Workshop February 2023
We are organizing an in-person workshop on Schramm-Loewner evolution and related topics at UPenn on Saturday and Sunday, Feb. 18 and 19, 2023. If you are interested in participating, please register here. The event is supported by the NSF Career award 2046514.
We are not able to promise financial support to any participants (except speakers). If we have money available after the workshop we can reimburse the cost of additional participants, with priority to junior participants and those without other sources of funding.
Information for parking and restaurant recommendation near Penn.
Confirmed Speakers
Greg Lawler (Chicago)
Vlad Margarint (CU Boulder)
Eveliina Peltola (Aalto and Bonn)
Pu Yu (MIT)
Yizheng Yuan (Cambridge)
Dapeng Zhan (Michigan State)
Schedule
Saturday, Feb. 18
Venue: Amado Recital Hall, Irvine Auditorium, 3401 Spruce St.
10:30-11:00am: Registration
11-11:45am: Eveliina
11:45am-2:00pm: Lunch break
2:00-2:45pm: Vlad
3:00-3:30pm: Tea break
3:30-4:15pm: Yizheng
4:30-5:00pm: Tea break
5:00-5:45pm: Pu
Sunday, Feb. 19
Venue: Room A6, David Rittenhouse Laboratory, 209 S 33rd St.
09:30 am Continental breakfast
10:00-10:45am: Greg
11:00-11:45am: Dapeng
Abstracts
Greg Lawler: A sampling of SLE results (some old, some new)
SLE I will revisit two older papers of mine about partition function approaches to SLE in multiply connected domains and show how those ideas inform solutions to some more recent problems.
Vlad Margarint: Continuity in kappa and approximations in SLE theory
In this talk, I will cover a study of the Loewner Differential Equation using Rough Path techniques, and beyond. The Loewner Differential Equation describes the evolution of a family of conformal maps. We rephrase this in terms of singular Rough Differential Equations (RDE). In this context, it is natural to study questions on the stability, and approximations of solutions of this equation. I will present a result on the continuity of the dynamics and related objects in the parameter kappa. The first approach will be based on Rough Path Theory, and the second approach will be based on a constructive method of independent interest: the square-root interpolation of the Brownian driver of the Loewner Differential Equation. I will also touch on the second question, which is the approximation of solutions of this singular RDE. I will present another approximation method with some further applications to the simulations of the SLE traces. In the final part, I will touch on some recent results on the Multiple SLE with Dyson Brownian motion driver, as well as on future potential investigations in this direction.
Eveliina Peltola: On large deviations of SLEs, real rational functions, and zeta-regularized determinants of Laplacians
When studying large deviations (LDP) of Schramm-Loewner evolution (SLE) curves, a ''Loewner energy", and "Loewner potential'', that describe the rate function for the LDP, were recently introduced. While these objects were originally derived from SLE theory, they turned out to have several intrinsic, and perhaps surprising, connections to various fields. I will discuss some of these connections and interpretations towards Brownian loops, semiclassical limits of certain correlation functions in conformal field theory, and rational functions with real critical points (Shapiro-Shapiro conjecture in real enumerative geometry). (Based on joint work with Yilin Wang - IHES, France.)
Pu Yu: Radial mating-of-trees and the reversibility of whole plane SLE_\kappa for \kappa>8
The reversibility of whole plane SLE_\kappa for \kappa<8 has been established by Miller-Sheffield and Zhan, and in a recent work by Viklund and Wang, it has been conjectured that the reversibility also holds for \kappa>8 via studying the Loewner energy. Based on a joint work with Ang, we construct radial analog of the mating-of-trees coupling between space-filling SLE and Liouville quantum gravity by Duplantier-Miller-Sheffield, from which we prove the reversibility of whole plane SLE_\kappa for \kappa>8.
Yizheng Yuan: Regularity of SLE: Up-to-constant variation and modulus of continuity
I will present several sharp results on the path regularity of SLE. In particular, I will discuss the modulus of continuity, law of iterated logarithm, and variation regularity. Previous works have determined the optimal Hölder and p-variation exponent; our results improve them by providing the optimal (up to constant) gauge function with the correct logarithmic correction. As a key step in the proof, we obtain sharp estimates on the lower tail of the Minkowski content.
I will also explain how to obtain the analogous bounds for discrete models converging to SLE, and illustrate it on the LERW. In particular, we obtain convergence of the LERW in a stronger path topology.
This talk is based on a joint paper with Nina Holden.
Dapeng Zhan: SLE$_\kappa(\rho)$ bubble measures
For $\kappa>0$ and $\rho>-2$, we construct a $\sigma$-finite measure, called a rooted SLE$_\kappa(\rho)$ bubble measure, on the space of curves in the upper half plane $\mathbb H$ started and ended at the same boundary point, which satisfies some SLE$_\kappa(\rho)$-related domain Markov property, and is the weak limit of SLE$_\kappa(\rho)$ curves in $\mathbb H$ with the two endpoints both tending to the root. For $\kappa\in(0,8)$ and $\rho\in ((-2)\vee(\frac\kappa 2-4),\frac\kappa 2-2)$, we derive decomposition theorems for the rooted SLE$_\kappa(\rho)$ bubble with respect to the Minkowski content measure of the intersection of the rooted SLE$_\kappa(\rho)$ bubble with $\mathbb R$, and construct unrooted SLE$_\kappa(\rho)$ bubble measures.