The aim of the project was to advance the mathematical understanding of the

analytic and probabilistic aspects of the Loewner energy, in particular, how it

links random conformal geometry, complex analysis and Teichmuller theory. The

project was developed from previous results by Y. Wang (original PI) who showed

the Loewner energy to be the large deviation rate function of SLE0+ and also the

Kahler potential of the Weil-Petersson Teichmuller space. In addition to providing

support for conference travel and for graduate students in this area, the project

resulted in the following mathematical results.

1. A study of the radial SLE_\infty large deviations [1] and an expression for the

 rate function as a geometrically defined quantity associated with foliations on the

 Riemann sphere [3]. These results revealed unexpected symmetries of the rate

 function.

2. A study of the large deviations of SLE0+ [2], confirming conjectures in relating

 the multichordal Loewner energy to determinants of Laplacians and Brownian loop

 measure, which also led to an unexpected link to real rational functions and

 provided a new proof of the Shapiro-Shapiro conjecture on enumerative geometry. 

3. A result relating the Loewner energy to the renormalized energy of moving

 frames on the sphere [5].

4. An identification of a a subclass of Weil-Petersson quasicircles (piecewise

 geodesic Jordan curve) for which both the Loewner driving function and the Kahler

 structure have simple expression. Moreover, this class coincides with the family of

 Jordan curves optimizing the Loewner energy under the constraint of passing

 through given points [6]. We used shear coordinates to describe this class and

 gave characterizations of the Weil-Petersson metric tensor and symplectic form

 in terms of shears [7].

5. A holographic expression of the Loewner energy expressed as a renormalized

 volume in the hyperbolic 3-space.

To disseminate the results obtained from the project and to reach a broader

 readership, a survey paper reviewing the results on various large deviation

 principles of SLE was published [4].

[1] Large deviations of radial SLE infinity
Morris Ang, Minjae Park, Yilin Wang
Electron. J. Probab., Vol. 25, paper no. 102, 1-13 (2020)​

[2] Large deviations of multichordal SLE0+, real rational functions, and zeta-regularized determinants of Laplacians
Eveliina Peltola, Yilin Wang
J. Eur. Math. Soc. (JEMS), published online.

[3] The Loewner-Kufarev energy and foliations by Weil-Petersson quasicircles
Fredrik Viklund, Yilin Wang (submitted)

[4] Large deviations of Schramm-Loewner evolutions: A survey
Yilin Wang
Probability Surveys, vol. 19: 351-403 (2022)

[5] The Loewner Energy via Moving Frames and Surfaces of Finite Renormalised Area Bounding Weil-Petersson Curves
Alexis Michelat, Yilin Wang (submitted)

[6] Piecewise geodesic Jordan curves I: weldings, explicit computations, and Schwarzian derivatives
Don Marshall, Steffen Rohde, Yilin Wang (preprint)

[7] Circle homeomorphisms with square summable diamond shears
Dragomir Saric, Yilin Wang, Catherine Wolfram (submitted)

[8] Universal Liouville action as a renormalized volume and its gradient flow
Martin Bridgeman, Kenneth Bromberg, Franco Vargas Pallete, Yilin Wang (preprint)

Last Modified: 08/08/2023
Modified by: Scott R Sheffield