SUMMER SCHOOL
Analysis on Metric Spaces

Juha Heinonen's paper "Geometric Embeddings of Metric Spaces" and his book "Lectures on Analysis on Metric Spaces" are good surveys of the area covered by some of the papers below.

  1. Sobolev met Poincare.
     by P. Hajlaz and P. Koskela
    Mem. of Amer. Math. Soc 145 (2000) no 688
    (See also Hajlasz and Koskela, Sobolev meets Poincare, C.R. Acad. Sci. Paris, 1. 320, Serie I, p. 1211-1215, 1995 and P. Koskela, Upper gradients and Poincare inequalities, MR 2023123.)

    Prove Theorems 2.1, 2.3, 3.1, 3.3, and as much of Chapters 4-7 as possible. Heinonen's book also has much of this material.
    [presenter: Williams]

  2. Separated nets in Euclidean space and Jacobians of bi-Lipschitz maps.
     by D. Burago and B. Kleiner
    Geom. Funct. Anal. 8 (1998) no 2. 273--282
    and
    Lipschitz maps and nets in Euclidean space.
     by C. McMullen
    Geom. Funct. Anal. 8 (1998) no 2. 304--314

    Prove the main results of both these papers.
    [presenter: Maki]

  3. The Poincare inequality is an open ended condition.
     by S. Keith and X. Zhong
    see Keith's page

    Prove this very new well known result
    [presenter: Hinde]

  4. Measurable differential structures and the Poincare inequality.
     by S. Keith
    see Keith's page

    Prove the main result.
    [presenter: Vagharshakyan]

  5. Metric and geometric quasiconformality in Ahlfors regular Loewner spaces.
     by J. Tyson
    Conformal Geometry and Dynamics. 5 (2001), pages 21--73

    Prove as much of Theorem 10.9 as possible.
    [presenter: Azzam]

  6. The quasiconformal Jacobian problem.
     by M. Bonk, J. Heinonen and E. Saksman
    Contemp. Math. 355 (2004) pages 77--96

    Prove the main result
    [presenter: Luthy]

  7. An A1 weight not comparable to any quasiconformal Jacobian.
     by C. J. Bishop,
    in Canary, Gilman, Heinonen, and Masur, Proceedings of the 2005 Ahlfors Bers Colloquium
    see Bishop's page
    and
    Plane with A_{infty} weighted metric not bi-Lipschitz.
    embeddable in R^N.
    by T. Laakso
    Bull. Lond. Math. Soc. 34 (2002) no 6., 667-676
    [presenter: Meyerson]

  8. Newtonian spaces: An extension of Sobolev spaces to metric measure spaces.
     by N. Shanmugalingam Rev. Mat. Iberoam. 16 (2000) no 2. 243-279

    Presenter should be able to cover the full paper in 2h.
    [presenter: Maples]

  9. Quasisymmetric rigidity of Sierpinski carpets.
     by M. Bonk, S. Merenkov 63 (1988), no. 1, 79-97.
    see Merenkov's page

    Presenter should be able to discuss all of this short paper, assuming its references [1], [2], and [4]
    [presenter: Gautam]

  10. Distortion of Hausdorff measures and improved Painleve removability for quasiregular mappings.
     by K. Astala, A. Clop, J. Mateu, J. Orobitg, and I. Uriarte-Tuero,
    see arXiv:math.CV/0609327

    Cover most of the paper, except possibly the last counterexample.
    [presenter: Kovac]

  11. Rigidity of Schottky Sets.
     by M. Bonk, B. Kleiner, S. Merenkov
    63 (1988), no. 1, 79-97.
    see Merenkov's page

    Do Theorems 1.1, 1.2, and 1.3
    [presenter: Hakobyan]

  12. Removable sets for Sobolev spaces
     by P. Koskela
    Ark. Math. 37 (1999), 291-304
    and
    Removability theorems for Sobolev functions and quasiconformal maps.
    by P. Jones, S. Smirnov
    Ark. Mat. 38 (2000) n0. 2, 263--279

    Do some combination of these two papers.
    [presenter: Tecu]

  13. Quasisymmetric parametrizations of two dimensional metric spheres
    by B. Kleiner, M. Bonk
    Inventiones Math 150 (2002), 127-183

    Extract something good for 2 hours.
    [presenter: Bateman]