We extend rectified flow to infinite-dimensional Hilbert space.
problem Extending rectified flow to infinite-dimensional spaces.
method Established a rigorous functional formulation using the superposition principle for continuity equations.
result Demonstrated superior performance compared to existing models.
Mean curvature flow is not a gradient flow on two nondegenerate metric spaces.
problem Whether mean curvature flow is a gradient flow on nondegenerate metric spaces of simple closed plane curves.
method Examined two nondegenerate metric spaces: uniformness-preserving and curvature-weighted structures.
result Mean curvature flow is not a gradient flow on either metric space.
Study of σk flow in Minkowski space for noncompact hypersurfaces.
problem Understanding the behavior of σk curvature flow in Minkowski space. method Proving flow existence and convergence to self-expander after rescaling.
result The flow converges to a self-expander after rescaling.
Reconstruct flows from their orbit spaces using group actions.
problem Reconstructing flows from their orbit spaces.
method Using group actions and pseudo-Anosov flows.
result Reconstruct flows from their orbit spaces.
Study on curvature flow in Minkowski space for cocompact hypersurfaces.
problem Investigating curvature flow in Minkowski space for cocompact hypersurfaces.
method Investigation of the cocompact inverse \(σ_k\) curvature flow in Minkowski space.
result Longtime existence and convergence of the curvature flow established.
The paper develops flows for tori and spheres, addressing complex geometries.
problem Learning flows on tori and spheres for complex geometries.
method Recursive flows starting from circles, intervals, or spheres.
result Expressive and numerically stable flows on tori and spheres.
Ancient solutions found for mean curvature flows in hyperbolic spaces.
problem Mean curvature flows of isoparametric submanifolds in hyperbolic spaces.
method Study mean curvature flows of isoparametric submanifolds in hyperbolic spaces.
result Ancient solutions always exist for these flows.
Study of curvature flow in Minkowski space converging to a hyperboloid.
problem Understanding curvature flows in Minkowski space.
method Analyzing fully nonlinear curvature flows of noncompact spacelike hypersurfaces in Minkowski space.
result The flow converges to the future timelike hyperboloid after rescaling.
New measure defined for Brakke flow, linking classical and new definitions.
problem Defining and characterizing the Brakke flow.
method Introduced a space-time-Grassmann measure to characterize the flow.
result Equivalence between classical and new definitions of the Brakke flow.
Classifies self-similar curve shortening flows in hyperbolic 2-space.
problem Classifying self-similar curve shortening flows in hyperbolic 2-space.
method Analyzes and classifies solutions in hyperbolic 2-space.
result Completes the classification of self-similar curve shortening flows in constant curvature model spaces in 2-dimensions.
Ricci flow solves curvature-bound initial spaces to smooth manifolds.
problem Initial spaces with bounded curvature.
method Ricci flow with Alexandrov curvature bounds.
result Flow converges to a smooth manifold isometric to initial space.
Brakke flow support is parabolically rectifiable
problem Support of Brakke flow is parabolically rectifiable
method Developed approach to Brakke flow as space-time-Grassmann measure
result Standard convergence of Brakke flows is equivalent to space-time-Grassmann Radon measures
New geometric flow equations describe how space-time dimensions change.
problem Understanding how the number of space-time dimensions affects geometry.
method Developed D-flow equations to model the variation of space-time geometries.
result Solutions for D-flow equations on D-dimensional spheres and Freund-Rubin Compactification.
New flow for capillary surfaces converges to spherical caps.
problem Optimizing capillary surfaces in space forms.
method Constrained mean curvature flow.
result Flow converges to spherical caps globally.
Study on stability of mean curvature flow in hyperbolic space.
problem Stability of volume preserving mean curvature flow in hyperbolic space.
method Analysis of initial conditions and flow behavior in hyperbolic space.
result The flow converges exponentially to an umbilical sphere under certain conditions.
Study vector fields and flows on singular spaces like submanifolds.
problem Understanding vector fields and flows on singular spaces.
method Integrate derivations of the C∞-ring of global smooth functions into flows. result Derivations integrate to smooth flows on subcartesian spaces.
The paper extends entropy formulas to super Ricci flows on metric measure spaces.
problem Entropy formulas for super Ricci flows on metric measure spaces.
method Extending Perelman's W-entropy and Shannon entropy power to super Ricci flows. result Equivalence between volume non-local collapsing property and lower boundedness of W-entropy on RCD(0,N) spaces. CPFM integrates dimensionality reduction and reconstruction with flow networks.
problem Learning coupled continuous flows for data and embeddings.
method Coupled flow matching framework with Gromov-Wasserstein objective and dual-conditional flow network.
result CPFM preserves and recovers residual information in latent space.
We study the heat equation on time-dependent metric measure spaces (as well as the dual and the adjoint heat equation) and prove existence, uniqueness and regularity. Of particular interest are properties which characterize the underlying space as a super Ricci flow as previously introduced by the second author. Our ma…
A new method learns latent space normalizing flow for approximate inference in generator models.
problem Approximate inference in generator models with complex posterior distributions.
method Jointly learns latent space normalizing flow and generator model using MCMC-based maximum likelihood.
result The short-run Langevin flow approximates the posterior and aligns with the normalizing flow prior.
Inverse mean curvature flow converges to a disk in hyperbolic space.
problem Understanding flow behavior in hyperbolic geometry.
method Inverse mean curvature flow with free boundary on geodesic spheres.
result Flow converges to a totally geodesic disk.
Gradient flow in parameters equals linear interpolation in outputs.
problem Understanding and optimizing training algorithms in deep learning.
method Proving equivalence between gradient flow in parameter space and linear interpolation in output space, and deriving formulas for global minima.
result Gradient flow in parameters can be transformed into linear interpolation in outputs, leading to global minima.
This research tackles ordering latent variables in normalizing flows.
problem Learning low-dimensional, meaningful representations in latent space.
method Nested dropout normalizing flows with ordered latent variables.
result A trade-off exists between flow likelihood and ordering quality.
We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the L2-space produces the same evolution as the gradient flow of the relative entropy in the L2-Wasserstein space. This means that the heat flow is well defined by either one of the two gradient fl…
Higher KdV flows on spaces of closed equicentroaffine plane curves are studied and it is shown that the flows are described as certain multi-Hamiltonian systems on the spaces. Multi-Hamiltonian systems describing higher mKdV flows are also given on spaces of closed Euclidean plane curves via the geometric Miura transfo…
The paper proves new Minkowski inequalities for flows in warped spaces.
problem Proving new Minkowski inequalities for flows in warped spaces.
method Locally constrained inverse curvature flows in Riemannian warped spaces.
result New Minkowski inequalities are derived for flows in warped spaces.
Study on Ricci flows of awesome homogeneous spaces, proving finite extinction time.
problem Understanding the long-time behavior of Ricci flows on homogeneous spaces.
method Analyzing Ricci flows on non-compact manifolds, focusing on finite extinction time.
result Ricci flows on non-contractible spaces have finite extinction time, confirming conjecture.
Avoids noncompact hypersurfaces from touching in evolving flows.
problem Preventing noncompact hypersurfaces from touching in evolving flows.
method Analyzes mean curvature flow and weak set flows in Euclidean and Riemannian spaces.
result Proves that noncompact hypersurfaces remain disjoint in evolving flows.
Study of geometric analysis on asymmetric metric spaces, including heat flow and Sobolev spaces.
problem Analysis of geometric properties on asymmetric metric measure spaces.
method Introduction of upper gradients, q-Laplacian, and q-heat flow in asymmetric settings. result Extension of concepts from symmetric to asymmetric metric measure spaces.
The paper studies flow lines on Higgs bundle moduli spaces, classifying them via secant varieties.
problem Classifying flow lines on moduli spaces of Higgs bundles.
method Gradient flow lines for L2 norm of Higgs field, Morse-theoretic compactification, secant varieties. result Flow lines have an algebro-geometric classification via secant varieties.
We propose an investigation of stability of vacua in string theory by studying their stability with respect to a (suitable) world-sheet renormalization group (RG) flow. We prove geometric stability of (Euclidean) anti-de Sitter (AdS) space (i.e., Hn) with respect to the simplest RG flow in closed string the…
Study on convergence rate of weighted Yamabe flow.
problem Weighted Yamabe problem on smooth metric measure spaces.
method Weighted Yamabe flow and its convergence rate analysis.
result Study and analysis of convergence rate of the weighted Yamabe flow.
Synthetic Ricci flows defined for metric measure spaces.
problem Characterizing Ricci flows for non-smooth spaces.
method Heat flow, optimal transport, volume asymptotics.
result Equivalent characterizations of weighted Ricci flow.
Study connects landslide flow to integrable systems for harmonic maps.
problem Understanding the holonomy of complex landslide flow.
method Integrable systems approach to harmonic maps into symmetric spaces.
result Holonomy of complex landslide flow derived from harmonic map holonomy.
Lipschitz regularity proved for harmonic map heat flows into CAT(0) spaces.
problem Proving Lipschitz regularity for harmonic map heat flows into CAT(0) spaces.
method Elliptic approximation method
result Every weak solution of the harmonic map heat flow into CAT(0) spaces is Lipschitz continuous in both space and time.
Survey on Ricci flow on spaces with conical singularities.
problem Analyzing Ricci flow on spaces with isolated conical singularities.
method Ricci de Turck flow preserving conical singularities, stability of Ricci flat metrics, preservation of positive scalar curvature under certain conditions.
result Ricci flat metrics with isolated conical singularities are stable and positive scalar curvature is preserved under the flow.
Study geometric flows with varying parameters and prove continuous dependence.
problem Continuous dependence of flows on parameters in geometric settings.
method Derived suitable topologies for vector fields and flows, proved new continuous dependence.
result Proved continuous dependence of flows on parameters in a general topological space.
Fractal Flow enhances normalizing flows with interpretable latent space and hierarchical modeling.
problem High-dimensional density estimation and generative modeling challenges.
method Integrates topic modeling (LDA) and fractal strategy into normalizing flows.
result Achieves latent clustering, controllable generation, and superior estimation accuracy.
Proves Thom's conjecture for parabolic flows on Hilbert spaces.
problem Gradient flows on infinite-dimensional spaces and geometric flows with symmetry.
method Analytic functions, Hilbert spaces, Yang-Mills Flow, Ricci flow, critical points, Lojasiewicz inequality.
result Gradient conjecture holds for parabolic flows on Hilbert spaces, including flows with gauge symmetry.
Riemannian flows model curved spaces better.
problem Normalizing flows misspecified on curved spaces.
method Riemannian continuous normalizing flows using ODE solutions.
result Improves modeling on spheres, torii, and hyperbolic spaces.
The study classifies flows of finite curvature in 3D space.
problem Classifying flows of finite curvature in 3D space.
method Partial classification of eternal mean convex flows.
result Topologically nonplanar flows must exit a catenoid.
Topological Flow Matching: A Generative Modeling Framework for Structured Spaces
problem Handling structured spaces in generative modeling
method Introducing topological flow matching
result Captures the structure of the underlying domain while preserving desirable properties
Given a solution of the (backwards) Ricci flow one can construct a so called canonical soliton metric on space-time, introduced by E. Cabezas-Rivas and P. Topping. We observe that for a mean curvature flow within a (backwards) Ricci flow background, the space-time track of the mean curvature flow yields a canonical sol…
Develops new synthetic Ricci flow concepts for metric measure spaces.
problem No specific problem stated; focuses on new mathematical concepts.
method Formulated in terms of dynamic convexity and local concavity of entropy, and global/short-time asymptotic transport cost estimates.
result Shows these properties characterise smooth (weighted) Ricci flows.
Integrable flows on null curves in anti-de Sitter 3-space studied.
problem Analyzing integrable flows on null curves in anti-de Sitter 3-space.
method Formulated integrable flows related to the KdV hierarchy on null curves exploiting the geometry of anti-de Sitter 3-space.
result Explicitly found closed stationary solutions in terms of periodic solutions of a Lamé equation.
Study of flow in hyperbolic space with capillary boundary.
problem Optimizing hypersurfaces with capillary boundary in hyperbolic space.
method Mean curvature flow with capillary boundary, preserving volume and energy.
result Flow converges to a truncated umbilical hypersurface.
Recently, we have studied evolution of a family of Finsler metrics along Finsler Ricci flow and proved its convergence in short time. Here, existence of solutions to the so called Hamilton Ricci flow on Finsler spaces is studied and a short time solution is found. To this end the Finslerian Ricci-DeTurck flow on Finsle…
The study proves compactness and structure of Ricci flow limits.
problem Understanding the structure of Ricci flow limits.
method Weak compactness theorem and structure theory development.
result Ricci flow limit spaces have a regular part with smooth convergence and a singular set of high codimension.