Research

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My recent work has concerned four topics: geometric embeddings, Loewner carpets, conformal dimension, existence of Poincare inequalities, self-improvement properties of Poincaré inequalities and the local geometry of Lipschitz differentiability spaces. The basic approach in my recent work is to take a foundational concept in analysis on metric spaces (say modulus, Poincaré inequality, conformal dimension), study its basic properties using new tools and then apply these improved insights to various problems.

The research involves also the study of discrete and computational problems. Specifically, the questions on conformal dimension involve investigating problems at the interface of discrete and continuous, and how continuous structures can be obtained as limits of discrete objects. Indeed, some new continuous invariants may be recovered from such a limiting process. In a separate direction, geometry plays a central role in computational geometry and path planning problems. There I apply continuous methods to give algorithms for interesting geometric problems. This perspective of computability and optimization also is a crucial tool in the study of continuous invariants, such as conformal dimension.

Below, after the list of coauthors, I give a more detailed description of my work split by topics in some blog posts.

List of Coauthors:

- Nageswari Shanmugalingam
- Pietro Poggi-Corradini
- Jeff Cheeger
- Pekka Koskela
- Jan Maly
- Gianmarco Giovannardi
- Gareth Speight
- Elefterios Soultanis
- Panu Lahti
- Riikka Korte
- James Gill
- Estibalitz Durand-Cartagena
- Jasun Gong
- Guy C. David
- Juha Lehrbäck
- Antti Vähäkangas
- Valentin Polishchuk
- David Kirkpatrick
- Khanh Nguyen
- Zheng Zhu

## Duality for Modulus of Curves

This project studies two fundamental notions in analysis on metric spaces: modulus and capacity. While these notions are ancient, there are new results one can obtain regarding them. One specific type of question asks when the modulus of curves is in some sense dual to a modulus of a family of separating surfaces. It is…

## Structure of Sobolev Spaces

Motivated by a “curvewise” approach, this project seeks to develop new tools and a finer understanding of certain structures and issues concerning Sobolev Spaces. Indeed, in a doubling space with a Poincaré inequality the theory is very well understood. However in a setting which is just complete and separable, much can be done. Previously the…

## Geometric embeddings and rigidity

Existence of Bi-Lipschitz embeddings can be a useful tool in studying the geometry of the space. In the first paper on quantitative bi-Lipschitz embeddings I gave a positive answer employing a detailed study of collapsing theory on the space. We were able to construct bi-Lipschitz embeddings for subsets of manifold, orbifolds and group quotients and…

## Discrete and computational geometry

I am also engaging in research problems with computer scientists in computational geometry. This work mostly involved convex optimization and path planning. The most interesting contribution was on finding geometric flows via dual programs. This introduced an idea using potentials, which has also played a significant role in my research on continuous modulus. The goal…

## Bounded Variation functions and Isoperimetry

Bounded variation functions form an important class in metric measure spaces. In this work we studied their infinitesimal regularity, which has recently played a role in resolving problems in theoretical computer science. Further developing tools to study this function class is useful both from a theoretical perspective but also may yield useful insights and tools…

## Loewner carpets and conformal dimension

Motivated by questions such as Cannon’s conjecture I am studying the possibilities for conformal dimension and its attainment. Indeed, even for simple fractals such as the Sierpiński carpet, the attainment question of conformal dimension remains unknown. This number measures how small a dimension can a given space be deformed into. If it is attained, then…

## Poincaré inequalities and geometry

Poincaré inequalities and Loewner conditions are intricately connected to the study of functions on the space and the geometry of the space. However, their characterization has been challenging. My work in this area started from the study of the infinitesimal geometry of RNP-differentiability spaces, which lead to a new way of characterizing spaces satisfying Poincaré…

## Self-improvement phenomena

Self-improvement phenomena are important both for theoretical and practical reasons. In analysis of metric spaces, Keith and Zhong on the one hand and Lewis on the other proved foundational self-improvement results. Their results are hard, and the realization of this project has been that all of these self-improvement results can be cast in the same…