CAREER: Heat Semigroups and Strichartz Estimates on Fractals

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。本项目旨在研究分数维上的薛定谔方程。1925年，物理学家欧文·薛定谔提出了一个用波来描述量子粒子行为的方程。薛定谔方程通常是在假设粒子在理想的、光滑的环境中运动的情况下研究的。然而，自然现象经常发生在非光滑的环境中，例如高度多孔的介质(例如海绵或过滤器)或错综复杂的分支结构(例如大型网络)。其中一些特征可以使用称为分形图的数学模型来复制。这个项目解决了在这些不流畅的环境中提出的重大数学挑战。该项目将为研究广泛开放的问题奠定数学基础，例如电子在分形环境中的行为。该项目的教育部分旨在使学生以及职前和在职教师注意到诸如分数和概率等主题。该项目强调为当地拉美裔社区提供服务，并向德克萨斯州服务不足的中学提供服务。使用视觉上吸引人的模型，如分形图，将吸引更广泛的公众对拟议研究的数学方面的关注。该研究旨在产生数学工具来研究分数维环境下的薛定谔方程。所获得的结果将为分形学的分析和概率领域提供新的知识，并将服务于例如在渗流系统中运动的量子粒子的数学建模。在概率论、位势理论、泛函分析和偏微分方程的十字路口，将发展新的技术来构造合适的基于热半群的函数空间，证明特征函数乘积的新估计，并推导出关于分数维上的线性薛定谔方程解的非平凡色散(Strichartz)估计。该项目还将分析随机初始数据对解的存在和正则性的影响，并将探索量子概率和量子动力学对分形学的可能概念。教育部分力求将研究生和本科生的研究和教育结合起来。包括研究生、本科生、职前和在职教师在内的一门新的可持续研讨会课程将产生概率和分形学主题的创新教材。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). This project aims to study the Schrödinger equation on fractals. In 1925, the physicist Erwin Schrödinger introduced an equation describing the behavior of quantum particles in terms of waves. The Schrödinger equation is usually studied under the assumption that the particle moves in an idealized, smooth environment. However, natural phenomena often occur in non-smooth settings such as highly porous media (for example, sponges or filters) or intricately branching structures (for example, large networks). Some of these features can be replicated using mathematical models called fractals. This project addresses the significant mathematical challenges posed in these non-smooth settings. The project will lay mathematical groundwork to study widely open questions, such as the behavior of electrons in a fractal environment. The educational component of the project seeks to bring topics such as fractals and probability to the attention of students as well as pre-service and in-service teachers. The project emphasizes service to the local Hispanic community and outreach to under-served secondary schools in Texas. The use of visually appealing models such as fractals will attract the attention of the broader public to the mathematical aspects of the proposed research.The research aims to produce mathematical tools to study the Schrödinger equation in fractal settings. The results obtained will contribute new knowledge to the fields of analysis and probability on fractals and will serve, for instance, in the mathematical modeling of quantum particles traveling in a percolating system. New techniques at the crossroads of probability, potential theory, functional analysis, and partial differential equations will be developed to construct suitable heat-semigroup based function spaces, to prove novel estimates for products of eigenfunctions, and to derive non-trivial dispersive (Strichartz) estimates of solutions to the linear Schrödinger equation on fractals. The project will also analyze the effects of random initial data on the existence and regularity of solutions, and will explore possible notions of quantum probability and quantum dynamics on fractals. The educational component seeks to integrate research and education at graduate and undergraduate levels. A new sustainable seminar course involving graduate students, undergraduate students, and pre-service and in-service teachers will result in innovative teaching materials on the subjects of probability and fractals.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.