Rami Luisto

Research

My mathematical research interests are largely in the field of quasiconformal geometry, metric geometry, quantitative topology and in regularity of Sobolev and BV mappings.

In the field of quasiconformal and metric geometry my spesific research questions are focused on branched covers, quasiregular mappings and mappings of finite distortion between Euclidean domains. In particular, I am interested in the so called branch set of a mapping, along with the interplay of the structure of the branch set and the local & global behaviour of the mapping. I'm also interested in the structure of BLD-mappings between (sub)Riemannian manifolds and more general path-metric spaces.

In machine learning and AI I am currently focused on studying topics related to computer vision.

To keep up with the current advances both in my field and the surrounding fields I use arxivist. I warmly recommend a daily dosage.

Preprints

• On BLD-mappings with small distortion (Joint with Aapo Kauranen and Ville Tengvall). Preprint. [arxiv:1909.08860]
• On proper branched coverings and a question of Vuorinen (Joint with Aapo Kauranen and Ville Tengvall). Preprint. [arxiv:1904.12645]
• Characterization of branched covers with simplicial branch sets (Joint with Eden Prywes.) Preprint. [arxiv:1809.10874]
• A non-Euclidean story or: how to persist when your geometry doesn't. Preprint. [arxiv:2003.13758]

Publications

1. Weak regularity of the inverse under minimal assumptions (Joint with Stanislav Hencl and Aapo Kauranen). Archive for Rational Mechanics and Analysis volume 238, pages 185-213 (2020). [arxiv:1804.03449]
2. Mappings of finite distortion: compactness of the branch set (Joint with Aapo Kauranen and Ville Tengvall). Journal d'Analyse Mathematique (to appear). [arxiv:1709.08724]
3. A Newman property for BLD-mappings. Conform. Geom. Dyn. 23 (2019), 135-146. [arXiv:1709.08438]
4. Stoïlow's theorem revisited (Joint with Pekka Pankka.) Expositiones Mathematicae 38 (2020), no. 3, 303-318. [arxiv:1701.05726]
5. A characterization theorem for BLD-mappings between metric spaces, The Journal of Geometric Analysis (2017) 27:2081. [arXiv:1509.01832]
6. Note on local-to-global properties of BLD-mappings, Proceedings of the AMS 144 (2016), no.2, 599-607. [arXiv:1406.0320]
7. Rigidity of extremal quasiregularly elliptic manifolds (Joint with Pekka Pankka). Groups, Geometry & Dynamics 10 (2016), no. 2, 723-732. [arXiv:1307.7940]

Theses

• Interactions between quasiregular- and BLD-mappings Doctoral dissertation, University of Helsinki, 2016. PDF , e-thesis
• BLD-kuvausten rajakäyttäytymisestä ja haarajoukoista (On the limit behaviour and branch sets of BLD-mappings) Licentiate thesis, (in finnish) University of Helsinki, 2014. PDF , e-thesis
• On the non-existence of BLD-mappings between manifolds. Master's thesis, University of Helsinki, 2011. PDF , e-thesis
• Kvasihyperbolinen metriikka (Quasihyperbolic metric) Bachelor's thesis, (in finnish) University of Helsinki, 2009. PDF