# Professor Olga Maleva PhD

## In 'Staff and Students'

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Professor of Mathematics

## Contact details

- Address
- School of Mathematics

Watson Building

University of Birmingham

Edgbaston

Birmingham

B15 2TT

UK

Olga's research concerns differentiability of Lipschitz mappings between finite and infinite dimensional spaces, geometric measure theory and the geometry of exceptional sets. Olga is particularly interested in a range of topics related to the converse to the classical theorem of Rademacher. Namely, she has been working on establishing finer measure-theoretic regularity properties (such as porosity, rectifiability, Hausdorff/Minkowski dimensions etc) of Universal differentiability sets and sets on which Lipschitz mappings behave in the worst possible way, as well as the behaviour of typical Lipschitz functions.

Olga has published research papers in leading mathematical journals and has been awarded research grants by the European Commission and the Royal Society and the Engineering and Physical Sciences Research Council. Olga has been invited speaker to a number of international research conferences and has given numerous research seminars in the UK and abroad.

## Qualifications

- PhD in Mathematics, The Weizmann Institute of Science, Israel, 2003
- MSc equivalent in Mathematics with distinction, Saint Petersburg State University, Russia, 1998

## Biography

Olga Maleva defended her MSc Thesis in 1998 at the Saint Petersburg State University, Russia and went on to study for a PhD in the area of Geometric and Functional Analysis at the Weizmann Institute of Science in Israel.

After the completion of her PhD at the end of 2003, Olga was awarded the Marie Curie Intra-European fellowship which she held for two years at the University College London. She then worked for one year as a Director of Studies at Emmanuel College, University of Cambridge, and after that on EPSRC grant together with Professor David Preiss at the University of Warwick, before moving to the present permanent post in 2008.

In 2008, Olga joined the University of Birmingham as a Lecturer in Pure Mathematics at the University of Birmingham. She was promoted to Senior Lecturer in 2014, and to Reader in Pure Mathematics in 2017. She is teaching various courses in Analysis to undergraduate and graduate students.

## Teaching

### Semester 1

LH/LM Functional and Fourier Analysis

## Postgraduate supervision

Olga is interested in supervising PhD students in the following areas:

- Geometric Analysis
- Problems in Analysis with measure theoretic flavour

## Research

### Research themes

- Differentiability of Lipschitz mappings
- Null sets
- Geometric measure theory
- Nonlinear quotient mappings
- Geometry of Banach spaces
- Regularity of mappings in Banach spaces

### Research activity

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.

## Other activities

Editorial adviser for the Journal and Bulletin of the London Mathematical Society.

## Publications

### Recent publications

#### Article

Hutchins, R & Maleva, O 2024, 'On the structural decomposition of planar Lipschitz quotient mappings', *Pure Appl. Funct. Anal.*, vol. 8, no. 6, pp. 1747-1766. <http://yokohamapublishers.jp/

Maleva, O & Villanueva-Segovia, C 2021, 'Best constants for Lipschitz quotient mappings in polygonal norms', *Mathematika*, vol. 67, no. 1, pp. 116-144. https://doi.org/

Dymond, M & Maleva, O 2020, 'A dichotomy of sets via typical differentiability', *Forum of Mathematics, Sigma*, vol. 8, e41. https://doi.org/

Maleva, O & Preiss, D 2019, 'Cone unrectifiable sets and non-differentiability of Lipschitz functions', *Israel Journal of Mathematics*, vol. 232, no. 1, pp. 75-108. https://doi.org/

Maleva, O & Dymond, M 2016, 'Differentiability inside sets with Minkowski dimension one', *Michigan Mathematical Journal*, vol. 65, no. 3, pp. 613-636. https://doi.org/

Maleva, O & Preiss, D 2015, 'Directional upper derivatives and the chain rule formula for locally Lipschitz functions on Banach spaces', *Transactions of the American Mathematical Society*, vol. 368, no. 7, pp. 4685-4730. https://doi.org/

Dore, M & Maleva, O 2012, 'A compact universal differentiability set with Hausdorff dimension one', *Israel Journal of Mathematics*, vol. 191, pp. 889–900. https://doi.org/

Doré, M & Maleva, O 2011, 'A universal differentiability set in Banach spaces with separable dual', *Journal of Functional Analysis*, vol. 261, no. 6, pp. 1674-1710. https://doi.org/

Doré, M & Maleva, O 2010, 'A compact null set containing a differentiability point of every Lipschitz function', *Mathematische Annalen*, vol. 351, no. 3, pp. 633-663. https://doi.org/

Maleva, O 2007, 'Metric derived numbers and continuous metric differentiability via homeomorphisms, Banach Spaces and their Applications in Analysis', *de Gruyter Proceedings in Mathematics*, pp. 307-330.

Maleva, O 2007, 'Unavoidable sigma-porous sets', *Journal of the London Mathematical Society*, vol. 76, no. 2, pp. 467-478. https://doi.org/

Maleva, O 2005, 'Components of level sets of uniform co-Lipschitz functions on the plane', *Proceedings of the American Mathematical Society*, vol. 133, no. 03, pp. 841-851. https://doi.org/

Kun, G, Mathe, A & Maleva, O 2005, 'Metric characterization of pure unrectifiability', *Real Analysis Exchange*, vol. 31, no. 1, pp. 195-214.

Maleva, O 2005, 'On Lipschitz ball noncollapsing functions and uniform co-Lipschitz mappings of the plane', *Abstract and Applied Analysis*, vol. 2005, no. 5, pp. 543-562. https://doi.org/

#### Preprint

Dymond, M & Maleva, O 2021 'Typical Lipschitz mappings are typically non-differentiable' arXiv. <https://arxiv.org/