The Experts below are selected from a list of 5907 Experts worldwide ranked by ideXlab platform
Yusuke Kuno - One of the best experts on this subject based on the ideXlab platform.
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the logarithms of dehn twists
Quantum Topology, 2014Co-Authors: Nariya Kawazumi, Yusuke KunoAbstract:Let Σ be an oriented connected compact surface of genus g (≥ 1) with 1 boundary component. Choose a basepoint ∗ ∈ ∂Σ. We denote π := π1(Σ, ∗) and H := H1(Σ;Q). The Simple loop going around the boundary in the opposite direction defines an element ζ ∈ π. Any Simple Closed Curve C ⊂ Σ defines the right handed Dehn twist tC along C as an element of the mapping class group of the surface Σ relative to the boundary ∂Σ. The classical formula says the action |tC | of the Dehn twist tC on the homology group H is given by |tC | = 1H − [C] ⊗ [C] ∈ Hom(H,H), where [C] ∈ H is the homology class of C with a fixed orientation, and we identify H ⊗H = Hom(H,H), Y ⊗ Z → (X → (X · Y )Z), by the Poincare duality. Our result generalizes this formula to the action of tC on the completed group ring Qπ, where the completion is induced by the augmentation ideal Iπ ⊂ Qπ. Massuyeau [10] introduced the notion of a symplectic expansion of the group π, which provides an isomorphism of pairs of complete Hopf algebras
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the logarithms of dehn twists
arXiv: Geometric Topology, 2010Co-Authors: Nariya Kawazumi, Yusuke KunoAbstract:By introducing an invariant of loops on a compact oriented surface with one boundary component, we give an explicit formula for the action of Dehn twists on the completed group ring of the fundamental group of the surface. This invariant can be considered as ``the logarithms" of Dehn twists. The formula generalizes the classical formula describing the action on the first homology of the surface, and Morita's explicit computations of the extended first and the second Johnson homomorphisms. For the proof we use a homological interpretation of the Goldman Lie algebra in the framework of Kontsevich's formal symplectic geometry. As an application, we prove the action of the Dehn twist of a Simple Closed Curve on the $k$-th nilpotent quotient of the fundamental group of the surface depends only on the conjugacy class of the Curve in the $k$-th quotient.
Wendelin Werner - One of the best experts on this subject based on the ideXlab platform.
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conformal invariance of planar loop erased random walks and uniform spanning trees
arXiv: Probability, 2001Co-Authors: Gregory F Lawler, Oded Schramm, Wendelin WernerAbstract:We prove that the scaling limit of loop-erased random walk in a simply connected domain $D$ is equal to the radial SLE(2) path in $D$. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invariant. Assuming that the boundary of the domain is a $C^1$ Simple Closed Curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano Curve, where the tree is wired along a proper arc $A$ on the boundary, is the chordal SLE(8) path in the closure of $D$ joining the endpoints of $A$. A by-product of this result is that SLE(8) is almost surely generated by a continuous path. The results and proofs are not restricted to a particular choice of lattice.
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conformal invariance of planar loop erased random walks and uniform spanning trees
arXiv: Probability, 2001Co-Authors: Gregory F Lawler, Oded Schramm, Wendelin WernerAbstract:We prove that the scaling limit of loop-erased random walk in a simply connected domain $D$ is equal to the radial SLE(2) path in $D$. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invariant. Assuming that the boundary of the domain is a $C^1$ Simple Closed Curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano Curve, where the tree is wired along a proper arc $A$ on the boundary, is the chordal SLE(8) path in the closure of $D$ joining the endpoints of $A$. A by-product of this result is that SLE(8) is almost surely generated by a continuous path. The results and proofs are not restricted to a particular choice of lattice.
Giusi Vaira - One of the best experts on this subject based on the ideXlab platform.
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maximal solution of the liouville equation in doubly connected domains
Journal of Functional Analysis, 2019Co-Authors: Michal Kowalczyk, Angela Pistoia, Giusi VairaAbstract:Abstract In this paper we consider the Liouville equation Δ u + λ 2 e u = 0 with Dirichlet boundary conditions in a two dimensional, doubly connected domain Ω. We show that there exists a Simple, Closed Curve γ ⊂ Ω such that for a sequence λ n → 0 and a sequence of solutions u n it holds u n log 1 λ n → H , where H is a harmonic function in Ω ∖ γ and λ n 2 log 1 λ n ∫ Ω e u n d x → 8 π c Ω , where c Ω is a constant depending on the conformal class of Ω only.
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maximal solution of the liouville equation in doubly connected domains
arXiv: Analysis of PDEs, 2018Co-Authors: Michal Kowalczyk, Angela Pistoia, Giusi VairaAbstract:In this paper we consider the Liouville equation $\Delta u +\lambda^2 e^{\,u}=0$ with Dirichlet boundary conditions in a two dimensional, doubly connected domain $\Omega$. We show that there exists a Simple, Closed Curve $\gamma\subset \Omega$ such that for a sequence $\lambda_n\to 0$ and a sequence of solutions $u_{n}$ it holds $\frac{u_{n}}{\log\frac{1}{\lambda_n}}\to H$, where $H$ is a harmonic function in $\Omega\setminus\gamma$ and $\frac{\lambda_n^2}{\log\frac{1}{\lambda_n}}\int_\Omega e^{\,u_n}\,dx\to 8\pi c_\Omega$, where $c_\Omega$ is a constant depending on the conformal class of $\Omega$ only.
Nariya Kawazumi - One of the best experts on this subject based on the ideXlab platform.
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the logarithms of dehn twists
Quantum Topology, 2014Co-Authors: Nariya Kawazumi, Yusuke KunoAbstract:Let Σ be an oriented connected compact surface of genus g (≥ 1) with 1 boundary component. Choose a basepoint ∗ ∈ ∂Σ. We denote π := π1(Σ, ∗) and H := H1(Σ;Q). The Simple loop going around the boundary in the opposite direction defines an element ζ ∈ π. Any Simple Closed Curve C ⊂ Σ defines the right handed Dehn twist tC along C as an element of the mapping class group of the surface Σ relative to the boundary ∂Σ. The classical formula says the action |tC | of the Dehn twist tC on the homology group H is given by |tC | = 1H − [C] ⊗ [C] ∈ Hom(H,H), where [C] ∈ H is the homology class of C with a fixed orientation, and we identify H ⊗H = Hom(H,H), Y ⊗ Z → (X → (X · Y )Z), by the Poincare duality. Our result generalizes this formula to the action of tC on the completed group ring Qπ, where the completion is induced by the augmentation ideal Iπ ⊂ Qπ. Massuyeau [10] introduced the notion of a symplectic expansion of the group π, which provides an isomorphism of pairs of complete Hopf algebras
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the logarithms of dehn twists
arXiv: Geometric Topology, 2010Co-Authors: Nariya Kawazumi, Yusuke KunoAbstract:By introducing an invariant of loops on a compact oriented surface with one boundary component, we give an explicit formula for the action of Dehn twists on the completed group ring of the fundamental group of the surface. This invariant can be considered as ``the logarithms" of Dehn twists. The formula generalizes the classical formula describing the action on the first homology of the surface, and Morita's explicit computations of the extended first and the second Johnson homomorphisms. For the proof we use a homological interpretation of the Goldman Lie algebra in the framework of Kontsevich's formal symplectic geometry. As an application, we prove the action of the Dehn twist of a Simple Closed Curve on the $k$-th nilpotent quotient of the fundamental group of the surface depends only on the conjugacy class of the Curve in the $k$-th quotient.
Gregory F Lawler - One of the best experts on this subject based on the ideXlab platform.
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conformal invariance of planar loop erased random walks and uniform spanning trees
arXiv: Probability, 2001Co-Authors: Gregory F Lawler, Oded Schramm, Wendelin WernerAbstract:We prove that the scaling limit of loop-erased random walk in a simply connected domain $D$ is equal to the radial SLE(2) path in $D$. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invariant. Assuming that the boundary of the domain is a $C^1$ Simple Closed Curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano Curve, where the tree is wired along a proper arc $A$ on the boundary, is the chordal SLE(8) path in the closure of $D$ joining the endpoints of $A$. A by-product of this result is that SLE(8) is almost surely generated by a continuous path. The results and proofs are not restricted to a particular choice of lattice.
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conformal invariance of planar loop erased random walks and uniform spanning trees
arXiv: Probability, 2001Co-Authors: Gregory F Lawler, Oded Schramm, Wendelin WernerAbstract:We prove that the scaling limit of loop-erased random walk in a simply connected domain $D$ is equal to the radial SLE(2) path in $D$. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invariant. Assuming that the boundary of the domain is a $C^1$ Simple Closed Curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano Curve, where the tree is wired along a proper arc $A$ on the boundary, is the chordal SLE(8) path in the closure of $D$ joining the endpoints of $A$. A by-product of this result is that SLE(8) is almost surely generated by a continuous path. The results and proofs are not restricted to a particular choice of lattice.