Study K3 surfaces and their metrics, focusing on dynamics.
problem Understanding dynamics on K3 surfaces.
method Interactions between K3 surface geometry and Ricci-flat metrics, dynamical study of automorphisms.
result Positive entropy automorphisms on K3 surfaces.
Theory of point vortices extended to closed surfaces.
problem Extending point vortex dynamics to closed surfaces.
method Unified theory of point vortex dynamics on the plane, sphere, and closed surfaces.
result Comprehensive guide to point vortex dynamics on closed surfaces with genus zero and vanishing total vorticity.
Modeling implied volatility surface dynamics with Hawkes kernels.
problem Understanding and predicting high-frequency dynamics of the implied volatility surface.
method Hawkes modeling of the volatility surface, with coefficients governing skew and convexity.
result Simple conditions on Hawkes kernel coefficients ensure no-arbitrage and reduce parameter estimation.
We study the dynamics of the vector field on an open surface given by the gradient of a Green's function. This dynamical approach enables us to show that this field induces an invariant decomposition of the surface as the union of a disk and a 1-skeleton that encodes the topology of the surface. We analyze the structur…
Investigates multifractal scaling in critical dynamics of random surfaces.
problem Analyzing multifractal scaling in critical dynamics of random surfaces.
method Examined multifractal scaling in various conformal field theories on random surfaces.
result Higher moments of time variations of the order parameter exhibit multifractal scaling.
This paper studies the critical dynamics of random surfaces, focusing on area and genus evolution.
problem Understanding the time evolution of random surfaces and their genus.
method Analyzes the dynamics of area and genus using Cox-Ingersoll-Ross process and critical phenomena.
result The genus of surfaces evolves into two phases: planar surfaces and foamy surfaces.
New method connects veering triangulations to dynamic pairs.
problem Understanding veering triangulations and their properties.
method Shearing decomposition of veering triangulations.
result Canonically associated dynamic pairs of branched surfaces.
Dynamic functional time-series methods improve forecast accuracy for foreign exchange implied volatility surfaces.
problem Forecasting implied volatility surfaces in foreign exchange markets.
method Dynamic functional principal component analysis and multivariate functional time-series methods.
result Dynamic univariate functional time-series method shows the greatest improvement in forecast accuracy.
Various problems of geometry, topology and dynamical systems on surfaces as well as some questions concerning one-dimensional dynamical systems lead to the study of closed surfaces endowed with a flat metric with several cone-type singularities. Such flat surfaces are naturally organized into families which appear to b…
Zeta functions extended to nonorientable surfaces, order of vanishing computed.
problem Computing dynamical zeta functions for nonorientable surfaces.
method Simple argument extending microlocal proofs to nonorientable case.
result Order of vanishing of zeta function is the first Betti number.
This paper finds a global surface of section in dynamically convex L(p,p-1) using ECH.
problem Finding a global surface of section in dynamically convex L(p,p-1).
method Using Embedded Contact Homology (ECH).
result Relates periods of the surface of section to the first ECH spectrum.
Construct intrinsic Langevin dynamics for rigid inclusions on curved surfaces.
problem Stochastic dynamics of rigid inclusions on curved surfaces.
method Cartan's method of moving frames, Hamiltonian equations, intrinsic Langevin equations, Fokker-Planck equation.
result Extracted overdamped equations for accurate simulations of diffusion processes.
Study shows dynamics of rank 1 orbifolds in flat surfaces.
problem Characterize dynamics of rank 1 affine invariant orbifolds.
method Analyzes M-isoperiodic foliations and their ergodic properties.
result Leaves of the isoperiodic foliation are either all closed or all dense.
Study characterizes bladder motion using dynamic MRI and statistical analysis.
problem Limited volume coverage in dynamic MRI sequences hinders 3D shape reconstruction.
method 3D dense velocity measurements, LDDMM framework, statistical characterization, mean curvature changes, surface deformation analysis.
result Stable shape descriptor for characterizing bladder surface dynamics.
We propose a general framework for constructing and describing infinite type flat surfaces of finite area. Using this method, we characterize the range of dynamical behaviors possible for the vertical translation flows on such flat surfaces. We prove a sufficient condition for ergodicity of this flow and apply the cond…
Lecture notes introduce Abelian differentials and their flat surfaces, focusing on families and Teichmüller dynamics.
problem Study of Abelian differentials and their geometric properties.
method Associate flat surfaces to Abelian differentials and analyze their families under GL2+(R) action. result Properties of orbit of Abelian differentials under Teichmüller dynamics.
Absolutely partially hyperbolic surface endomorphisms have a coherent center foliation.
problem Understanding the dynamics of absolutely partially hyperbolic surface endomorphisms.
method Showed the existence of a center foliation and leaf conjugacy to the linearization.
result Absolutely partially hyperbolic surface endomorphisms have a dynamically coherent center foliation.
New minimal surfaces found from vortex crystals.
problem Minimal surfaces and vortex crystals.
method Gluing helicoids into minimal surfaces.
result New minimal surfaces and vortex crystals discovered.
Study the exponential map on surfaces using fluid dynamics.
problem Exponential map of volume-preserving diffeomorphisms on closed surfaces.
method Fluid dynamical proof of Ebin--Misiołek--Preston theorem and extension of Shnirelman's rigidity result.
result Exponential map is a nonlinear Fredholm mapping of index zero and Fredholm quasiregular.
Study of dynamics on cubic surfaces and their connection to Painlevé 6 Equation.
problem Understanding the dynamics of automorphism groups on cubic surfaces.
method Analyzing holomorphic automorphisms and character varieties.
result Several open questions about the dynamics of automorphism groups.
Study on liquidity dynamics in Uniswap v3 pools using statistical methods.
problem Characterize liquidity in Uniswap v3 pools.
method Functional principal component analysis (FPCA) and dynamic factor methods.
result Liquidity dynamics in Uniswap v3 pools are well-captured by a low-order Legendre polynomial basis.
The study examines dynamics on SU(2)-representation varieties for surfaces and non-orientable surfaces.
problem Dynamics of group actions on SU(2)-representation varieties of surfaces and non-orientable surfaces.
method Description and analysis of group actions generated by Dehn twists on SU(2)-representation varieties.
result Explicit invariant rational functions on SU(2)-representation varieties for specific cases of surfaces and non-orientable surfaces.
Proves identities linking curve lengths and orthogeodesics on hyperbolic surfaces.
problem Exploring relationships between curve lengths and orthogeodesics on hyperbolic surfaces.
method Partitioned orthogeodesics into sets based on dynamical behavior, relating them to geodesics on orbifold surfaces.
result Extends a result to surfaces with cusps, showing how to extend a previous result.
Study of flows on circle bundles over translation surfaces, showing decay of correlations.
problem Ergodic properties of flows on circle bundles over translation surfaces.
method Generalizing Heisenberg nilflows to more general base surfaces, showing relatively mixing.
result Showed that such flows exhibit decay of correlations in the orthogonal complement of functions constant along fibers.
Study homeomorphisms on fine curve graph of surfaces, revealing new types of dynamics.
problem Understanding dynamics of homeomorphisms on fine curve graphs of surfaces.
method Analyzing the action of homeomorphisms on the fine curve graph and relating to classical curve graphs.
result Homeomorphisms induce parabolic isometries, and all positive reals are realized as asymptotic translation lengths.
Study on stability of hyperkähler flow in 4-manifolds.
problem Stability of hyperkähler flow in 4-manifolds.
method Extending results from mean curvature flow for minimal surfaces to hyperkähler flow.
result Obtained a dynamic stability theorem for hyperkähler flow.
New spectral invariants recover Calabi invariant for surface dynamics.
problem Understanding Hamiltonian homeomorphisms and spectral invariants.
method Defining new spectral invariants for Lagrangian links in surfaces.
result Our invariants recover the Calabi invariant and resolve open questions.
This paper investigates dynamics that persist under isotopy in classes of orientation-preserving homeomorphisms of orientable surfaces. The persistence of periodic points with respect to periodic and strong Nielsen equivalence is studied. The existence of a dynamically minimal representative with respect to these relat…
We focus on the topology and dynamics of minimal sets and Levi-flats in surfaces of general type. Our method relies on the ergodic theory of Riemann surfaces laminations: we use harmonic measures and Lyapunov exponents. Our first result establishes that minimal sets have large Hausdorff dimension when a leaf is simply …
Explains a 2D color exchange invariant correspondence to 3D linking numbers.
problem Understanding color exchange invariants in 2D dynamics and their 3D geometric interpretation.
method Visualizes invariants as linking of lines on a special surface with Arf-Kervaire invariant one, and interprets it as an obstruction to continuous transformation.
result Interprets a 2D color exchange invariant as a 3D linking number, providing a topological explanation.
Study connects flow dynamics to 3D geometry via surface intersections.
problem Relating flow dynamics to geometric properties of 3-manifolds.
method Relates pseudo-Anosov flow dynamics to hyperbolic geometry via curve graphs.
result Established a link between flow invariants and geometric features of 3-manifolds.
Constructs algorithms to recognize and classify 2D surfaces.
problem Recognizing and classifying 2D surfaces in dynamic systems.
method Discrete topological structures and algorithms for simplicial and CW-complexes.
result Determines the topological type of 2-manifolds.
A new method for 3D surface registration using dynamic programming.
problem Elastic shape registration of 3D surfaces.
method Optimization over a subset of reparametrizations using dynamic programming.
result Proposes an algorithm that produces a solution closer to optimal than gradient-based methods.
In this paper, we explain how generalized dynamical r-matrices can be obtained by (quasi-)Poisson reduction. New examples of Poisson structures and Poisson groupoid actions naturally appear in this setting. As an application, we use a generalized dynamical r-matrix induced by the gauge fixing procedure to give a new fi…
Proves effective slope gaps for lattice surfaces.
problem Proving effective slope gaps for lattice surfaces.
method Proves effective slope gap distribution for square torus and general lattice surfaces.
result Effective slope gap distribution result for lattice surfaces.
Continuum-wise hyperbolicity is exactly the pseudo-Anosov dynamics with spine singularities.
problem Classification of continuum-wise hyperbolic surface homeomorphisms
method Proving a complete structural classification
result Every cwF-hyperbolic homeomorphism is pseudo-Anosov with spine singularities Study of marginally trapped surfaces in a perturbed Schwarzschild spacetime.
problem Understanding marginally trapped surfaces in perturbed Schwarzschild spacetime.
method Developed a method to study spacelike surfaces in a double null coordinate system.
result For every incoming null hypersurface nearly spherically symmetric, there exists a unique embedded marginally trapped surface.
Study of Torelli groups of partitioned surfaces with bounds and asymptotic lengths.
problem Understanding Torelli groups of partitioned surfaces.
method Topological and dynamical analysis of Torelli groups of partitioned surfaces.
result Asymptotic translation lengths of Torelli groups of partitioned surfaces behave almost like the reciprocal of the Euler characteristic of the surface.
The paper defines biquandles for groups and constructs a homomorphism.
problem Describing dynamical systems in configuration systems.
method Defined biquandles on groups Gnk and constructed a homomorphism. result Homomorphism from surface singular braid monoid to Gn2. We study the mean curvature flow of hypersurfaces in Rn+1, with initial surfaces sufficiently close to the standard n-dimensional sphere. The closeness is in the Sobolev norm with the index greater than 2n+1 and therefore it does not impose restrictions of the mean curvature of the initial surface. W…
Study Schrödinger evolution on surfaces in 3D contact sub-Riemannian manifolds.
problem Analyzing the Schrödinger evolution on surfaces embedded in 3D contact sub-Riemannian manifolds.
method Relating self-adjointness of the Schrödinger operator to geometric invariants of the foliation.
result Classification of self-adjoint extensions yielding disjoint dynamics.
Generative diffusion models forecast implied vol surfaces without arbitrage issues.
problem Forecasting arbitrage-free implied volatility surfaces using historical data with path-dependent dynamics.
method Generative diffusion model (DDPM) with conditional training on market variables, including EWMAs and returns. Dynamic penalty scheme based on SNR to enforce arbitrage-free surfaces.
result Superior performance in volatility forecasting compared to existing methods.
Survey explores translation surfaces from geometric and topological perspectives.
problem Underutilized aspects of translation surfaces in dynamical viewpoint.
method Equivalent definitions, moduli spaces, period map, holonomy map.
result Highlights foundational, yet underrated, aspects of translation surfaces theory.
Twisted SL2C local systems on surfaces of finite type appear often in geometry and physics. Most of them arise geometrically as local systems of charts for pleated hyperbolic structures. Bonahon and Thurston's "shear-bend coordinates" parameterize these local systems of charts. On a surface …
Curves can bound only finitely many developable surfaces.
problem Bounding developable surfaces with nonvanishing mean curvature.
method Proof of finiteness for developable surfaces with prescribed boundary curves.
result Generic curves bound only finitely many developable surfaces.
A dynamical analog of the prime ideals for simple non-commutative rings is introduced. We prove a factorization theorem for the dynamical ideals. The result is used to classify the surface knots and links in the smooth 4-dimensional manifolds.
Classifies minimal surfaces of finite genus in 3D space.
problem Classifying minimal surfaces of finite genus in 3D space.
method Four-step classification, lamination techniques, dynamics theorem, and topology bounds.
result Classification of minimal surfaces of genus zero and infinite topology.
New connection found between complex polynomials and surface homeomorphisms.
problem Investigating the existence of generalized pseudo-Anosov maps from quadratic polynomials.
method Developed a new connection between dynamics of quadratic polynomials and surface homeomorphisms, focusing on Hubbard trees.
result Identified conditions for constructing generalized pseudo-Anosov maps from quadratic polynomials.