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  • (F 2-23)
    • Welcome to the course! Here is our syllabus.

Course Calendar

Date Content References Slides
F 2-23 Models from statistical mechanics Kemp., Ch.1; Lawler, An introduction to the stochastic Loewner evolution Slides
W 3-1 Statistical mechanics intuition, cont. Conformal maps I. Tran's paper; Duren, 2.1-2.2; Lawler, Conformally Invariant Processes, 3.2 Slides
F 3-3 Conformal maps II: Loewner's proof of the 3rd coefficient. Duren, 2.3, 3.1-3.5. See also Lawler Conformally Invariant Processes 3.6 for proof of the Caratheodory convergence theorem.
F 3-10 Martin boundary, SLE_0, SLE_0(\rho) and Loewner energy Berestycki-Norris notes, chapter 1
W 3-15 Brownian motion, harmonic functions, harmonic measure and the Green's function Berestycki-Norris notes, chapters 2 and 3; see also Lawler Conformally Invariant Processes 2.4 for a more detailed treatment of the Green's function
F 3-17 Green's func, cont.; Compact H-hulls, their mapping out functions and estimates Berestycki-Norris notes 3.3, chapter 4, 5.1
F 3-24 Continuity and differentiability estimate for g_K; half-plane capacity Berestycki-Norris notes 5.2-5.3, 6.2
W 3-29 Two notions of capacity, Loewner transform, Loewner equation Berestycki-Norris notes Finish 6.2, 6.1, 7.1-7.2 Slides
F 3-31 A Loewner transform example and its Loewner energy; alternative definition of local growth; inverting Loewner transform Berestycki-Norris notes 7.3, 8.1
F 4-7 Continuity of Loewner transform and inverse, SLE and Schramm's principle Berestycki-Norris notes 8.2-8.3, 9.1
W 4-12 Rohde-Schramm Theorem, SLE in other domains; Review of Ito integral Berestycki-Norris notes 9.2-9.3, Oksendal chapter 3
F 4-14 Ito's formula for semimartingales, the Bessel SDE and time changes of local martingales Oksendal 3.3, 4.1-4.2, Kemppainnen 2.2-2.3, Lawler Conformally Invariant Processes ch.1
F 4-21 Analysis of Bessel SDE; hitting of SLE_\kappa on real line Berestycki-Norris chapter 10
W 4-26 Phases of SLE, transience; statement of Hausdorff dimension Berestycki-Norris chapter 11; Rohde-Schramm Basic properties of SLE section 7
F 4-28 SLE one point function; upper bound on Hausdorff dim Kemppainen sec. 5.6.4, 5.3.6; Beffara; for helpful review of Hausdorff dimension and its relation to other dimensions (box dimension, etc), see Przytycki-Urbanski Conformal fractals chapter 8
F 5-5 May holiday, no class
W 5-10 SLE large deviations in uniform topology I Guskov A large deviation principle for the Schramm-Loewner evolution in the uniform topology, sections 1-4 (see also Wang Large deviations of Schramm-Loewner evolutions: A survey, Wang The energy of a deterministic Loewner chain: Reversibility and interpretation via SLE_{0+}, and Peltola-Wang Large deviations of multichordal SLE_{0+}, real rational functions, and zeta-regularized determinants of Laplacians. See also Tran-Yuan A support theorem Prop. 1.4 for the referenced result on the closeness of SLE to any finite-energy curve.)
F 5-12 SLE large deviations in uniform topology II Guskov A large deviation principle, sections 4-6
F 5-19 Student presentations
W 5-24 Topological support of SLE Tran-Yuan, sections 2 and 3
F 5-26 Finish topological support of SLE for k<=4; Conformal invariance of function spaces Tran-Yuan section 4; Berestycki-Norris 15.1
F 6-2 Bridge decomposition of SLE_{8/3} (student presentation); Review of Gaussian free field Alberts & Duminil-Copin; Berestycki-Norris 15.2, Berestycki-Powell 1.6
W 6-7 Loop erasure of planar BM (student presentation); SLE_4 as a contour line of the GFF Zhan; Berestycki-Norris 15.3-4, Schramm-Sheffield
F 6-9 SLE facts we didn't get to; video lecture by Nina Holden: Conformal welding of Liouville quantum gravity disks and an application to the integrability of SLE Misc
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Potential papers for student presentations

Presentations may be 30-50 minutes in length, and should focus on main ideas from a paper. Since there will not be time to give many detailed proofs, presentations should outline key results and key steps in the argument. The literature is vast, and students are also very welcome to suggest papers not on this list (including relevant works they themselves have done). Top

References

Papers mentioned in class as well as lecture notes/textbooks used as basis for lecture material. Top