CAREER: Heat Semigroups and Strichartz Estimates on Fractals

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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.