SUMMER SCHOOL
Stochastic Loewner Equation and Applications

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Topics for the participants' lectures

Every participant will lecture on one of the following papers. Some papers are longer and will be discussed by two of the participants. Some of the papers will only be read partially, more detailed instructions are below. The papers are listed neither in the order of increasing difficulty nor in the order in which they will necessarily be presented.

Every participant will lecture for 2 hours, split into two lectures at different times of the schedule.

Every participant is expected to submit a 4-6 page summary of his or her topic prior to the summer school. The preferred format is either TeX or LateX. A sample file can be retrieved here.

  1. The Loewner differential equation and slit mappings by D. Marshall and S. Rohde (available from Rohde's website)

    We are mainly interested in the fact that Hoelder 1/2 continuous driving force generates slits, but if time allows the converse can also be discussed. Section 2 needs some side reading on quasiconformal maps etc., some of that can be presented in the lecture but probably not all. [Sunhi Choi]

  2. Basic properties of SLE by S. Rohde and O. Schramm (available from math.PR/0106036)

    Part I: The second paragraph of the abstract of the paper starts "The present paper attempts a first systematic study of SLE. It is proved that for all κ not equal 8 the SLE trace is a path." The purpose of the first part of this series is to prove this statement, which is basically Theorem 5.1. This requires most of the article prior to Theorem 5.1, but some parts such as Lemma 3.3 can be skipped. [Brady Newdelman]

  3. Basic properties of SLE by S. Rohde and O. Schramm (available from math.PR/0106036)

    Part II: The second paragraph of the abstract continues "for κ in [0,4] it is a simple path; for κ in (4,8) it is a self-intersecting path; and for κ>8 it is space filling." The second part of the series is to discuss this sentence. This requires mostly presentation of sections 6 and 7 of the article. [Jon Handy]

  4. Aggregation in the Plane and Loewner's Equation by L. Carleson and N. Makarov. (available from Commun. Math. Phys. or Makarov's webpage)

    This paper is some parallel development to SLE in the early days of SLE. Indeed, Schramm's work on SLE had been inspired in part by a lecture of Carleson on this subject. The paper has a beautiful discussion in the introduction, of which some would be certainly nice in the presentation, though maybe not all remarks in the introduction can be elaborated on in detail. Likewise, it would be nice if the real variable approach in section 3 was touched upon, but maybe this cannot be done in full detail. [Nick Crawford]

  5. Critical Percolation in the plane. I. Conformal invariance and Cardy's formula. II. Continuum scaling limit. by S. Smirnov. (available from Smirnov's webpage ("long version") or directly in dvi format)

    The presenter should focus on Sections 1 and 2. Lemma 4.1 is needed and can be looked up in Grimmett's book on percolation (11.70 in second edition, 9.70 in first edition). The last part of the paper ("continuum scaling limit") is very sketchy and contains only a fraction of the full story (which has recently been worked out in great detail by F. Camia and C. Newman math.PR/0504036). The presenter may wish to consult the paper Critical exponents for two-dimensional percolation by S. Smirnov and W. Werner [Math. Res. Lett. 8 (2001), no. 5-6, 729-744] (see its Math Review) where the critical percolation exponents were derived on the basis of Smirnov's result. [Gabor Pete]

  6. The harmonic explorer and its convergence to SLE(4) by O. Schramm and S. Sheffield (available from math.PR/0310210)

    The presenter should focus on the proof of convergence in Hausdorff topology (this can be done in some detail) and only sketch the extension to the stronger convergence discussed in the second part of the paper. [Nam-Gyu Kang]

  7. Conformal invariance of planar loop-erased random walks and uniform spanning trees by G. Lawler, O. Schramm and W. Werner (can be retrieved from Annals of Probability or math.PR/0112234)

    Part I: The presenter should extract only the material pertaining to the proof of conformal invariance of the loop-erased random walk (as stated in Theorem 1.1). That is, up to page 33 only (still a lot), save for everything on pages 1-33 that is not geared towards Theorem 1.1. [Aubrey Clayton]

  8. Conformal invariance of planar loop-erased random walks and uniform spanning trees by G. Lawler, O. Schramm and W. Werner (can be retrieved from Annals of Probability or math.PR/0112234)

    Part II: The presenter should cover the parts of the paper that have to do with the uniform spanning tree (UST) and/or the UST Peano curve. The crux is the description of the scaling limit of these object. For the UST it may be useful to take a look back at the seminal paper of O. Schramm Scaling limits of loop-erased random walks and uniform spanning trees [Israel J. Math. 118 (2000), 221-288] (see math.PR/9904022) where SLE was introduced. [Peter Ralph]

  9. Conformal restriction, highest-weights representations and SLE by R. Friedrich and W. Werner (available from math.PR/0301018)

    The announcement in Compte Rendue by the same authors has a few extra sentences explaining better what is going on. Many of the exponents appearing in SLE theory have been conjectured by physicists before. An important way to do these conjectures is via representation theory of the Virasoro (Lie) algebra. This paper is the unique paper in the summer school talking about Virasoro algebras. The presenter should make clear how the Virasoro approach works to get "5/8" and how this can be understood and explained once one has the SLE theory. Probably one has to do a bit reading on the side (and explaining on the side) to understand just enough of the highest weight representation theory to make sense of the Virasoro approach. The SLE basics in the paper should of course be sketched, but the presenter should be aware that others are discussing much of the SLE part as well. [Shannon Starr]

  10. Conformal restriction: the chordal case by G. Lawler, O. Schramm and W. Werner (available from Journal of AMS or math.PR/0209343 )

    The paper comes in two parts: (1) The two-sided restriction measures and their connection to SLE8/3 and (2) the one-sided restriction, SLE(κ,ρ) processes and the connection with the outer boundary of Brownian motion. The presenter should try to cover the following topics -- listed in the decreasing order of importance: (a) Two sided restriction measures and the special role SLE8/3 plays for them. (b) Locality of SLE6. (c) The equivalence of outer boundaries of Brownian motion and SLE6 as stated in Theorems 9.1, 9.3 and 9.4. (d) Brownian bubbles. (e) One sided restriction and SLE(κ,ρ). [Jesse Goodman]

  11. Hausdorff dimensions for SLE6 by V. Beffara (available from Annals of Probability or math.PR/0204208)

    The presenter should focus on Sections 0-3 of the paper. The principal result is the value of Hausdorff dimension of the trace of SLE6 stated in Theorem 1. The paper can be read and presented in more or less linear fashion. The presenter may wish to take a look at the generalization of this work to SLEκ available from math.PR/0211322. [Tai Melcher]

  12. Values of Brownian intersection exponents, parts I, II and III by G. Lawler, O. Schramm and W. Werner (available form arxiv -- part I, part II, and part III)

    The presented should extract the relevant results about Brownian intersection exponents, cascade relations and their solutions in terms of the "U-function," which provides a (not-yet understood) link to quantum gravity approach to conformal invariance by physicists. There is some overlap with topic no. 10. [Noam Berger]

  13. Conformal invariance of domino tiling by R. Kenyon (available from Annals of Probability)

    Kenyon's work represents an independent approach to conformal invariance in 2D combinatorial models. The presenter should focus on the case of domino tilings (i.e., ignore generalizations to polyominos whenever possible). It may be necessary to go back to the original papers by Kasteleyn (ref. [12]) for the explanation of the "Kasteleyn's method." [Jason Asher]