Tapio Rajala
Ph.D., University of Jyväskylä, 2009
Title of Docent, University of Jyväskylä, 2012
Academy Research Fellow
University of Jyväskylä

Research interests

I am interested in geometric analysis and geometric measure theory in metric measure spaces. Here are some of the topics I have recently studied.

My research is currently funded by the Academy of Finland, Academy Research Fellow project Local and global structure of metric measure spaces with Ricci curvature lower bounds (summer 2014 - summer 2019) and Academy project Geometric Aspects of Sobolev Space Theory (summer 2018 - summer 2022). Previously my research was funded by the Academy of Finland, 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.


Conference: Subriemannian geometry and beyond (19th-23rd February, 2018)
27th Jyväskylä Summer School and MAnET-workshop (7th-18th August, 2017)
Workshop: Optimal transport meets density functional theory (31st May - 7th June, 2017)

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Papers and notes

These are the preliminary versions of my publications. Notice that they might differ from the published ones. Some information related to my publications can also be found in my profile at MathSciNet (requires surbscription) and at Google Scholar.


  1. A density result on Orlicz-Sobolev spaces in the plane
    (with W. A. Ortiz), Preprint, 2018, 11 pp.
  2. Non-existence of optimal transport maps for the multi-marginal repulsive harmonic cost
    (with A. Gerolin and A. Kausamo), Preprint, 2018, 13 pp.
  3. Optimal transport maps on Alexandrov spaces revisited
    (with T. Schultz), Preprint, 2018, 16 pp.
  4. Planar W1,1-extension domains
    (with P. Koskela and Y. Zhang), Preprint, 2017, 44 pp.
  5. A geometric characterization of planar Sobolev extension domains
    (with P. Koskela and Y. Zhang), Preprint, 2015, 33 pp.


  6. Duality theory for multimarginal optimal transport with repulsive costs in metric spaces
    (with A. Gerolin and A. Kausamo), ESAIM Control Optim. Calc. Var., to appear.
  7. Restricting open surjections
    (with J. Jaramillo and E. Le Donne), Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, to appear.
  8. A density result for homogeneous Sobolev spaces on planar domains
    (with D. Nandi and T. Schultz), Potential Anal., to appear.
  9. Removable sets for intrinsic metric and for holomorphic functions
    (with S. Kalmykov and L. V. Kovalev), J. Anal. Math., to appear.


  10. Isometric embeddings of snowflakes into finite-dimensional Banach spaces
    (with E. Le Donne and E. Walsberg), Proc. Amer. Math. Soc. 146 (2018), no. 2, 685-693.
  11. Products of snowflaked Euclidean lines are not minimal for looking down
    (with M. Joseph), Anal. Geom. Metr. Spaces, 5 (2017), no. 1, 78-97.
  12. Ahlfors-regular distances on the Heisenberg group without biLipschitz pieces
    (with E. Le Donne and S. Li), Proc. Lond. Math. Soc., 115 (2017), 348-380.
  13. A density problem for Sobolev spaces on Gromov hyperbolic domains
    (with P. Koskela and Y. Zhang), Nonlinear Anal., 154 (2017), 189-209.
  14. Local homogeneity and dimensions of measures
    (with A. Käenmäki and V. Suomala), Ann. Sc. Norm. Super. Pisa Cl. Sci., 16 (2016), no. 4, 1315-1351.
  15. Optimal maps and exponentiation on finite dimensional spaces with Ricci curvature bounded from below
    (with N. Gigli and K.-T. Sturm), J. Geom. Anal., 26 (2016), no. 4, 2914-2929.
  16. Tangent lines and Lipschitz differentiability spaces
    (with F. Cavalletti), Anal. Geom. Metr. Spaces, 4 (2016), no. 1, 85-103.
  17. L estimates in optimal mass transportation
    (with H. Jylhä), J. Funct. Anal., 270 (2016), 4297-4321.
  18. Failure of the local-to-global property for CD(K,N) spaces
    Ann. Sc. Norm. Super. Pisa Cl. Sci., 16 (2016), 45-68.
  19. A function whose graph has positive doubling measure
    (with T. Ojala), Proc. Amer. Math. Soc., 144 (2016), no. 2, 733-738.
  20. Euclidean spaces as weak tangents of infinitesimally Hilbertian metric measure spaces with Ricci curvature bounded below
    (with N. Gigli and A. Mondino), J. Reine Angew. Math., 705 (2015), 233-244.
  21. Riemannian Ricci curvature lower bounds in metric measure spaces with σ-finite measure
    (with L. Ambrosio, N. Gigli and A. Mondino), Trans. Amer. Math. Soc., 367 (2015), no. 7, 4661-4701.
  22. Failure of topological rigidity results for the measure contraction property
    (with C. Ketterer), Potential Anal., 42 (2015), no. 3, 645-655.
  23. Assouad dimension, Nagata dimension, and uniformly close metric tangents
    (with E. Le Donne), Indiana Univ. Math. J., 64 (2015), no. 1, 21-54.
  24. Radon-Nikodym property and area formula for Banach homogeneous group targets
    (with V. Magnani), Int. Math. Res. Notices, 2014 (2014), no. 23, 6399-6430.
  25. Non-branching geodesics and optimal maps in strong CD(K,∞)-spaces
    (with K.-T. Sturm), Calc. Var. Partial Differential Equations, 50 (2014), no. 3-4, 831-846.
  26. Slopes of Kantorovich potentials and existence of optimal transport maps in metric measure spaces
    (with L. Ambrosio), Ann. Mat. Pura Appl., 193 (2014), no. 1, 71-87.
  27. Local multifractal analysis in metric spaces
    (with A. Käenmäki and V. Suomala), Nonlinearity, 26 (2013), no. 8, 2157–2173.
  28. Improved geodesics for the reduced curvature-dimension condition in branching metric spaces
    Discrete Contin. Dyn. Syst., 33 (2013), no. 7, 3043-3056.
  29. Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm
    J. Funct. Anal., 263 (2012), no. 4, 896-924.
  30. Local Poincaré inequalities from stable curvature conditions on metric spaces
    Calc. Var. Partial Differential Equations, 44 (2012), no. 3-4, 477-494.
  31. Thin and fat sets for doubling measures in metric spaces
    (with T. Ojala and V. Suomala), Studia Math., 208 (2012), no. 3, 195-211.
  32. Existence of doubling measures via generalised nested cubes
    (with A. Käenmäki and V. Suomala), Proc. Amer. Math. Soc., 140 (2012), no. 9, 3275-3281.
  33. Generalized Hausdorff dimension distortion in Euclidean spaces under Sobolev mappings
    (with A. Zapadinskaya and T. Zürcher), J. Math. Anal. Appl., 384 (2011), no. 2, 468-477.
  34. Generalized dimension distortion under mappings of sub-exponentially integrable distortion
    (with A. Zapadinskaya and T. Zürcher), Ann. Acad. Sci. Fenn. Math., 36 (2011), no. 2, 553-566.
  35. Planar Sobolev homeomorphisms and Hausdorff dimension distortion
    Proc. Amer. Math. Soc., 139 (2011), no. 5, 1825-1829.
  36. Comparing the Hausdorff and packing measures of sets of small dimension in metric spaces
    Monatsh. Math., 164 (2011), no. 3, 313-323.
  37. Weakly controlled Moran constructions and iterated functions systems in metric spaces
    (with M. Vilppolainen), Illinois J. Math., 55 (2011), no. 3, 1015-1051.
  38. Upper conical density results for general measures on Rn
    (with M. Csörnyei, A. Käenmäki and V. Suomala), Proc. Edinb. Math. Soc., 53 (2010), no. 2, 311-331.
  39. Packing dimension and Ahlfors regularity of porous sets in metric spaces
    (with E. Järvenpää, M. Järvenpää, A. Käenmäki, S. Rogovin and V. Suomala),
    Math. Z., 266 (2010), no. 1, 83-105.
  40. Large porosity and dimension of sets in metric spaces
    Ann. Acad. Sci. Fenn. Math., 34 (2009), no. 2, 565-581.
  41. Packing dimension of mean porous measures
    (with D. Beliaev, E. Järvenpää, M. Järvenpää, A. Käenmäki, S. Smirnov and V. Suomala),
    J. Lond. Math. Soc. (2), 80 (2009), no. 2, 514-530.


  42. Porosity and dimension of sets and measures
    (Introductory part of the Ph.D. thesis), Rep. Univ. Jyväskylä Dept. Math. Stat. 119 (2009).
  43. A note on the level sets of Hölder continuous functions on the real line
    (A short note from 2008), 6 pp.
  44. Dimension of mean porous measures
    Real Analysis Exchange 2007, 31st Summer Symposium Conference, 257-262.
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In the spring 2019 I will teach again Functional Analysis.

I have previously taught the courses
Alexandrov Spaces (2018), Advanced Measure Theory (2016), Optimal Mass Transportation (2014), Functional Analysis (2014), Basics in Number Theory (2013), Number Theory (2010), Introduction to Mathematics (2010), Symbolic Mathematics (2008 and 2009)

and acted as a teaching assistant on the courses
Measure and Integration Theory 1 & 2 (2007), Linear Algebra and Geometry 1 (2006), Analysis 2 (2006), Approbatur 3 (2005), Analysis 1 (2004), Euclidean Spaces (2004), Vectors and Matrices (2003).

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- last update 6.11.2018 -