CAREER: Visualizing Mathematical Structures in High-Dimensional Space

虽然传达数学概念的最有效方法是使用真实世界的对象或虚拟的操作工具，但创建超越三维的数学现象的插图一直是一项特别具有挑战性的任务，因此许多数学现象只存在于数学家的脑海中。本研究旨在探讨计算机图形技术是否可以帮助专家数学家和公众可视化和交流高维数学对象及其变形的问题。这项研究的最终目标是建立一种机制，通过这种机制，专业的人类观众可以操纵他们只能看到部分的高维几何物体，即通过切片或投影到二维或三维。本研究的理论贡献将影响和改进数学可视化的方法，特别是图形可视化、计算机辅助设计和大规模空间可视化，而项目成果将对数学家研究高维对象的能力产生直接和变革性的影响，并在亲自、演讲和档案作品中交流他们所学到的东西。该项目的成功最终将转化为在高维几何发挥重要作用的纯数学和应用数学领域的更快发展，并为大学和K-12学生提供数学可视化工具集，以方便他们的几何课程。该项目将使包括开源软件在内的研究成果免费提供，并将通过绘制和展示路易斯维尔大学天文馆的教学动画向公众传播数学科学。该项目还包括针对K-12、本科生和研究生的综合教育和推广活动。该研究将探索一种交互式可视化范例，该范例利用嵌入在高维中的能量驱动的自变形对象模型，辅以专家人类观众的低维类比，以指导变形走向最终目标。研究人员将首先为高维物体分配变形能量，以便通过约束或操纵在我们的维度中看到和熟悉的配置方面，以原则性和良好的方式控制未见和不熟悉的配置方面。通常情况下，数学模拟涉及在大量迭代中反复执行的大量矢量化操作。该项目将利用硬件支持的并行性来加速数学模拟，并提取连续项因一个关键变化而不同的关键时刻，以表示和分析各种数学演变。通过结合引导松弛法和加速计算，这项研究可能会对建立关于几何和拓扑问题类别的直觉做出新的贡献，否则这些问题几乎不可能进行交流、感知和传播；并且可以潜在地进一步通过计算机方法，特别是通过视觉和计算空间可视化工具的力量，帮助和赋予人类理解能力的整个概念。该项目的所有成果，包括技术报告、研究文章、教育和推广活动的链接、开源软件和教学动画，将从该项目的网站（http://www.cecsresearch.org/vcl/nsf1651581/）获得。

While the most efficient method for communicating math concepts is the use of real world objects or virtual manipulatives, creating illustrations of mathematical phenomena beyond three dimensions has been a particularly challenging task and many mathematical phenomena have thus only existed in the mathematician's mind. This research seeks to investigate the question of whether computer graphics techniques can help expert mathematicians and general public to visualize and communicate the higher-dimensional mathematical objects and their deformations. The ultimate goal of this research is to establish a mechanism by which an expert human viewer can manipulate a higher-dimensional geometric object that they can only see in part, i.e., via a slice or projection into two or three dimensions. Theoretical contributions of this research will impact and improve methods in mathematical visualization, particularly graph visualization, computer aided design, and large-scale spatial visualization, while the project deliverables will have direct and transformative impact on the ability of mathematicians to study higher-dimensional objects, and to communicate what they have learned in person, in presentations, and in archival works. The success of this project will ultimately translate into more rapid advancement in areas of pure and applied mathematics where higher dimensional geometry plays an important role, and provide mathematical visualization tool sets to facilitate college and K-12 students in their geometry courses. The project will make research outcomes including open source software freely available, and will disseminate mathematical sciences to the general public by rendering and presenting pedagogical animations at the University of Louisville Planetarium. The project also includes integrated educational and outreach activities for K-12, undergraduate, and graduate students. The research will explore an interactive visualization paradigm that makes use of energy-driven self-deformable object models embedded in higher dimensions, supplemented by reduced-dimensional analogies for expert human viewers to guide the deformations towards their final goals. The investigators will begin by assigning a deformation energy to the higher-dimensional object, so that the aspects of the configuration that are unseen and unfamiliar can be controlled in a principled and well-posed manner by constraints or manipulations on the aspects of the configuration that are seen and familiar in our dimensions. Often times mathematical simulations are concerned with heavily vectorized operations performed over and over in a large number of iterations. The project will exploit hardware-enabled parallelism to accelerate mathematical simulations, and to extract key moments where successive terms differ by one critical change to represent and analyze various mathematical evolutions. By combining guided relaxation method and accelerated computation, this research can potentially make a novel contribution to building intuition about classes of geometric and topological problems that otherwise would be nearly impossible to communicate, perceptualize, and disseminate; and can potentially further the entire concept of the assistance and empowerment of human understanding by computer methods, specifically via the power of visual and computational spatial visualization tools. All outcomes of this project, including technical reports, research articles, links to educational and outreach activities, open source software, and pedagogical animations will be accessible from the project's web site (http://www.cecsresearch.org/vcl/nsf1651581/).