SUMMER SCHOOL
SLE, conformal welding, and random planar maps.

  1. Stochastic Loewner evolution, Course notes 2006
    by M. Zinsmeister
    lecture notes.
    Part 1 Contact Huy Tran for instructions
    [presenter: Eveliina Peltola, Helsinki]

  2. Stochastic Loewner evolution, Course notes 2006
    by M. Zinsmeister
    lecture notes.
    Part 2 Contact Huy Tran for instructions
    [presenter: Jianping Jiang, U Arizona]

  3. Half-plane capacity and conformal radius
    by S. Rohde, C. Wong
    Proc. Amer. Math. Soc. 142 (2014), 931-938
    Prove the main theorem 1.2
    [presenter: Angel Chavez, U Arizona]

  4. Quasiconformal variation of slit domains
    by C. Earle, A. Epstein
    Proc. Amer. Math. Soc. 129 (2001), 3363-3372
    Prove Theorem 1, and theorems 2 and 3 in the analytic case.
    [presenter:Kirill Lazebnik, SUNY Stony Brook]

  5. The Loewner equation and Lipschitz graphs
    by S. Rohde, H. Tran, M. Zinsmeister
    preprint
    The main theorem of the paper is Theorem 1.1. The presenter should emphasize what conditions to get the existence of the curve.
    [presenter: O"mer Faruk Tekin, UCLA ]

  6. Space-filling curves and phases of the Loewner equation
    by J. Lind, S. Rohde
    Indiana Univ. Math. J. 61 (2012), 2231-2249
    Prove the main theorem 1.1 (section 3.2) and elaborates theorem 1.3 (section 4) if time permits.
    [presenter: Jhih-Huang Li, Geneva ]

  7. Loewner curvature
    by J. Lind, S. Rohde
    preprint 2014
    Do sections 2, 3, 4.1 and Theorem 5.1.
    [presenter: Kyle Kinneberg, UCLA]

  8. Backward SLE and the symmetry of welding
    by S. Rohde, D. Zhan
    arXiv:1307.2532 .
    The main theorem is Theorem 1.1. The presenter may look at the authors' slide at https://www.ipam.ucla.edu/publications/iagws4/iagws4_11254.pdf. Do sections 2, 3, and 6.
    [presenter: Ben Mackey, Michigan SU]

  9. Random curves, scaling limits and Loewner evolutions
    by A. Kemppainen, S. Smirnov
    arXiv:1212.6215
    -Prove the main theorem 1.3 (section 3). If time allows, discuss applications to statistical physics models.
    [presenter: Alexander Glazman, Geneva ]

  10. Conformal welding and Koebe's theorem
    by C. Bishop
    Ann. of Math. (2) 166 (2007), 613-656
    Do the introduction and prove Theorem 4 and possibly Theorem 2.
    [presenter: Larissa Richards, Toronto]

  11. Random confor- mal weldings
    by K. Astala, P. Jones, A. Kupiainen, E. Saksman
    Acta Math. 207 (2011), 203-254
    and : Random curves by confor- mal welding
    by K. Astala, P. Jones, A. Kupiainen, E. Saksman
    C. R. Math. Acad. Sci. Paris 348 (2010), 257-262.
    The presenter should give an overview of the first paper (based on the second). We care about the construction (section 3). Give an overview of section 4 and do section 5.
    [presenter: Miika Nikula, Helsinki]

  12. Recurrence of distributional limits of finite planar graphs
    by I. Benjamini, O. Schramm
    Electron. J. Probab. 6 (2001), 13 pp.
    Prove the main theorem in the triangulation case (Proposition 2.1). Discuss the ring lemma and skip the proof of lemma 2.3. The presenter can mention a quantitative version of lemma 2.3 in the paper of Gurel-Gurevich and Nachmias
    [presenter: Gerandy Brito, U Washington]

  13. Uniform infinite planar triangulations
    by O. Angel and O. Schramm
    Comm. Math. Phys. 241 (2003), 191213
    Main parts to present are Sections 3 and 4.
    [presenter: Krystal Taylor, IMA]

  14. On the Riemann surface type of random planar maps
    J.T. Gill, S. Rohde
    Rev. Mat. Iberoam. 29 (2013), 1071-1090
    Do the proof of Theorem 1.1 and application to UIPT.
    [presenter: Joe Adams, SUNY Stony Brook]

  15. Recurrence of planar graph limits
    O. Gurel-Gurevich, A. Nachmias,
    Ann. of Math. (2) 177 (2013), 761-781
    We are interested in the recurrence of random walk in UIPT. Do sections 3, 4 and 5 (rough).
    [presenter: Martin Tassy, Brown]

  16. SLE Coordinate Changes
    O. Schramm, D. Wilson,
    New York J. Math. 11 (2005), pages 659 to 669
    Define SLE(kappa, roh) and prove Theorem 3 for the case m=1. Also do Theorem 4.6 in Section 4 of Rohde and Zhang's paper.
    [presenter: Laurie Field, U Chicago]

  17. Liouville quantum gravity and KPZ formula
    B. Duplantier, S. Sheffield,
    Invent Math (2011), 333-393
    Show Proposition 1.1 which makes sense of random measure mu = exp(gamma*h) where h is a Gaussian Free Field. This random measure is related to the boundary measure constructed in Kari Astala, Peter Jones, A. Kupianinen, Eero Saksman's paper. (That boundary measure is constructed from the trace of Gaussian Free Field to the boundary).
    [presenter: Zhiqiang Li, UCLA]