The earthquake flow is asymmetric and cannot be extended to an SL(2,R) action.
problem The asymmetry of Thurston's earthquake flow and its implications.
method Analysis of orbifold automorphisms and measured geodesic laminations.
result The earthquake flow does not extend to an SL(2,R) action and lacks continuous self-symmetries.
Let $\cT$ be Teichmüller space of a closed surface of genus at least 2. For any point $c\in \cT$, we describe an action of the circle on $\cT\times \cT$, which limits to the earthquake flow when one of the parameters goes to a measured lamination in the Thurston boundary of $\cT$. This circle action shares some of the …
Extends earthquake and horocycle flows to new measures.
problem Ergodic theory of earthquake flow on measured laminations.
method Generalizes shear coordinates to arbitrary measured laminations.
result Classifies ergodic measures for P action on bundle of quadratic differentials.
Proof of Thurston's earthquake theorem using Anti-de Sitter geometry.
problem Proving Thurston's earthquake theorem for orientation-preserving homeomorphisms.
method Using the bi-invariant geometry of Anti-de Sitter three-space.
result Provided a proof of Thurston's earthquake theorem.
The Teichmuller unipotent flow can be defined concretely on certain moduli spaces of singular flat surfaces by shearing polygonal presentations of the surfaces. Thurston's earthquake flow on moduli spaces of hyperbolic surfaces is more mysterious. Both flows have deep and important connections to other areas of mathema…
The paper generalizes Thurston's earthquake map to cluster algebras of finite type.
problem Tackling Thurston's earthquake map in the context of cluster algebras of finite type.
method Introducing a cluster algebraic generalization of Thurston's earthquake map, defined by gluing exponential maps.
result Proves an analogue of the earthquake theorem for cluster algebras of finite type, showing the cluster earthquake map is a homeomorphism.
Study on earthquake metric on Teichmüller space, proving properties and new completions.
problem Understanding the earthquake metric on Teichmüller space.
method Proofs of properties, new completions, and interpretation of the metric.
result Coincidence of various completions for the earthquake metric.
Paper defines and studies measures related to earthquakes and best Lipschitz maps.
problem Understanding Thurston's conjecture about maps and measures on hyperbolic surfaces.
method Examining Lie algebra valued transverse measures and their relation to earthquakes.
result Defines and shows correspondence between best Lipschitz maps and earthquakes.
This paper adapts Thurston's earthquake metric to Riemann surfaces with marked points.
problem Defining a norm and metric on Teichmüller spaces for surfaces of arbitrary genus.
method Adapting Thurston's earthquake norm to Riemann surfaces with marked points and using complex Legendre transforms.
result Establishes a complete analogue of Thurston's earthquake norm in the conformal setting.
A measured laminations on the universal hyperbolic solenoid § is, by our definition, a leafwise measured lamination with appropriate continuity for the transverse variations. An earthquakes on theuniversal hyperbolic solenoid § is uniquely determined by a measured lamination on §; it is a leafwise earthquake with…
In this paper we study the typical speed of a generic earthquake trajectory leaving compact sets in the moduli space of the once-punctured torus. Mirzakhani showed that the earthquake flow is measurably equivalent to the horocyclic flow, which has been studied extensively. Our main result shows that the earthquake flow…
Continuity of earthquake flow map transfers Teichmüller dynamics results.
problem Transfer results from Teichmüller dynamics to earthquake flow.
method Analyze continuity of earthquake flow map and its inverse.
result Transfer results from Teichmüller dynamics to earthquake flow.
We study the convergence of earthquake paths and horocycle paths in the Gardiner-Masur compactification of Teichmüller space. We show that an earthquake path directed by a uniquely ergodic or simple closed measured geodesic lamination converges to the Gardiner-Masur boundary. Using the embedding of flat metrics into th…
Method computes centers of Poisson and skein algebras for loops on surfaces.
problem Computing centers of Poisson and skein algebras associated to loops on surfaces.
method Systematic method using Goldman and Wolpert's Poisson algebras and Turaev's skein algebras.
result Computed centers of various Poisson and skein algebras for finite type hyperbolic surfaces.
The landslide flow, introduced in [5], is a smoother analog of the earthquake flow on Teichmüller space which shares some of its key properties. We show here that further properties of earthquakes apply to landslides. The landslide flow is the Hamiltonian flow of a convex function. The smooth grafting map sgr taking …
The dynamics of earthquake flow equidistributes geodesics on hyperbolic surfaces.
problem Equidistribution of geodesics on hyperbolic surfaces.
method Dynamics of the earthquake flow.
result The dynamics of the earthquake flow equidistributes geodesics on hyperbolic surfaces.
Spatially-aware model improves earthquake hazard assessment accuracy.
problem Misrepresentation of seismic effects across diverse landscapes.
method Causal Bayesian network with Gaussian Processes and normalizing flows.
result Achieves up to 35.2% AUC improvement over existing methods.
For a compact surface X0, Thurston introduced a compactification of its Teichmüller space T(X0) by completing it with a boundary PML(X0) consisting of projective measured geodesic laminations. We introduce a similar bordification for the Teichmüller space T(X0) of a noncompact Ri…
Unique geodesics selected by energy minimization in Teichmüller space.
problem Finding a unique geodesic between points in Teichmüller space.
method Energy minimization of harmonic map rays, extending Thurston boundary.
result Selection of a unique Thurston geodesic through points in Teichmüller space.
A Lie bracket defined on the linear span of the free homotopy classes of undirected closed curves was discovered in stages passing through Thurston's earthquake deformations, Wolpert's corresponding calculations with Hamiltonian vector fields and Goldman's algebraic treatment of the latter leading to a Lie bracket on t…
Geodesic envelopes stay uniformly bounded in specific Teichmüller spaces.
problem Understanding the behavior of geodesics in Teichmüller spaces.
method Identifying extremal geodesics, computing Fenchel-Nielsen twisting, and estimating earthquake path lengths.
result Width of geodesic envelopes is uniformly bounded in specific Teichmüller spaces.
Study on Vaisman metrics on Kodaira-Thurston surface using pluriclosed flow.
problem Characterizing and preserving Vaisman metrics on Kodaira-Thurston surface.
method Characterization of T2-invariant Vaisman metrics, analysis of pluriclosed flow behavior. result Pluriclosed flow preserves Vaisman condition on Kodaira-Thurston surface, including non-constant scalar curvature.
New flows represent Thurston norm ball faces, differing by veering mutations.
problem Dynamic representation of Thurston norm ball faces by distinct flows.
method Combining veering triangulations and mutations to represent faces by multiple flows.
result Non-fibered faces can be represented by two distinct flows differing by veering mutations.
The Teichmüller space TS(b) of hyperbolic metrics on a surface S with fixed lengths at the boundary components is symplectic. We prove that any sum of infinitesimal earthquakes on S that is tangent to TS(b) is Hamiltonian, by providing a Hamiltonian L. Such fun…
This paper uses combinatorial Ricci flow to tackle Thurston's triangulation conjecture.
problem Thurston's triangulation conjecture for hyperbolic 3-manifolds.
method Combinatorial Ricci flow approach to prove convergence and geometric decompositions.
result Combinatorial Ricci flow converges if and only if the triangulation is geometric.
New method predicts spatial events like hurricanes and earthquakes with uncertainty.
problem Quantifying uncertainty in natural hazard predictions.
method Representing spatial point clouds as empirical measures, constraining prediction sets to spatial data manifold, using Wasserstein distance.
result Achieves near-nominal coverage and lower energy/manifold distances compared to baselines.
Study earthquake deformations on a once-punctured torus.
problem Understanding earthquake deformations on Teichmüller space.
method Two methods: linear recurrence relations and hyperbolic geometry.
result Algebraic and geometric interpretations of earthquake deformations.
Deep neural networks predict earthquake locations with high accuracy.
problem Predicting the location of earthquakes with high precision.
method Recurrent Convolutional Neural Networks (R-CNN) model that accounts for spatio-temporal dependencies.
result Neural networks model outperforms baseline models in predicting earthquakes with ROC AUC 0.975 and PR AUC 0.0890.
Constructs Anosov flows in hyperbolic 3-manifolds, disproving a conjecture.
problem Proving existence of infinitely many distinct Anosov flows in certain 4-manifolds.
method Using Cannon-Thurston maps and pseudo-Anosov quasigeodesic flows.
result Infinitely many distinct Anosov flows in some 4-manifolds.
Study shows house buyers in Christchurch value earthquake risk differently based on time since 2011 quake.
problem Understanding how house buyers' perception of earthquake risk changes over time.
method Used a hedonic price model to analyze house prices in Christchurch over three periods.
result Buyers value earthquake risk differently based on the time since the 2011 Christchurch earthquake.
The Euler class conjecture links geometric structures to integral points on the Thurston norm ball.
problem Determining if integral points on the Thurston norm dual ball correspond to geometric structures.
method Examining various geometric, topological, and dynamical structures on 3-manifolds.
result Integral points on the Thurston norm dual ball correspond to the Euler class of taut foliations and other structures.
We show that the analog of Hamilton's Ricci flow in the combinatorial setting produces solutions which converge exponentially fast to Thurston's circle packing on surfaces. As a consequence, a new proof of Thurston's existence of circle packing theorem is obtained. As another consequence, Ricci flow suggests a new algo…
Deviance Voronoi residuals improve earthquake insurance risk assessment.
problem Assessing earthquake insurance risk using spatio-temporal point process models.
method Extended Voronoi residuals and created simulation-based approach.
result Proposed formula for country-wide minimum capital test.
We prove that the bijective correspondence between the space of bounded measured laminations MLb(H) and the universal Teichmüller space T(H) given by λ↦Eλ∣S1 is a homeomorphism for the Fréchet topology on MLb(H) and the Teichmüller topology on T(H), where $E^λ…
EQShapelets detect earthquakes with high accuracy and interpretability.
problem Automated detection and cataloging of earthquakes.
method Time-series shape-based approach embedded in machine learning.
result EQShapelets detected all cataloged and 281 uncataloged events with lower false detection rate.
We give a short proof of the fact that bounded earthquakes of the unit disk induce quasisymmetric maps of the unit circle. By a similar method, we show that symmetric maps are induced by bounded earthquakes with asymptotically trivial measures.
Constructs graph manifolds with many Anosov flows.
problem Finding graph manifolds supporting multiple Anosov flows.
method Cutting geodesic flows, pulling back to finite covers, and gluing compatible pairs of flows.
result Constructs graph manifolds with at least n Anosov flows for any n.
We show that if M is a hyperbolic 3-manifold which admits a quasigeodesic flow, then pi_1(M) acts faithfully on a universal circle by homeomorphisms, and preserves a pair of invariant laminations of this circle. As a corollary, we show that the Thurston norm can be characterized by quasigeodesic flows, thereby generali…
This paper was first written in 1990, but was never published. In it, the author presents a novel approach to the study of constant curvature spacetimes in 2+1 dimensions. A parameterization of flat 2+1-dimensional domains of dependence is given in terms of measured geodesic laminations. There is also an interesting re…
A machine learning surrogate model predicts earthquake-induced building responses.
problem Expensive FE model simulations for earthquake damage estimation.
method SVD-based earthquake characterization and machine learning model training.
result Deep neural network provides most accurate predictions of building responses.
New benchmark for earthquake forecasting models shows current neural point processes are not yet suitable.
problem Lack of a modern benchmark for evaluating neural point process models in earthquake forecasting.
method Curated and standardized earthquake catalog, evaluation protocols, and datasets.
result None of the tested NPPs outperformed the classical ETAS model.
The paper confirms a conjecture about foliating almost Fuchsian manifolds with CMC surfaces.
problem Confirming a conjecture about foliating almost Fuchsian manifolds with CMC surfaces.
method Proving the long-time existence and convergence of a modified mean curvature flow.
result The CMC foliation conjecture is confirmed for a subclass of almost Fuchsian manifolds.
The paper proves a theorem about earthquake extensions of vector fields on circles.
problem Proving a theorem about earthquake extensions of vector fields on circles.
method Using the geometry of the dual of Minkowski three-space and Half-pipe three-geometry.
result A generalization of Kerckhoff's and Gardiner's infinitesimal earthquake theorems to a broader setting.
We prove an "Earthquake Theorem" for hyperbolic metrics with geodesic boundary on a compact surfaces S with boundary: given two hyperbolic metrics with geodesic boundary on a surface with k boundary components, there are 2k right earthquakes transforming the first in the second. An alternative formulation arises…
This paper contains a generalization of the convex ideal case of the Thurston-Andreev theorem when the genus is greater than 1. The heart of the paper concerns taking formal angle data on a surface and ``conformally flowing'' this formal angle data to uniquely associated uniform angle data. This flow turns out to be th…
Earthquake signal detection is at the core of observational seismology. A good detection algorithm should be sensitive to small and weak events with a variety of waveform shapes, robust to background noise and non-earthquake signals, and efficient for processing large data volumes. Here, we introduce the Cnn-Rnn Earthq…
Fractional combinatorial flow improves surface conformal structures.
problem Improving discrete conformal structures on surfaces.
method Introducing a fractional combinatorial Calabi flow for discrete conformal structures on surfaces.
result Longtime existence and global convergence of the fractional combinatorial Calabi flow for various surface types.
Bayesian neural networks improve earthquake rupture prediction and uncertainty estimation.
problem Insufficient data for earthquake rupture studies.
method Used Bayesian neural networks to model earthquake rupture simulations.
result Improved F1-score of 0.8334 compared to plain NN, indicating better performance.