Scientists / Open questions

Wendelin Werner

Probability; stochastic processes; conformal invariance

2006 Fields Medal

24 questions

Completing the Ermakov–Tóth–Werner scaling-limit program for the one-dimensional coalescing ideal gas: particle paths for general profile functions and the proof of the coalescence-time invariance principle
Bounded-type thin local sets of the 2D Gaussian free field: maximality of nested CLE_4 carpets and removal of the topological regularity condition
Sharp uniform-in-location bounds for the conditional shape law of planar Brownian complement components (Werner, 1994)
Axiomatic characterization of non-simple conformal loop ensembles and a loop-soup construction for $\kappa \in (4,8)$
Extending the CLE-plus-Brownian-excursion construction and identification of $\mathrm{SLE}_\kappa(\rho)$ beyond the simple-loop regime $\kappa\in(8/3,4]$ , $\rho>-2$
Almost-sure localization size, critical parameters, and stationary fluctuations for stuck walks
Discrete-to-continuum convergence of the asymmetric and uniform CLE exploration procedures of Werner and Wu
Werner's stochastic Green formula for planar Brownian loops: absolute winding mass in the Wiener sausage, cutoff-profile robustness, and the class of admissible loops
Computing the constant $\alpha(\kappa)$ in Kemppainen–Werner's theorem $\nu^i=\nu^e=\alpha(\kappa)\,\nu^{\mathrm{cle}}$ for the nested simple conformal loop ensembles on the Riemann sphere
Measurable moving targets for Werner's one-dimensional Brownian comparison: extending the circle-valued avoidance inequality beyond continuous targets to rederive Beurling's projection theorem for arbitrary compact sets within the one-dimensional framework
Winding asymptotics of reflected Brownian motion in a three-dimensional cone in the null-recurrent regime
Open problems from Werner's conformally invariant measure on self-avoiding loops: the polygon scaling limit, Brownian-loop structure, and Riemann surfaces
Near-critical percolation interfaces: convergence as curves and the Loewner driving function (massive SLE $_6$ )
Sharp small-time asymptotics for the occupation time of cones by Brownian motion (the Meyre–Werner conjectures (13)–(14))
Open problems from 'Conformal restriction, highest-weight representations and SLE': completing the rigorous SLE–conformal field theory correspondence
Exact winding shells of planar Brownian motion: $L^p$ and almost-sure convergence beyond the $L^2$ shell theorem, and the second moment of the Brownian loop's winding shells
Almost-sure integral tests for occupation times of half-lines beyond strictly stable processes, and the sharp hitting-time tail for perturbed reflecting Brownian motion (the Shi–Werner 1995 open extensions)
Scaling limits of the Gaussian free field local-time pseudo-metric: the CLE $_4$ loop metric and Brownian-time identities on Riemann surfaces
The true self-repelling motion: rigorous dynamics, the critical dimension d=2, and universality of the 3/2 local variation
Scaling limits of critical FK $_q$ percolation and the fuzzy Potts model: proving convergence to CLE $_{\kappa'}$ and the divide-and-color arm exponents
Random soup carpets beyond symmetry: the converse to subcriticality, critical behaviour at the carpet phase transition, and general shape laws
Direct construction, discrete approximation, and Mobius-invariant dynamics for the Curien-Werner Markovian hyperbolic triangulation
Convergence of the two-dimensional self-avoiding walk to chordal $\mathrm{SLE}_{8/3}$ and the conjectured intersection exponents
In-law limit for the largest disc covered by the planar Wiener sausage

Back to open problems

Completing the Ermakov–Tóth–Werner scaling-limit program for the one-dimensional coalescing ideal gas: particle paths for general profile functions and the proof of the coalescence-time invariance principle

Bounded-type thin local sets of the 2D Gaussian free field: maximality of nested CLE_4 carpets and removal of the topological regularity condition

Sharp uniform-in-location bounds for the conditional shape law of planar Brownian complement components (Werner, 1994)

Axiomatic characterization of non-simple conformal loop ensembles and a loop-soup construction for κ∈(4,8)\kappa \in (4,8)κ∈(4,8)

Extending the CLE-plus-Brownian-excursion construction and identification of SLEκ(ρ)\mathrm{SLE}_\kappa(\rho)SLEκ​(ρ) beyond the simple-loop regime κ∈(8/3,4]\kappa\in(8/3,4]κ∈(8/3,4], ρ>−2\rho>-2ρ>−2

Almost-sure localization size, critical parameters, and stationary fluctuations for stuck walks

Discrete-to-continuum convergence of the asymmetric and uniform CLE exploration procedures of Werner and Wu

Werner's stochastic Green formula for planar Brownian loops: absolute winding mass in the Wiener sausage, cutoff-profile robustness, and the class of admissible loops

Computing the constant α(κ)\alpha(\kappa)α(κ) in Kemppainen–Werner's theorem νi=νe=α(κ) νcle\nu^i=\nu^e=\alpha(\kappa)\,\nu^{\mathrm{cle}}νi=νe=α(κ)νcle for the nested simple conformal loop ensembles on the Riemann sphere

Measurable moving targets for Werner's one-dimensional Brownian comparison: extending the circle-valued avoidance inequality beyond continuous targets to rederive Beurling's projection theorem for arbitrary compact sets within the one-dimensional framework

Winding asymptotics of reflected Brownian motion in a three-dimensional cone in the null-recurrent regime

Open problems from Werner's conformally invariant measure on self-avoiding loops: the polygon scaling limit, Brownian-loop structure, and Riemann surfaces

Near-critical percolation interfaces: convergence as curves and the Loewner driving function (massive SLE6_66​)

Sharp small-time asymptotics for the occupation time of cones by Brownian motion (the Meyre–Werner conjectures (13)–(14))

Open problems from 'Conformal restriction, highest-weight representations and SLE': completing the rigorous SLE–conformal field theory correspondence

Exact winding shells of planar Brownian motion: LpL^pLp and almost-sure convergence beyond the L2L^2L2 shell theorem, and the second moment of the Brownian loop's winding shells

Almost-sure integral tests for occupation times of half-lines beyond strictly stable processes, and the sharp hitting-time tail for perturbed reflecting Brownian motion (the Shi–Werner 1995 open extensions)

Scaling limits of the Gaussian free field local-time pseudo-metric: the CLE4_44​ loop metric and Brownian-time identities on Riemann surfaces

The true self-repelling motion: rigorous dynamics, the critical dimension d=2, and universality of the 3/2 local variation

Scaling limits of critical FKq_qq​ percolation and the fuzzy Potts model: proving convergence to CLEκ′_{\kappa'}κ′​ and the divide-and-color arm exponents

Random soup carpets beyond symmetry: the converse to subcriticality, critical behaviour at the carpet phase transition, and general shape laws

Direct construction, discrete approximation, and Mobius-invariant dynamics for the Curien-Werner Markovian hyperbolic triangulation

Convergence of the two-dimensional self-avoiding walk to chordal SLE8/3\mathrm{SLE}_{8/3}SLE8/3​ and the conjectured intersection exponents

In-law limit for the largest disc covered by the planar Wiener sausage

Axiomatic characterization of non-simple conformal loop ensembles and a loop-soup construction for $\kappa \in (4,8)$

Extending the CLE-plus-Brownian-excursion construction and identification of $\mathrm{SLE}_\kappa(\rho)$ beyond the simple-loop regime $\kappa\in(8/3,4]$ , $\rho>-2$

Computing the constant $\alpha(\kappa)$ in Kemppainen–Werner's theorem $\nu^i=\nu^e=\alpha(\kappa)\,\nu^{\mathrm{cle}}$ for the nested simple conformal loop ensembles on the Riemann sphere

Near-critical percolation interfaces: convergence as curves and the Loewner driving function (massive SLE $_6$ )

Exact winding shells of planar Brownian motion: $L^p$ and almost-sure convergence beyond the $L^2$ shell theorem, and the second moment of the Brownian loop's winding shells

Scaling limits of the Gaussian free field local-time pseudo-metric: the CLE $_4$ loop metric and Brownian-time identities on Riemann surfaces

Scaling limits of critical FK $_q$ percolation and the fuzzy Potts model: proving convergence to CLE $_{\kappa'}$ and the divide-and-color arm exponents

Convergence of the two-dimensional self-avoiding walk to chordal $\mathrm{SLE}_{8/3}$ and the conjectured intersection exponents