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.

From summer 2014 until summer 2019 my research is funded by the Academy of Finland, Academy Research Fellow project Local and global structure of metric measure spaces with Ricci curvature lower bounds. Previously my research was funded by the Academy of Finland, postdoctoral project Geometric properties of sets and measures: densities, rectifiability and constructions from the summer 2011 until 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.

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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 geometric characterization of planar Sobolev extension domains
    (with P. Koskela and Y. Zhang), Preprint, 2015, 33 pp.


  2. Isometric embeddings of snowflakes into finite-dimensional Banach spaces
    (with E. Le Donne and E. Walsberg), Proc. Amer. Math. Soc., to appear.
  3. Ahlfors-regular distances on the Heisenberg group without biLipschitz pieces
    (with E. Le Donne and S. Li), Proc. Lond. Math. Soc., to appear.


  4. A density problem for Sobolev spaces on Gromov hyperbolic domains
    (with P. Koskela and Y. Zhang), Nonlinear Anal., 154 (2017), 189-209.
  5. 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.
  6. 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.
  7. Tangent lines and Lipschitz differentiability spaces
    (with F. Cavalletti), Anal. Geom. Metr. Spaces, 4 (2016), no. 1, 85-103.
  8. L estimates in optimal mass transportation
    (with H. Jylhä), J. Funct. Anal., 270 (2016), 4297-4321.
  9. Failure of the local-to-global property for CD(K,N) spaces
    Ann. Sc. Norm. Super. Pisa Cl. Sci., 16 (2016), 45-68.
  10. A function whose graph has positive doubling measure
    (with T. Ojala), Proc. Amer. Math. Soc., 144 (2016), no. 2, 733-738.
  11. 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.
  12. 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.
  13. Failure of topological rigidity results for the measure contraction property
    (with C. Ketterer), Potential Anal., 42 (2015), no. 3, 645-655.
  14. Assouad dimension, Nagata dimension, and uniformly close metric tangents
    (with E. Le Donne), Indiana Univ. Math. J., 64 (2015), no. 1, 21-54.
  15. Radon-Nikodym property and area formula for Banach homogeneous group targets
    (with V. Magnani), Int. Math. Res. Notices, 2014 (2014), no. 23, 6399-6430.
  16. 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.
  17. 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.
  18. Local multifractal analysis in metric spaces
    (with A. Käenmäki and V. Suomala), Nonlinearity, 26 (2013), no. 8, 2157–2173.
  19. Improved geodesics for the reduced curvature-dimension condition in branching metric spaces
    Discrete Contin. Dyn. Syst., 33 (2013), no. 7, 3043-3056.
  20. Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm
    J. Funct. Anal., 263 (2012), no. 4, 896-924.
  21. Local Poincaré inequalities from stable curvature conditions on metric spaces
    Calc. Var. Partial Differential Equations, 44 (2012), no. 3-4, 477-494.
  22. Thin and fat sets for doubling measures in metric spaces
    (with T. Ojala and V. Suomala), Studia Math., 208 (2012), no. 3, 195-211.
  23. Existence of doubling measures via generalised nested cubes
    (with A. Käenmäki and V. Suomala), Proc. Amer. Math. Soc., 140 (2012), no. 9, 3275-3281.
  24. 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.
  25. 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.
  26. Planar Sobolev homeomorphisms and Hausdorff dimension distortion
    Proc. Amer. Math. Soc., 139 (2011), no. 5, 1825-1829.
  27. Comparing the Hausdorff and packing measures of sets of small dimension in metric spaces
    Monatsh. Math., 164 (2011), no. 3, 313-323.
  28. Weakly controlled Moran constructions and iterated functions systems in metric spaces
    (with M. Vilppolainen), Illinois J. Math., 55 (2011), no. 3, 1015-1051.
  29. 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.
  30. 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.
  31. Large porosity and dimension of sets in metric spaces
    Ann. Acad. Sci. Fenn. Math., 34 (2009), no. 2, 565-581.
  32. 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.


  33. Porosity and dimension of sets and measures
    (Introductory part of the Ph.D. thesis), Rep. Univ. Jyväskylä Dept. Math. Stat. 119 (2009).
  34. A note on the level sets of Hölder continuous functions on the real line
    (A short note from 2008), 6 pp.
  35. Dimension of mean porous measures
    Real Analysis Exchange 2007, 31st Summer Symposium Conference, 257-262.
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In the spring 2016 I am giving a course on Advanced Measure Theory.

I have previously taught the courses
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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Prime numbers, factoring and other computations

As a hobby I let my home computers use their idle time, among other things, in trying to factor special numbers, mainly Fermat numbers. These are numbers of the form Fm=22m+1 where m is a natural number. For a list of all the known factors of the Fermat numbers, see Wilfrid Keller's web page.

There are three projects which I have participated that try to find factors of Fermat numbers. A coordinated effort using Lenstra's elliptic curve method is done in the context of the huge GIMPS-project. This project concentrates on finding factors for Fermat numbers smaller than F30.

For larger Fermat numbers trial division is not done using elliptic curves but with more direct methods. The search for factors of larger Fermat numbers is coordinated by Luigi Morelli at FermatSearch.

The third project is the Proth prime search -project at PrimeGrid. The purpose of this project is to find prime numbers of the form k*2n+1. Any prime that is found is then tested to see if it divides a Fermat number or a generalized Fermat number.

Factoring results

So far I have discovered the following Fermat number factors.
2864929972774011*241+1 divides 2239 + 1 (7.12.2012, using mmff on an NVidia GTX580 graphics card)
143918649*24654+1 divides 224652 + 1 (9.10.2012, sieving with NewPGen, divisibility testing with PFGW)
1784180997819127957596374417642156545110881094717*216+1 divides 2214+1 (3.2.2010, ECM using mprime)

At the time of finding the factor for F14 it was the smallest Fermat number which was proven to be composite but for which no prime factor had been found. It was proven composite already in 1963 by John Selfridge and Alexander Hurwitz. During my search for Fermat number divisors I have found several divisors of generalized Fermat numbers. Most of them can be found from the tables at another webpage maintained by Wilfrid Keller.

Other results

Occasionally I have tried to refresh my programming skills and have written tiny programs to computationally attack some problems in number theory and to find large primes of certain special form, such as Cunningham chains and prime tuplets.

As one example of such computations I have recently confirmed that the equation n!+1 = m2 has no integer solutions for 7 < n < 1010. This problem is known as the Brocard's problem. The approach was to test if the number n!+1 is a quadratic residue (mod p) with a collection of large primes p. The same approach was used by Berndt and Galway in 2000 to confirm that there were no solutions for 7 < n < 109. (Note that the upper limit could be easily pushed further up. My computations on a slow laptop took less than a day.)

As another example: I have checked that the equation an+bm = ck does not have any integer solutions with the constraints a, c, b coprime, 1 < a,b,c < 10001, 2 < n < 10001, m,k > 2 and ck < 2280000. This is related to the Beal conjecture. (Testing this range took roughly 21 hours on the above mentioned laptop.)

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- last update 2.4.2017 -