CAREER: Heat Semigroups and Strichartz Estimates on Fractals

该奖项全部或部分根据2021年美国救援计划法案（公法117-2）资助。本计画的目的是研究薛定谔方程在分形上的应用。1925年，物理学家埃尔温·薛定谔引入了一个方程，描述了量子粒子在波中的行为。薛定谔方程通常在假设粒子在理想化的光滑环境中运动的情况下进行研究。然而，自然现象通常发生在非光滑环境中，例如高度多孔的介质（例如海绵或过滤器）或复杂的分支结构（例如大型网络）。其中一些特征可以用称为分形的数学模型来复制。该项目解决了在这些非光滑设置中所带来的重大数学挑战。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引起学生以及职前和在职教师对分形和概率等主题的注意。该项目强调为当地西班牙裔社区提供服务，并向得克萨斯州服务不足的中学提供服务。使用分形等视觉上吸引人的模型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形环境中的薛定谔方程。所获得的结果将为分形的分析和概率领域提供新的知识，例如，将用于量子粒子在量子系统中运动的数学建模。在概率论，势理论，泛函分析和偏微分方程的十字路口的新技术将被开发，以构建合适的热半群为基础的函数空间，以证明新的估计产品的特征函数，并获得非平凡的色散（Eschhartz）估计的解决方案的线性薛定谔方程的分形。该项目还将分析随机初始数据对解的存在性和规律性的影响，并将探索量子概率和量子动力学对分形的可能概念。教育部分力求将研究生和本科生的研究和教育结合起来。一个新的可持续的研讨会课程，涉及研究生，本科生，职前和在职教师将导致创新的教学材料的概率和分形的主题。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.