- Differentiability of Lipschitz mappings
- Null sets
- Geometric measure theory
- Nonlinear quotient mappings
- Geometry of Banach spaces
- Regularity of mappings in Banach spaces
The main directions in Olga's research to date are Lipschitz quotient mappings and null sets in Banach spaces.
Her current research addresses the questions to determine the sigma-ideal generated by sets of points of non-differentiability of Lipschitz functions on normed spaces.
In a recent joint paper joint with Preiss, Olga has developed a new method of finding a derivative assignment of composition of Lipschitz functions, at each point outside a null set. This is despite the fact that the image under the inner function may be a subset of the set of points where the outer function is not differentiable.
In a series of papers joint with Dymond (her previous PhD student) and with Doré (her previous postdoctoral advisee) Olga finds how to construct extremely small sets with the universal differentiability property: every function satisfying the Lipschitz condition on the space is differentiable at a point from this set. The methods used for this purpose combine ideas from analysis and descriptive set theory. This circle of questions relates analytic properties of Lipschitz maps and the geometry of null sets.
Olga's earlier work was focused on problems in non-linear analysis in normed spaces, and in particular, on Lipschitz quotient mappings. Lipschitz quotient mappings generalise both linear projections and bi-Lipschitz deformations and exhibit highly non-trivial topological properties even in the finite-dimensional case, which leads to an intricate interplay between the geometry of these mappings and the non-linear structure of Banach spaces. In particular, Olga has undertaken an in-depth study of the topology of level sets and point preimages under Lipschitz quotient mappings.
Subsequently Olga found an unexpected application of Lipschitz quotient mappings in the study of differentiability in infinite-dimensional spaces. She applied the theory of Lipschitz quotient mappings in order to construct in infinite-dimensional Banach spaces null subsets which 'almost contain' every Lipschitz curve. The null sets in question have the property that there exists a Lipschitz function differentiable at no point of the set. This surprising result covers as a particular case a theorem by Lindenstrauss, Preiss and Tiser.
Part of Olga's research has been and is devoted to extending the notion of differentiability to functions on metric spaces without linear structure. In particular, a new, intrinsic metric characterisation of purely 1-unrectifiable subsets of the finite-dimensional Euclidean space, which yields a remarkable metric version of the Besicovitch-Federer projection theorem, was obtained in a joint work with Kun and Máthé; further results were obtained in joint work with Duda.