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 isContinue reading “Duality for Modulus of Curves”

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 theContinue reading “Structure of Sobolev Spaces”

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 andContinue reading “Geometric embeddings and rigidity”

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 goalContinue reading “Discrete and computational geometry”

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 toolsContinue reading “Bounded Variation functions and Isoperimetry”

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, thenContinue reading “Loewner carpets and conformal dimension”

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 PoincaréContinue reading “Poincaré inequalities and geometry”

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 sameContinue reading “Self-improvement phenomena”