approach potentially extendable to higher-dimensional manifolds. Additional applications include an improved comparison inequality between intrinsic and extrinsic distances for submanifolds of $\mathbb{R}^N$, lower bounds for the first eigenvalue of the Laplacian on algebraic varieties and explicit bounds on how much smooth varieties can be deformed without changing their diffeomorphism type.

math.NA↗

arXiv2025

Local uniqueness of the Black String with small circle size

In this article we study uniqueness of the Black String, i.e. the product of 4-dimensional Schwarzschild space with a circle of length L. In arXiv:2410.20967, this was reduced to a non-linear elliptic PDE, and we use this setup to show that for small L the Black String is infinitesimally rigid as a Ricci-flat metric. Using a fixed point theorem, we prove that this implies local uniqueness, i.e. there exist no other Ricci-flat metrics near the Black String, and we give bounds for the size of the neighborhood in which the Black String is unique. We compare this with the toy problem of a scalar field satisfying an elliptic equation that was already solved in arXiv:2410.20967 using different methods. In this case we can use the fixed point theorem method to prove not just a local but a global statement.

hep-th↗

arXiv2025

Examples of real stable bundles on K3 surfaces

Motivated by gauge theory on manifolds with exceptional holonomy, we construct examples of stable bundles on K3 surfaces that are invariant under two involutions: one is holomorphic; and the other is anti-holomorphic. These bundles are obtained via the monad construction, and stability is examined using the Generalised Hoppe Criterion of Jardim-Menet-Prata-S\'a Earp, which requires verifying an arithmetic condition for elements in the Picard group of the surfaces. We establish this by using computer aid in two critical steps: first, we construct K3 surfaces with small Picard group-one branched double cover of $\mathbb{P}^1 \times \mathbb{P}^1$ with Picard rank $2$ using a new method which may be of independent interest; and second, we verify the arithmetic condition for carefully chosen elements of the Picard group, which provides a systematic approach for constructing further examples.

math.AG↗

arXiv2024

Towards a uniqueness theorem for static black holes in Kaluza-Klein theory with small circle size

Kaluza-Klein theory, by which we mean vacuum gravity in 5-dimensions, with asymptotics that are a product of a circle with Minkowski spacetime, has a variety of different static black hole solutions; localized black holes and the homogeneous and inhomogeneous black strings. There is currently no uniqueness theorem for the solutions, and for fixed circle size multiple solutions with the same mass co-exist. Intuitively for small circle sizes we might expect the theory truncates to become 4-dimensional, and correspondingly the only black holes are the homogeneous black strings. Thus we conjecture that for fixed mass and sufficiently small circle size, the only black holes are homogeneous ones. Here we give evidence that this is indeed the case. Firstly we introduce a toy scalar field model with a potential that allows tachyonic behaviour. Putting this theory on a product of Minkowski with a circle gives an analogous set of static homogeneous and inhomogeneous solutions to that of the black holes. We prove that solutions must be homogeneous for small circle sizes, the analog of our conjecture for this toy model. A weaker statement that is straightforward to derive is a bound on how inhomogeneous a solution can be - putting this scalar theory in a large but finite cavity, a norm of the wavefunction of the Kaluza-Klein modes can be shown to vanish in the small circle limit. Turning to the full gravitational theory, we employ a metric ansatz that imposes static axisymmetry, encompasses the homogeneous and inhomogeneous black strings (but not the localized solutions) and allows us to measure inhomogeneity of a solution. Employing a finite cavity and imposing boundary conditions that are compatible with homogeneity we show a similar result; the norm of certain Kaluza-Klein modes is bounded by the circle size, providing evidence that our conjecture is true.

hep-th↗

arXiv2024

Harmonic $1$-forms on real loci of Calabi-Yau manifolds

We numerically study whether there exist nowhere vanishing harmonic $1$-forms on the real locus of some carefully constructed examples of Calabi-Yau manifolds, which would then give rise to potentially new examples of $G_2$-manifolds and an explicit description of their metrics. We do this in two steps: first, we use a neural network to compute an approximate Calabi-Yau metric on each manifold. Second, we use another neural network to compute an approximately harmonic $1$-form with respect to the approximate metric, and then inspect the found solution. On two manifolds existence of a nowhere vanishing harmonic $1$-form can be ruled out using differential geometry. The real locus of a third manifold is diffeomorphic to $S^1 \times S^2$, and our numerics suggest that when the Calabi-Yau metric is close to a singular limit, then it admits a nowhere vanishing harmonic $1$-form. We explain how such an approximate solution could potentially be used in a numerically verified proof for the fact that our example manifold must admit a nowhere vanishing harmonic $1$-form.

math.DG↗

arXiv2023

Spin(7)-instantons on Joyce's first examples of compact Spin(7)-manifolds

We construct $Spin(7)$-instantons on one of Joyce's compact $Spin(7)$-manifolds. The underlying compact $Spin(7)$-manifold given by Joyce is the same as in Lewis' construction of $Spin(7)$-instantons. However, our construction method and the resulting instantons are new. The compact $Spin(7)$-manifold is constructed by gluing a $Spin(7)$-orbifold and certain local model spaces around the orbifold singularities. We construct our instantons by gluing non-flat connections on the local model spaces to a flat connection on the $Spin(7)$-orbifold. We deliver more than $20,000$ new four-parameter families of examples of $Spin(7)$-instantons within the structure groups $SO(3), SO(4), SO(5), SO(7)$, and $SO(8)$.

math.DG↗

arXiv2023

Towards quantum-enabled cell-centric therapeutics

In recent years, there has been tremendous progress in the development of quantum computing hardware, algorithms and services leading to the expectation that in the near future quantum computers will be capable of performing simulations for natural science applications, operations research, and machine learning at scales mostly inaccessible to classical computers. Whereas the impact of quantum computing has already started to be recognized in fields such as cryptanalysis, natural science simulations, and optimization among others, very little is known about the full potential of quantum computing simulations and machine learning in the realm of healthcare and life science (HCLS). Herein, we discuss the transformational changes we expect from the use of quantum computation for HCLS research, more specifically in the field of cell-centric therapeutics. Moreover, we identify and elaborate open problems in cell engineering, tissue modeling, perturbation modeling, and bio-topology while discussing candidate quantum algorithms for research on these topics and their potential advantages over classical computational approaches.

quant-ph↗

arXiv2022

K3 surfaces with two involutions and low Picard number

Let $X$ be a complex algebraic K3 surface of degree $2d$ and with Picard number $\rho$. Assume that $X$ admits two commuting involutions: one holomorphic and one anti-holomorphic. In that case, $\rho \geq 1$ when $d=1$ and $\rho \geq 2$ when $d \geq 2$. For $d=1$, the first example defined over $\mathbb{Q}$ with $\rho=1$ was produced already in 2008 by Elsenhans and Jahnel. A K3 surface provided by Kond\={o}, also defined over $\mathbb{Q}$, can be used to realise the minimum $\rho=2$ for all $d\geq 2$. In these notes we construct new explicit examples of K3 surfaces over the rational numbers realising the minimum $\rho=2$ for $d=2,3,4$. We also show that a nodal quartic surface can be used to realise the minimum $\rho=2$ for infinitely many different values of $d$. Finally, we strengthen a result of Morrison by showing that for any even lattice $N$ of rank $1\leq r \leq 10$ and signature $(1,r-1)$ there exists a K3 surface $Y$ defined over $\mathbb{R}$ such that $\textrm{Pic} Y_\mathbb{C}=\textrm{Pic} Y \cong N$.

math.AG↗

arXiv2022

$G_2$-instantons on resolutions of $G_2$-orbifolds

We explain a construction of $G_2$-instantons on manifolds obtained by resolving $G_2$-orbifolds. This includes the case of $G_2$-instantons on resolutions of $T^7/\Gamma$ as a special case. The ingredients needed are a $G_2$-instanton on the orbifold and a Fueter section over the singular set of the orbifold which are used in a gluing construction. In the general case, we make the very restrictive assumption that the Fueter section is pointwise rigid. In the special case of resolutions of $T^7/\Gamma$, improved control over the torsion-free $G_2$-structure allows to remove this assumption. As an application, we construct a large number of $G_2$-instantons on the simplest example of a resolution of $T^7/\Gamma$ with hundreds of distinct ones among them. We also construct one new example of a $G_2$-instanton on the resolution of $(T^3 \times \text{K3})/\mathbb{Z}^2_2$.

math.DG↗

arXiv2022

Group invariant machine learning by fundamental domain projections

We approach the well-studied problem of supervised group invariant and equivariant machine learning from the point of view of geometric topology. We propose a novel approach using a pre-processing step, which involves projecting the input data into a geometric space which parametrises the orbits of the symmetry group. This new data can then be the input for an arbitrary machine learning model (neural network, random forest, support-vector machine etc). We give an algorithm to compute the geometric projection, which is efficient to implement, and we illustrate our approach on some example machine learning problems (including the well-studied problem of predicting Hodge numbers of CICY matrices), in each case finding an improvement in accuracy versus others in the literature. The geometric topology viewpoint also allows us to give a unified description of so-called intrinsic approaches to group equivariant machine learning, which encompasses many other approaches in the literature.

cs.LG↗

arXiv2022

Associative submanifolds in Joyce's generalised Kummer constructions

This article constructs examples of associative submanifolds in $G_2$-manifolds obtained by resolving $G_2$-orbifolds using Joyce's generalised Kummer construction. As the $G_2$-manifolds approach the $G_2$-orbifolds, the volume of the associative submanifolds tends to zero. This partially verifies a prediction due to Halverson and Morrison.

math.DG↗

arXiv2021

The Energy Footprint of Blockchain Consensus Mechanisms Beyond Proof-of-Work

Popular distributed ledger technology (DLT) systems using proof-of-work (PoW) for Sybil attack resistance have extreme energy requirements, drawing stern criticism from academia, businesses, and the media. DLT systems building on alternative consensus mechanisms, foremost proof-of-stake (PoS), aim to address this downside. In this paper, we take a first step towards comparing the energy requirements of such systems to understand whether they achieve this goal equally well. While multiple studies have been undertaken that analyze the energy demands of individual Blockchains, little comparative work has been done. We approach this research question by formalizing a basic consumption model for PoS blockchains. Applying this model to six archetypal blockchains generates three main findings: First, we confirm the concerns around the energy footprint of PoW by showing that Bitcoin's energy consumption exceeds the energy consumption of all PoS-based systems analyzed by at least three orders of magnitude. Second, we illustrate that there are significant differences in energy consumption among the PoSbased systems analyzed, with permissionless systems having an overall larger energy footprint. Third, we point out that the type of hardware that validators use has a considerable impact on whether PoS blockchains' energy consumption is comparable with or considerably larger than that of centralized, non-DLT systems.

cs.DC↗

arXiv2020

Improved Estimates for $G_2$-structures on the Generalised Kummer Construction

The resolution of the $G_2$-orbifold $T^7/\Gamma$, where $\Gamma$ is a suitably chosen finite group, admits a $1$-parameter family of $G_2$-structures with small torsion $\varphi^t$, obtained by gluing in Eguchi-Hanson spaces. It was shown by Joyce that $\varphi^t$ can be perturbed to torsion-free $G_2$-structures $\tilde{\varphi}^t$ for small values of $t$. Using norms adapted to the geometry of the manifold we give an alternative proof of the existence of $\tilde{\varphi}^t$. This alternative proof produces the estimate $\left|\left| \tilde{\varphi}^t-\varphi^t \right|\right|_{C^0} \leq ct^{5/2}$. This is an improvement over the previously known estimate $\left|\left| \tilde{\varphi}^t-\varphi^t \right|\right|_{C^0} \leq ct^{1/2}$. As part of the proof, we show that Eguchi-Hanson space admits a unique (up to scaling) harmonic form with decay, which is a result of independent interest.

math.DG↗