arXiv research

The thesis defines and proves invariants for manifolds of bounded geometry.

Study contact 3-manifolds using sub-Riemannian geometry, proving new properties of their Lipschitz homotopy groups.

Study quantifies convergence of Alexandrov spaces without collapsing.

We introduce the notion of good coverings of metric spaces, and prove that if a metric space admits a good covering, then it has the same locally Lipschitz homotopy type as the nerve complex of the covering. As an application, we obtain a Lipschitz homotopy stability result for a moduli space of compact Alexandrov spac…

Study 2D spaces with curvature, focusing on structure and approximations.

We construct a smooth compact n-dimensional manifold Y with one point singularity such that all its Lipschitz homotopy groups are trivial, but Lipschitz mappings Lip(S^n,Y) are not dense in the Sobolev space W^{1,n}(S^n,Y). On the other hand we show that if a metric space Y is Lipschitz (n-1)-connected, then Lipschitz …

We provide a sufficient condition for the nontriviality of the Lipschitz homotopy group of the Heisenberg group, $π_m^{Lip}(H_n)$, in terms of properties of the classical homotopy group of the sphere, $π_m(S^n)$. As an application we provide a new simplified proof of the fact that $π_n^{Lip}(H_n)\neq 0$, $n=1,2,...$, a…

This paper refines homotopy theory for cubical sets and uniform spaces.

Study limits of curved spaces with boundaries.

We develop some basic Lipschitz homotopy technique and apply it to manifolds with finite asymptotic dimension. In particular we show that the Higson compactification of a uniformly contractible manifold is mod $p$ acyclic in the finite dimensional case. Then we give an alternative proof of the Higher Signature Novikov …

Lipschitz and horizontal maps from an $n$-dimensional space into the $(2n+1)$-dimensional Heisenberg group $\H^n$ are abundant, while maps from higher-dimensional spaces are much more restricted. DeJarnette-Hajłasz-Lukyanenko-Tyson constructed horizontal maps from $S^k$ to $\H^n$ which factor through $n$-spheres and sh…

For a given null-cobordant Riemannian $n$-manifold, how does the minimal geometric complexity of a null-cobordism depend on the geometric complexity of the manifold? In [Gro99], Gromov conjectured that this dependence should be linear. We show that it is at most a polynomial whose degree depends on $n$. This constructi…

In the 1970s and again in the 1990s, Gromov gave a number of theorems and conjectures motivated by the notion that the real homotopy theory of compact manifolds and simplicial complexes influences the geometry of maps between them. The main technical result of this paper supports this intuition: we show that maps of di…