Research

I am interested in geometric mapping theory and geometric measure theory in metric measure spaces. Here are some of the topics that I have studied. The numbers refer to the list of publications.

BV- and Sobolev extension domains

Much of my recent research has been on the geometry of Sobolev extension domains. In the planar simply-connected case, we have found characterizations and necessary conditions for extension domains [55,60,61]. The relation between BV- and W^1,1-extension domains was clarified in the Euclidean setting in [48], giving for instance the bi-Lipschitz invariance of planar extension domains [49]. The relation between BV- and W^1,1-extension domains was studied in the metric spaces in [57]. One direction in understanding the extension domains is the study of removable sets [32,51], and another one is the density of other classes of function spaces in Sobolev spaces [32,38,45]. We have also studied the size of boundary and the set of two-sided points of extension domains [50,53,56].

BV- and Sobolev spaces in general metric spaces

BV- and Sobolev spaces play a crucial role also in analysis on metric spaces. One line of investigation is the generalization of the study of extension domains to more general settigns. Other questions include the behaviour of tangent spaces [29,52], characterizations of upper gradients in weighted Euclidean spaces [46] and weighted reflexive Banach spaces [63], decomposition of sets of finite perimeter in PI-spaces [41], and tensorization [54,62].

Metric removability, approximation, and intrinsic distances on domains

Questions on metric removability [39], approximation of domains [44], and estimates on intrinsic distances [42] for me are usually motivated by the previous two topics on BV- and Sobolev spaces. However, these are examples of questions in metric geometry where the tools are quite different from the ones used in the motivating problems and therefore the questions are studied independently. Other questions in metric geometry that I have studied include embeddings [33,34,35] and the relations betweeen different notions of dimensions and tangent spaces [22].

Synthetic notions of Ricci-curvature lower bounds

Using optimal mass transportation, Sturm, and Lott and Villani introduced notions of curvature-dimension conditions CD(K,N) that generalize the Ricci-curvature lower bound K and dimension upper bound N from Riemannian manifolds to general metric measure spaces. Later, a more restrictive conditions, called RCD(K,N) were introduced that allow only spaces that are more Riemannian-like. I have studied mostly the role of non-branching geodesics in these classes of spaces, in particular in connection to local Poincaré inequalities [15,16,17], existence of optimal maps [20,30], and the failure of local-to-global and topological rigidity properties [23,27]. I have also studied the definitions and local structure of RCD(K,N)-spaces, see [24,25].

Optimal mass transportation related to DFT

Optimal transportation can also be used in the study of the limiting behaviour in Density Functional Theory. This leads to multimarginal optimal mass transportation where the transport cost is repulsive (Coulomb). I have studied fundamental optimal mass transportation questions in this setting, such as duality theory [40], existence of optimal maps [37], and entropic regularization [43]. Related to this project is also the existence of good approximations to couplings. In one case this was studied in [59].

Existence of optimal transportation maps

In addition to the existence of optimal transportation maps in RCD(K,N)-spaces [20,30] and in the multi-marginal setting [37], I have studied the existence question in Alexandrov spaces [47], and in more abstract settings in metric measure spaces [19].

Fractal geometry and porosity

My Ph.D. thesis work was on conical densities and porosity and their relation to the dimension of sets and measures, see [3,4,5,6,7]. Recently, the dimension estimates for porous sets have re-entered my research in connection with the dimension estimates on the boundaries of Sobolev-extension domains [50]. I have also done work on multifractal analysis [18,31] and Moran constructions [8], that generalize the dimension analysis of self-similar sets in various directions.

Postdocs

2022 - 2023, Francesco Nobili
2022 - 2023, Emanuele Caputo
2021 - 2023, Carlos Mudarra
2021 - 2021, Ugo Bindini
2020 - 2021, Miguel García-Bravo
2018 - 2020, Danka Lučić
2018 - 2020, Enrico Pasqualetto
2016 - 2019, Augusto Gerolin

Ph.D. students

Jesse Koivu (working on his thesis on Sobolev spaces on metric measure spaces)
Jyrki Takanen defended his thesis On the boundaries of Sobolev extension domains in 2023.
Timo Schultz defended his thesis Existence of optimal transport maps with applications in metric geometry in 2020.
Anna Kausamo defended her thesis On the structure of multi-marginal optimal mass transportation in metric spaces in 2019.

Coauthors (45)

Luigi Ambrosio, Dmitry Beliaev, Ugo Bindini, Paolo Bonicatto, Miguel García-Bravo, Fabio Cavalletti, Emanuele Caputo, Marianna Csörnyei, Sylvester Eriksson-Bique, Augusto Gerolin, Nicola Gigli, Jesus Jaramillo, Esa Järvenpää, Maarit Järvenpää, Matthieu Joseph, Heikki Jylhä, Jesse Koivu, Antti Käenmäki, Sergey Kalmykov, Anna Kausamo, Christian Ketterer, Pekka Koskela, Leonid V. Kovalev, Enrico Le Donne, Sean Li, Danka Lučić, Valentino Magnani, Andrea Mondino, Debanjan Nandi, Tuomo Ojala, Walter A. Ortiz, Enrico Pasqualetto, Sari Rogovin, Timo Schultz, Stanislav Smirnov, Elefterios Soultanis, Karl-Theodor Sturm, Ville Suomala, Jyrki Takanen, Markku Vilppolainen, Erik Walsberg, Aleksandra Zapadinskaya, Yi Zhang, Zheng Zhu, Thomas Zürcher

Funding

My research has been funded by the Academy of Finland, Academy project Geometric Aspects of Sobolev Space Theory (summer 2018 - summer 2022), Academy Research Fellow project Local and global structure of metric measure spaces with Ricci curvature lower bounds (summer 2014 - summer 2019) and postdoctoral project Geometric properties of sets and measures: densities, rectifiability and constructions (summer 2011 - the end of 2013). Before this I was a postdoc at the Scuola Normale Superiore di Pisa working in the European Project Geometric Measure Theory in non Euclidean spaces.