# Quasisymmetric invariants and Analysis on metric spaces

Main funder

Funder's project number: 330048

Funds granted by main funder (€)

- 239 590,00

Funding program

Project timetable

Project start date: 01/09/2020

Project end date: 31/12/2020

Summary

Problems involving the geometry of fractals and of non-smooth metric spaces play a crucial role in mathematical physics and computer science. In mathematical physics, such geometries arise as models for natural and random phenomena, and in computer science finding efficient representations of objects has shown promise in developing efficient algorithms [AS10, I01,Alb19,CNK11]. In light of these applications, it is crucial to study the properties of these geometries, where much remains to be understood. In the last few decades a well-developed theory of analysis on metric spaces has arisen, and many new tools have been introduced. Often these tools, however, are only qualitative. Consider differentiation, where results hold infinitesimally instead of at a definite scale, limiting their applicability to problems in computer science and beyond that require explicit error bounds. The goal of this project is to convert these to quantitative results, holding at a definite scale with concrete bounds.

The proposal focuses on two main questions, which have been open for roughly twenty years, as well as a number of smaller connected problems. Firstly, is every measure on Euclidean space, that admits a Poincaré inequality, quantitatively absolutely continuous i.e. a Muckenhoupt weight? Secondly, do combinatorial Loewner spaces, that discretely resemble Loewner spaces, admit a quasisymmetric deformation to a Loewner space? We approach these problems by recognizing that existing theory leads to qualitative tools, such as curve families, p-harmonic functions and blow-up analysis, which when quantified imply rigidity and structures at a definite scale on the space.

In my prior work, I have explored various topics in analysis on metric spaces: the characterization of Poincaré inequalities using quantitative notions of connectivity, the infinitesimal regularity of bounded variation functions involving the study of minimizers of energy, self-improvement of Poincaré and Hardy inequalities, connections to uniformization, and the existence of curve families. The further development of these tools, together with other existing results, are directly related to the problems of this proposal and is of independent interest in the field. These problems have so far resisted efforts at solution and the new approach we propose presents unique opportunities to make progress towards their resolution. The results from my prior work will be useful in this endeavor.

The proposal focuses on two main questions, which have been open for roughly twenty years, as well as a number of smaller connected problems. Firstly, is every measure on Euclidean space, that admits a Poincaré inequality, quantitatively absolutely continuous i.e. a Muckenhoupt weight? Secondly, do combinatorial Loewner spaces, that discretely resemble Loewner spaces, admit a quasisymmetric deformation to a Loewner space? We approach these problems by recognizing that existing theory leads to qualitative tools, such as curve families, p-harmonic functions and blow-up analysis, which when quantified imply rigidity and structures at a definite scale on the space.

In my prior work, I have explored various topics in analysis on metric spaces: the characterization of Poincaré inequalities using quantitative notions of connectivity, the infinitesimal regularity of bounded variation functions involving the study of minimizers of energy, self-improvement of Poincaré and Hardy inequalities, connections to uniformization, and the existence of curve families. The further development of these tools, together with other existing results, are directly related to the problems of this proposal and is of independent interest in the field. These problems have so far resisted efforts at solution and the new approach we propose presents unique opportunities to make progress towards their resolution. The results from my prior work will be useful in this endeavor.