Quasisymmetric invariants and Analysis on metric spaces (Kvasi-invariantit)
Main funder
Funder's project number: 356861
Funds granted by main funder (€)
- 92 492,00
Funding program
Project timetable
Project start date: 01/09/2022
Project end date: 31/08/2023
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.
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.
