Schedule: All meetings are in 1011 Evans, on Fridays, from 10:30-12am.

• Friday, 16 September. Overview; capacity; convergence of conformal maps. (Peter Ralph)
  After a brief overview of SLE, some background necessary for Loewner's equation will be introduced: half-plane capacity, and caratheodory convergence. See, for instance, Chapter 3 of Conformally invariant processes in the plane, by G. Lawler.

• Friday, 23 September. Loewner's Equation. (Manjunath Krishnapur)
  Loewner's equation (the chordal version) will be introduced, and some properties of it proved: existence of the driving function, continuity of hulls, conditions for a family of hulls to be generated by a curve. See, for instance, Chapter 4 of Conformally invariant processes in the plane, by G. Lawler. Another source is The Loewner differential equation and slit mappings, by D. Marshall and S. Rohde, available from Rohde's website.

• Friday, 30 September. Stochastic Loewner's Equation (Ron Peled)
  The stochastic part, that Schramm added to LE, will be added, and some fundamental properties of the resultant object will be proved: transience and the extent of the self-touching and space-filling regimes. See, for instance, Chapter 6 of Conformally invariant processes in the plane, by G. Lawler, or Basic properties of SLE by S. Rohde and O. Schramm, available from math.PR/0106036.

• Friday, 7 October. Basic properties of SLE (Asaf Nachmias)
  The restriction and locality properties of kappa=8/3 and 6, respectively, will be introduced and proved.

• Friday, 14 October. Convergence of loop-erased random walk to SLE(2) (Aubrey Clayton)
  The convergence of the driving function for LERW to B(2t) will be shown, and a few words will be said about what more work needs to be done. See 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)

• Friday, 21 October. Critical Percolation in the Plane, part one. (Peter Ralph)
  We'll define the problem, and start on Smirnov's proof of the conformal invariance of crossing probablities for cricital site percolation on the triangular lattice, which extends Cardy's formula. Also, we'll probably mention SLE(6). See Smirnov's webpage ("long version") or dvi format.

• Friday, 28 October Critical Percolation in the Plane, part two. (Gabor Pete)
  Will continue with Smirnov's proof, and then say some words about why Smirnov's result implies the existance of a scaling limit for critical site percolation on the triangular lattice, as worked out by Camia and Newman. (see math.PR/0504036)

• Friday, 4 November Percolation theory: background. (Gabor Pete)
  Most of the pre-SLE bounds needed for Smirnov's proof of Cardy's formula and for the convergence of the exploration path to SLE_6, namely: the Russo-Seymour-Welsh estimates, the half-space 2-arm exponent, and an upper bound on the half-space 3-arm exponent will be proved.
• Friday, 11 November Veteran's Day. No seminar.
  
• Friday, 18 November One-arm exponent for critical 2D percolation. (Yun Long)
  The main result, that the probability of a "one-arm crossing event" in critical 2D percolation decays exponentially with exponent 5/48, will be proved. (see Electron. J. Probab. or arXiv:math.PR/0108211)

• Friday, 25 November Thanksgiving holiday. No seminar.

• Friday, 2 December Chordal restriction measures and Brownian Loop Soup (Manjunath Krishnapur)
  A random subset K of a simply connected domain H that touches the boundary of H at two points (say a and b) is said to have restriction property if the law of $K$ conditioned to remain in a simply connected open subset $D$ of $\H$ is identical to that of $\Phi(K)$, where $\Phi$ is a conformal map from $\H$ onto $D$ with $\Phi(a)=a$ and $\Phi(b)=b$. There is a one parameter family of such processes, two important examples being SLE(8/3) and the hull of a Brownian motion started at a and conditioned to exit at b.

  Brownian loop soup is a Poissonian realization of a conformally invariant measure on Brownian loops in the plane. To every SLE(k) (for k in a range of the parameter), one can add all loops in a loop soup (of a particular intensity) that the SLE intersects and get a hull distributed according to a particular (depending on k) Restriction measure. For background, see Conformal restriction: the chordal case by G. Lawler, O. Schramm and W. Werner (available from Journal of AMS or math.PR/0209343 )
  The Brownian Loop Soup by G. Lawler and W. Werner (available from math.PR/0304419)

• Friday, 9 December Chordal restriction measures and Brownian Loop Soup, continued. (Dapeng Zhan)
  See above.

Most of the participants in the seminar will choose a topic, and speak on it to the class. Here is our list of suggested topics; as people commit to topics and dates, we will construct a calendar. Much of this was originally compiled for a summer school organized by M. Biskup, C. Thiele, and J. Garnett of UCLA. The notes (a dvi file) from the summer school might be useful. Further suggestions are of course welcome.

1. Chapter 4 of Conformally Invariant Processes in the Plane, by G. Lawler.
   
   Presenter should introduce conformal invariance and Loewner's equation.
2. Chapter 6 of Conformally Invariant Processes in the Plane, by G. Lawler.
   
   Presenter should introduce SLE, and describe/prove some basic properties, such as transience, self-touching, and space-filling.
3. One-arm exponent for critical 2D percolation by G. Lawler, O. Schramm, W. Werner (avaliable from Electron. J. Probab. or arXiv:math.PR/0108211)
   
   Finds the exponent for the tail of the size of the cluster containing the origin in critical site percolation on the triangular grid, using its convergence to SLE(6).
4. 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, with the necessary percolation background from Grimmett's book. The last part of the paper ("continuum scaling limit") is very sketchy and contains only a fraction of the full story -- the presenter may also want to look at the recent paper of F. Camia and C. Newman (at math.PR/0504036) in which this is worked out in detail. 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.
5. 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 at most sketch the extension to the stronger convergence discussed in the second part of the paper.
6. Chordal restriction measures and Brownian Loop Soup

   A random subset K of a simply connected domain H that touches the boundary of H at two points (say a and b) is said to have restriction property if the law of $K$ conditioned to remain in a simply connected open subset $D$ of $\H$ is identical to that of $\Phi(K)$, where $\Phi$ is a conformal map from $\H$ onto $D$ with $\Phi(a)=a$ and $\Phi(b)=b$. There is a one parameter family of such processes, two important examples being SLE(8/3) and the hull of a Brownian motion started at a and conditioned to exit at b.

   Brownian loop soup is a Poissonian realization of a conformally invariant measure on Brownian loops in the plane. To every SLE(k) (for k in a range of the parameter), one can add all loops in a loop soup (of a particular intensity) that the SLE intersects and get a hull distributed according to a particular (depending on k) Restriction measure.

   Some appropriate papers are:
   Conformal restriction: the chordal case by G. Lawler, O. Schramm and W. Werner (available from Journal of AMS or math.PR/0209343 )
   The Brownian Loop Soup by G. Lawler and W. Werner (available from math.PR/0304419)