arXiv subjects

David McKinnon

Publications and source records attributed to David McKinnon.

At least 19 recordsLinked to original sources

arXiv2026

Modular curves and bad reduction

We prove results that imply, under various hypotheses, that every elliptic curve over a number field $k$ corresponding to a point on a modular curve has bad reduction at a certain prime $p$ of $\mathcal{O}_k$. For example, every elliptic curve with a cyclic torsion subgroup of order 20 defined over $\mathbb{Q}(\sqrt{-11})$ or $\mathbb{Q}(\sqrt{17})$ has bad reduction at all primes lying over $3$. The proofs of these statements are quite different, since $3$ is split in $\mathbb{Q}(\sqrt{-11})$ and inert in $\mathbb{Q}(\sqrt{17})$.

math.NT↗

arXiv2025

Dangerous Questions in Astronomy Education

As astronomy enters an era defined by global telescope networks, petabyte-scale surveys, and powerful computational tools, the longstanding goals of astronomy education, particularly introductory ``ASTRO101'', but equally encompassing both higher and lower level courses, warrant fresh examination. In June 2024, the AstroEdUNC meeting at UNC--Chapel Hill convened 100 astronomers, education researchers, and practitioners to synthesise community perspectives on the purpose, content, and delivery of astronomy education. Beginning with historical vignettes, the meeting's deliberations were organised into six interrelated themes: (1) Context, highlighting astronomy's evolution from classical charting to multi-messenger discovery and its role as a connective thread across STEM and the humanities; (2) Content, exploring how curricula can balance essential concepts with authentic investigations and leverage open-source and AI-augmented resources; (3) Skills, arguing that astronomy should foreground scientific literacy, computational fluency, and communication through genuine data-driven inquiry; (4) Engagement, advocating for active-learning strategies, formative assessment, and culturally inclusive narratives; (5) Beyond the Classroom, emphasising scaffolding, universal-design practices, and K--12/community partnerships; and (6) Astronomy Education Research, outlining priority areas for assessing knowledge, attitudes, and long-term outcomes. We provide concrete recommendations for future astronomy education research development, underscoring the need for approaches to education that are authentic while meeting the learning and life goal needs of the students, a vibrant community of practice and robust researcher-practitioner partnerships to ensure that introductory astronomy is pertinent, applicable and inspiring to a broad student population.

physics.ed-ph↗

arXiv2025

Curves of best approximation on wonderful varieties

We give an unconditional proof of the Coba conjecture for wonderful compactifications of adjoint type for semisimple Lie groups of type $A_n$. We also give a proof of a slightly weaker conjecture for wonderful compactifications of adjoint type for arbitrary Lie groups.

math.AG↗

arXiv2024

Affine-based Deformable Attention and Selective Fusion for Semi-dense Matching

Identifying robust and accurate correspondences across images is a fundamental problem in computer vision that enables various downstream tasks. Recent semi-dense matching methods emphasize the effectiveness of fusing relevant cross-view information through Transformer. In this paper, we propose several improvements upon this paradigm. Firstly, we introduce affine-based local attention to model cross-view deformations. Secondly, we present selective fusion to merge local and global messages from cross attention. Apart from network structure, we also identify the importance of enforcing spatial smoothness in loss design, which has been omitted by previous works. Based on these augmentations, our network demonstrate strong matching capacity under different settings. The full version of our network achieves state-of-the-art performance among semi-dense matching methods at a similar cost to LoFTR, while the slim version reaches LoFTR baseline's performance with only 15% computation cost and 18% parameters.

cs.CV↗

arXiv2024

Direct2.5: Diverse Text-to-3D Generation via Multi-view 2.5D Diffusion

Recent advances in generative AI have unveiled significant potential for the creation of 3D content. However, current methods either apply a pre-trained 2D diffusion model with the time-consuming score distillation sampling (SDS), or a direct 3D diffusion model trained on limited 3D data losing generation diversity. In this work, we approach the problem by employing a multi-view 2.5D diffusion fine-tuned from a pre-trained 2D diffusion model. The multi-view 2.5D diffusion directly models the structural distribution of 3D data, while still maintaining the strong generalization ability of the original 2D diffusion model, filling the gap between 2D diffusion-based and direct 3D diffusion-based methods for 3D content generation. During inference, multi-view normal maps are generated using the 2.5D diffusion, and a novel differentiable rasterization scheme is introduced to fuse the almost consistent multi-view normal maps into a consistent 3D model. We further design a normal-conditioned multi-view image generation module for fast appearance generation given the 3D geometry. Our method is a one-pass diffusion process and does not require any SDS optimization as post-processing. We demonstrate through extensive experiments that, our direct 2.5D generation with the specially-designed fusion scheme can achieve diverse, mode-seeking-free, and high-fidelity 3D content generation in only 10 seconds. Project page: https://nju-3dv.github.io/projects/direct25.

cs.CV↗

arXiv2024

Approximating rational points on surfaces

Let $X$ be a smooth projective algebraic variety over a number field $k$ and $P$ in $X(k)$. In 2007, the second author conjectured that, in a precise sense, if rational points on $X$ are dense enough, then the best rational approximations to $P$ must lie on a curve. We present a strategy for deducing a slightly weaker conjecture from Vojta's conjecture, and execute this strategy for the full conjecture for split surfaces.

math.AG↗

arXiv2023

JointNet: Extending Text-to-Image Diffusion for Dense Distribution Modeling

We introduce JointNet, a novel neural network architecture for modeling the joint distribution of images and an additional dense modality (e.g., depth maps). JointNet is extended from a pre-trained text-to-image diffusion model, where a copy of the original network is created for the new dense modality branch and is densely connected with the RGB branch. The RGB branch is locked during network fine-tuning, which enables efficient learning of the new modality distribution while maintaining the strong generalization ability of the large-scale pre-trained diffusion model. We demonstrate the effectiveness of JointNet by using RGBD diffusion as an example and through extensive experiments, showcasing its applicability in a variety of applications, including joint RGBD generation, dense depth prediction, depth-conditioned image generation, and coherent tile-based 3D panorama generation.

cs.CV↗

arXiv2023

Rational approximation on cubic hypersurfaces

We compute the constant of approximation for an arbitrary rational point on an arbitrary smooth cubic hypersurface $X$ over a number field $k$, provided that there is a $k$-rational line somewhere on $X$. In the process, we verify the Coba conjecture for $X$.

math.AG↗

arXiv2023

NeILF++: Inter-Reflectable Light Fields for Geometry and Material Estimation

We present a novel differentiable rendering framework for joint geometry, material, and lighting estimation from multi-view images. In contrast to previous methods which assume a simplified environment map or co-located flashlights, in this work, we formulate the lighting of a static scene as one neural incident light field (NeILF) and one outgoing neural radiance field (NeRF). The key insight of the proposed method is the union of the incident and outgoing light fields through physically-based rendering and inter-reflections between surfaces, making it possible to disentangle the scene geometry, material, and lighting from image observations in a physically-based manner. The proposed incident light and inter-reflection framework can be easily applied to other NeRF systems. We show that our method can not only decompose the outgoing radiance into incident lights and surface materials, but also serve as a surface refinement module that further improves the reconstruction detail of the neural surface. We demonstrate on several datasets that the proposed method is able to achieve state-of-the-art results in terms of geometry reconstruction quality, material estimation accuracy, and the fidelity of novel view rendering.

cs.CV↗

arXiv2022

Critical Regularizations for Neural Surface Reconstruction in the Wild

Neural implicit functions have recently shown promising results on surface reconstructions from multiple views. However, current methods still suffer from excessive time complexity and poor robustness when reconstructing unbounded or complex scenes. In this paper, we present RegSDF, which shows that proper point cloud supervisions and geometry regularizations are sufficient to produce high-quality and robust reconstruction results. Specifically, RegSDF takes an additional oriented point cloud as input, and optimizes a signed distance field and a surface light field within a differentiable rendering framework. We also introduce the two critical regularizations for this optimization. The first one is the Hessian regularization that smoothly diffuses the signed distance values to the entire distance field given noisy and incomplete input. And the second one is the minimal surface regularization that compactly interpolates and extrapolates the missing geometry. Extensive experiments are conducted on DTU, BlendedMVS, and Tanks and Temples datasets. Compared with recent neural surface reconstruction approaches, RegSDF is able to reconstruct surfaces with fine details even for open scenes with complex topologies and unstructured camera trajectories.

cs.CV↗

arXiv2022

Development and validation of an astronomy self-efficacy instrument for understanding and doing

This paper presents a new astronomy self efficacy instrument, composed of two factors, one relating to learning astronomy content, which we call astronomy personal self efficacy, and the other relating to the use of astronomical instrumentation, specifically the use of remote robotic telescopes for data collection. The latter is referred to as the astronomy instrumental self efficacy factor. The instrument has been tested for reliability and construct validity. Reliability testing showed that factor 1 had a Cronbachs alpha of 0.901 and factor 2 had a Cronbachs alpha of 0.937. Construct validity was established by computing one way analyses of variances, with the p value suitably protected, using independent variables peripherally related to the constructs. These analyses demonstrate that both scales possess high construct validity. The development of this astronomy specific instrument is an important step in evaluating self efficacy as a precursor to investigating the construct of science identity in the field of astronomy.

physics.ed-ph↗

arXiv2022

NeILF: Neural Incident Light Field for Physically-based Material Estimation

We present a differentiable rendering framework for material and lighting estimation from multi-view images and a reconstructed geometry. In the framework, we represent scene lightings as the Neural Incident Light Field (NeILF) and material properties as the surface BRDF modelled by multi-layer perceptrons. Compared with recent approaches that approximate scene lightings as the 2D environment map, NeILF is a fully 5D light field that is capable of modelling illuminations of any static scenes. In addition, occlusions and indirect lights can be handled naturally by the NeILF representation without requiring multiple bounces of ray tracing, making it possible to estimate material properties even for scenes with complex lightings and geometries. We also propose a smoothness regularization and a Lambertian assumption to reduce the material-lighting ambiguity during the optimization. Our method strictly follows the physically-based rendering equation, and jointly optimizes material and lighting through the differentiable rendering process. We have intensively evaluated the proposed method on our in-house synthetic dataset, the DTU MVS dataset, and real-world BlendedMVS scenes. Our method is able to outperform previous methods by a significant margin in terms of novel view rendering quality, setting a new state-of-the-art for image-based material and lighting estimation.

cs.CV↗

arXiv2020

On curves with high multiplicity on $\mathbb{P}(a,b,c)$ for $\min(a,b,c)\leq4$

On a weighted projective surface $\mathbb{P}(a,b,c)$ with $\min(a,b,c)\leq 4$, we compute lower bounds for the {\em effective threshold} of an ample divisor, in other words, the highest multiplicity a section of the divisor can have at a specified point. We expect that these bounds are close to being sharp. This translates into finding divisor classes on the blowup of $\mathbb{P}(a,b,c)$ that generate a cone contained in, and probably close to, the effective cone.

math.AG↗

arXiv2020

Approximating rational points on toric varieties

Given a smooth projective variety $X$ over a number field $k$ and $P\in X(k)$, the first author conjectured that in a precise sense, any sequence that approximates $P$ sufficiently well must lie on a rational curve. We prove this conjecture for smooth split toric surfaces conditional on Vojta's conjecture. More generally, we show that if $X$ is a $\mathbb{Q}$-factorial terminal split toric variety of arbitrary dimension, then $P$ is better approximated by points on a rational curve than by any Zariski dense sequence.

math.AG↗

arXiv2020

Codimension two integral points on some rationally connected threefolds are potentially dense

Let $V$ be a smooth, projective, rationally connected variety, defined over a number field $k$, and let $Z\subset V$ be a closed subset of codimension at least two. In this paper, for certain choices of $V$, we prove that the set of $Z$-integral points is potentially Zariski dense, in the sense that there is a finite extension $K$ of $k$ such that the set of points $P\in V(K)$ that are $Z$-integral is Zariski dense in $V$. This gives a positive answer to a question of Hassett and Tschinkel from 2001.

math.AG↗

arXiv2020

What fraction of an $S_n$-orbit can lie on a hyperplane?

Consider the $S_n$-action on $\mathbb{R}^n$ given by permuting coordinates. This paper addresses the following problem: compute $\max_{v,H} |H\cap S_nv|$ as $H\subset\mathbb{R}^n$ ranges over all hyperplanes through the origin and $v\in\mathbb{R}^n$ ranges over all vectors with distinct coordinates that are not contained in the hyperplane $\sum x_i=0$. We conjecture that for $n\geq3$, the answer is $(n-1)!$ for odd $n$, and $n(n-2)!$ for even $n$. We prove that if $p$ is the largest prime with $p\leq n$, then $\max_{v,H} |H\cap S_nv|\leq \frac{n!}{p}$. In particular, this proves the conjecture when $n$ or $n-1$ is prime.

math.CO↗

arXiv2018

The arithmetic puncturing problem and integral points

In this paper, we provide evidence to support a positive answer to a question of Hassett and Tschinkel. In particular, if an algebraic variety V has a dense set of rational points, they ask whether or not the set of D-integral points is potentially dense, where D is a set of codimension at least two. We give a positive answer to this question in many cases, including varieties whose generic linear section is a smooth rational curve, and certain K3 surfaces. We also discuss some stronger notions of integrality of points, and give some positive answers to some cases of the analogous question in the stronger context.

math.AG↗

arXiv2015

Exact synthesis of single-qubit unitaries over Clifford-cyclotomic gate sets

We generalize an efficient exact synthesis algorithm for single-qubit unitaries over the Clifford+T gate set which was presented by Kliuchnikov, Maslov and Mosca. Their algorithm takes as input an exactly synthesizable single-qubit unitary--one which can be expressed without error as a product of Clifford and T gates--and outputs a sequence of gates which implements it. The algorithm is optimal in the sense that the length of the sequence, measured by the number of T gates, is smallest possible. In this paper, for each positive even integer $n$ we consider the "Clifford-cyclotomic" gate set consisting of the Clifford group plus a z-rotation by $\fracπ{n}$. We present an efficient exact synthesis algorithm which outputs a decomposition using the minimum number of $\fracπ{n}$ z-rotations. For the Clifford+T case $n=4$ the group of exactly synthesizable unitaries was shown to be equal to the group of unitaries with entries over the ring $\mathbb{Z}[e^{i\fracπ{n}},1/2]$. We prove that this characterization holds for a handful of other small values of $n$ but the fraction of positive even integers for which it fails to hold is 100%.

quant-ph↗