Metric Geometry. We study metric spaces as objects of independent interest. Riemannian manifolds, metric spaces of finite cardinality, trees, Alexandrov spaces (geodesic metric spaces with curvature bounds), simplexes, fractals, Carnot groups, etc. What is their topology? What are their dimensions? What are their (shortest) paths? Do they have a curvature? What do we get if we quotient them, glue them, etc? What can we say about the limits of sequences of metric spaces? What are their isometries? Does any interesting group act on a metric space?

Uniformization and Mapping Theory.  Is a given metric space comparable in some controlled way to a more standard space? Is there a bi-Lipschitz/quasiconformal/quasisymmetric map between a given pair of spaces? Is the conformal dimension of a metric space realized (i.e. is the infimum of dimensions of all spaces that are quasisymmetric to X actually a minimum)? Is a given space rectifiable? (The latter sits in geometric measure theory.)

Analysis on Metric Spaces. Can we equip a metric space with a measure that is compatible with the metric? Can we talk about Sobolev spaces on these metric-measure spaces? To what extend does the Euclidean Sobolev theory persist on them? Does our space support Poincare inequalities? Can we talk about Laplacians and PDE on our spaces? Are Lipschitz functions on our space differentiable in a stronger sense than just Sobolev?

There are, of course, many other directions in which metric ideas develop—hyperbolic geometry, Sub-Riemannian geometry, coarse geometry, and geometric group theory among them.

I primarily work in geometric measure theory, mapping theory and analysis on metric spaces.

I share parts of my work through YouTube: YoungMeasures.