Paper studies dual Anomaly flow under T-duality.
problem Understanding dual Anomaly flow under T-duality.
method Introduced a family of monotone functionals to estimate the dilaton function.
result Detailed examples and reductions of the dual Anomaly flow.
Pinches flows in hyperbolic and de Sitter spaces with convex curvature.
problem Bounding dual flows in curved spaces.
method Proving pinching estimates with convex curvature function.
result Pinching estimates for dual flows in de Sitter space.
Study anisotropic inverse Gauss curvature flows and solve dual Orlicz Minkowski problems.
problem Solving dual Orlicz Minkowski problems for anisotropic flows.
method Anisotropic inverse Gauss curvature flows and stationary solutions.
result New existence results for dual Orlicz Minkowski problems for smooth measures.
Study of integral flows on Riemannian manifolds with focus on blow-up profiles and concentration-compactness.
problem Analyzing nonlinear integral flows on Riemannian manifolds with specific focus on blow-up profiles and concentration-compactness.
method Investigation of a family of nonlinear integral flows involving Riesz potentials, focusing on the Hardy-Littlewood-Sobolev (HLS) subcritical and critical regimes.
result Established convergence on unit spheres and certain locally conformally flat manifolds for the dual Yamabe flow.
New method solves a generalized Minkowski problem using a curvature flow.
problem Generalized Minkowski problem for smooth measures.
method Flow involving Gauss curvature and support function.
result Existence of solutions for the dual Orlicz-Minkowski problem.
New flows introduced for symplectic geometry.
problem No specific problem stated; focuses on new flows.
method Introduces several geometric flows on symplectic manifolds.
result Examples include the Hitchin gradient flow and dual Ricci flow.
Several results on existence and convergence of the Yang-Mills flow in dimension four are given. We show that a singularity modeled on an instanton cannot form within finite time. Given low initial self-dual energy, we then study convergence of the flow at infinite time. If an Uhlenbeck limit is anti-self-dual and has …
Study anisotropic flows without global terms and solve dual Orlicz Christoffel-Minkowski problems.
problem Anisotropic flows without global forcing terms and dual Orlicz Christoffel-Minkowski problems.
method Existence results for dual Orlicz Christoffel-Minkowski type problems via stationary solutions of anisotropic flows.
result Existence results for a class of dual Orlicz Christoffel-Minkowski type problems.
This paper solves the dual Minkowski problem for q-torsional rigidity.
problem The dual Minkowski problem for q-torsional rigidity.
method Introduced the p-th dual q-torsional measure and solved the p-th dual Minkowski problem for q-torsional rigidity using a Gauss curvature flow.
result Existence of smooth even and non-even solutions to the p-th dual Minkowski problem for q-torsional rigidity.
New method trains normalizing flows using entropy-regularized transport.
problem Training continuous normalizing flows efficiently.
method Formulates flows as gradients of scalar potentials, training only these potentials.
result Trains normalizing flows without explicit flow computation during training.
Paper studies inverse curvature flows and solves related geometric problems.
problem Inverse curvature flows and related geometric problems.
method Analyzes a class of expanding flows with specific speeds and proves existence and convergence.
result Proves the existence and convergence of flows under certain conditions, leading to new solutions to geometric problems.
By means of dual convex bodies, we obtain regularity of solutions to the expanding Gauss curvature flows with homogeneity degrees −p, 0<p<1. At the end, we remark that our method can also be used to obtain regularity of solutions to the shrinking Gauss curvature flows with homogeneity degrees less than one.
DFM simplifies CNF training without interpolants.
problem Efficiently training CNFs with computationally expensive ODE solving.
method DFM optimizes dual vector fields for bijective transformations.
result DFM outperforms CNF trained with FM or ML objectives.
Researchers develop Orlov-Schulman symmetries for self-dual conformal structures.
problem Developing symmetries for self-dual conformal structures.
method Explicit proof of compatibility with Lax-Sato flows, dressing scheme based on Riemann-Hilbert problem.
result Construction and proof of compatibility of Orlov-Schulman symmetries.
The study connects geodesic flows on Riemann surfaces to random walks on their dual graphs.
problem Understanding ergodicity of geodesic flows on infinite Riemann surfaces.
method Analyzing random walks on the dual graph of pants decompositions.
result Equivalence between ergodicity of geodesic flows and recurrence of random walks.
Study heat flows on time-dependent metric measure spaces, proving properties related to super-Ricci flows.
problem Characterize heat flows and their properties on time-dependent metric measure spaces.
method Prove existence, uniqueness, and regularity of heat equations and their duals on time-dependent metric measure spaces.
result Equivalence of dynamic convexity of Boltzmann entropy, monotonicity of Wasserstein distances, gradient estimates, and Bochner inequality.
Smooth even solutions found for a generalized convex geometry problem.
problem Dual Orlicz-Minkowski problem in convex geometry.
method Geometric flow involving Gauss curvature and normal vectors.
result Existence of smooth even solutions for smooth even measures.
Study on the flow of Hermitian-Yang-Mills on reflexive sheaves, proving limiting sheaf is isomorphic to dual of Harder-Narasimhan-Seshadri filtration.
problem Understanding the asymptotic behavior of Hermitian-Yang-Mills flow on reflexive sheaves.
method Analysis of the flow and proof of limiting properties.
result The limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the Harder-Narasimhan-Seshadri filtration.
This paper proves an C∞ closing lemma for Hamiltonian flows on symplectic 4-manifolds.
problem Proving the C∞ closing lemma for Hamiltonian flows on symplectic 4-manifolds. method Combining results from geodesic flows on Finsler surfaces with the dual lens map technique, extending to Hamiltonian flows with certain restrictions.
result Established the C∞ closing lemma for a large family of Hamiltonian flows on 4-dimensional symplectic manifolds. The paper explores the geometric properties of fluid flows and their symmetries.
problem Understanding the geometric properties of fluid flows and their symmetries.
method Analyzing the Euler equation and its relation to geodesic flows on groupoids of multiphase diffeomorphisms.
result Generalized flows, multiphase fluids, and vortex sheets are all geodesics on certain groupoids of multiphase diffeomorphisms.
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.
The paper solves curvature flow problems to prove sphere convergence and dual Minkowski solutions.
problem Proving sphere convergence and dual Minkowski solutions for curvature flow problems.
method A contracting flow of closed, convex hypersurfaces with speed frαK where K is the Gauss curvature, r is the distance from the hypersurface to the origin, and f is a positive and smooth function. result The flow exists for all time and converges smoothly to a soliton, which is a sphere centred at the origin if f≡1. We construct a discrete form of Hamilton's Ricci flow (RF) equations for a d-dimensional piecewise flat simplicial geometry, S. These new algebraic equations are derived using the discrete formulation of Einstein's theory of general relativity known as Regge calculus. A Regge-Ricci flow (RRF) equation is naturally asso…
The paper analyzes how GANs converge using dual metric flows.
problem Understanding the convergence dynamics of GANs.
method Investigates the convergence of GANs using dual metric flows, formal definitions, and proving convergence.
result GAN learning dynamics converge to a limit when learning rate is small.
Combinatorial Ricci flow finds hyperbolic metrics on 3-manifolds.
problem Finding complete hyperbolic metrics on cusped 3-manifolds.
method Analogue of surface and compact 3-manifold flows, minimizing co-volume, extending through singularities.
result Existence of complete hyperbolic metric is equivalent to flow convergence.
Defines T-duality and generalised Ricci flow relations using Courant algebroid relations.
problem Establishing compatibility between T-duality and generalised Ricci flow.
method Introducing Courant algebroid relations, invariant divergence operators, and generalised isometries.
result T-duality is compatible with generalised Ricci flow, and T-dual solutions are also solutions of generalised Ricci flow.
In [29], Plebanski reformulated the anti-self-dual Einstein equations with non-zero scalar curvature as a first order PDE for a connection in an SO(3)-bundle over the four-manifold. The aim of this article is to place this differential equation in a new framework, in which it is both elliptic and a stationary point of …
Paper solves a new Minkowski problem for a specific type of rigidity.
problem Solving a new Minkowski problem for a specific type of rigidity.
method Developed a nonlinear partial differential equation and used a curvature flow method.
result Existence of smooth non-even solutions to the p-th dual Minkowski problem for p < n-2.
Veering branched surfaces help construct geodesic flows on curved surfaces.
problem Constructing geodesic flows on negatively curved surfaces.
method Introduce veering branched surfaces and surgeries, then use them to construct veering triangulations that correspond to geodesic flows.
result Explicit constructions of veering branched surfaces corresponding to geodesic flows on negatively curved surfaces.
The paper finds circle packings with specific curvatures in hyperbolic geometry.
problem Finding circle packings with prescribed total geodesic curvatures and discrete Gaussian curvatures.
method Established existence and rigidity via variational principle, introduced combinatorial p-th Calabi flows.
result Introduced combinatorial p-th Calabi flows to find circle packings with prescribed curvatures.
For triangulated surfaces, we introduce the combinatorial Calabi flow which is an analogue of smooth Calabi flow. We prove that the solution of combinatorial Calabi flow exists for all time. Moreover, the solution converges if and only if Thurston's circle packing exists. As a consequence, combinatorial Calabi flow pro…
Develops risk measures on Lipschitz spaces for financial positions.
problem Lack of standard cash-additive methods in Lipschitz spaces.
method Proposes Lipschitz-free space, uses additivity along benchmark-deviation instruments.
result Derives dual representations for convex and coherent risk measures.
A notion of equivariant spectral flows for families of self-dual elliptic operators on Riemannian manifolds is purposed. As a consequence, a local version of a Lefschetz fix point theorem is proved for Toeplitz operators on odd-dimensional spin manifolds.
Paper presents a deep learning approach to AC Optimal Power Flow.
problem Nonlinear and nonconvex OPF problem in power systems.
method Combines deep learning with Lagrangian dual methods.
result Deep learning model achieves highly accurate predictions.
Symplectic homology matches dual capacities for convex domains.
problem Understanding symplectic capacities and Reeb flows on convex domains.
method Isomorphic filtered symplectic homology to dual singular homology.
result Gutt-Hutchings capacities match spectral invariants for convex domains.
New characterization of geodesic currents via curve functionals.
problem Characterize geodesic currents using curve functionals.
method Purely axiomatic and combinatorial approach.
result Characterization of curve functionals dual to geodesic currents.
New method improves generative modeling on convex domains using regularized mirror maps and Student-t priors.
problem Challenges in generative modeling on convex domains with heavy-tailed targets.
method Mirror Flow Matching with regularized mirror maps and Student-t priors.
result Empirically outperforms baselines and achieves competitive sample quality.
Introduces new info-geometric structure for dynamics on graphs and hypergraphs.
problem Modeling dynamics on discrete structures like graphs and hypergraphs.
method Introduces two dually flat structures: one on vertex space and another on edge space.
result Extends gradient flows to include nonequilibrium dynamics.
Unified geometric framework for quantum states using dual number algebras.
problem Representing quantum states in a geometrically unified way.
method Smooth embeddings into higher-order dual number algebras and algebraic flows.
result Established nilpotent dual algebras as a geometric landscape for quantum kinematics.
New formulas derived for scalar curvature in generalized Ricci flow.
problem Scalar curvature in generalized Ricci flow.
method Derivation of weighted scalar curvature monotonicity formulas and Perelman-type energy/entropy formulas.
result New convex Nash entropies and pseudolocality principles.
We establish a C1,α compactness theorem for the metrics with bounded self - dual Weyl tensor and Scalar curvature. The key step is to estimate the C1,α harmonic radius, where we use the blow up analysis as in \cite{Anderson90}. The result is motivated by, and may be applied to the Calabi flow on complex surfa…
Mirror flows converge to a limiting flow with a convex potential.
problem Incremental learning in mirror flows
method Rescaled trajectories converge to a limiting mirror flow
result Primal variable minimizes the loss over a time-dependent hypothesis set
Deformed σ-models linked to Ricci flow and Toda theories.
problem Understanding the relationship between deformed σ-models and geometric flows.
method Exploring trigonometric deformations of CP^n-1 models and their duals, linking to Ricci flow and Toda field theories.
result Trigonometric deformations of CP^n-1 models solve the Ricci flow equation and relate to Toda field theories.
Paper studies convergence of Yang-Mills-Higgs flow on Kähler manifolds.
problem Analyzing convergence of Yang-Mills-Higgs flow for twisted Higgs pairs.
method Proves convergence to a reflexive twisted Higgs sheaf outside a closed subset.
result Limiting twisted Higgs sheaf is isomorphic to the double dual of graded twisted Higgs sheaves.
Integrable flows on the Grassmannians Gr(N-1,N+1) are defined by the requirement of closedness of the differential N-1 forms ΩN−1 of rank N-1 naturally associated with Gr(N-1,N+1). Gauge-invariant parts of these flows, given by the systems of the N-1 quasi-linear differential equations, describe coisotropic deform…
Let X be a compact Hermitian surface, and g be any fixed Gauduchon metric on X. Let E be an Hermitian holomorphic vector bundle over X. On the bundle E, Donaldson's heat flow is gauge equivalent to a flow of holomorphic structures. We prove that this flow converges, in the sense of Uhlenbeck, to the double …
We modify the Laplacian coflow of co-closed G2-structures - dtdψ=Δψ where ψ is the closed dual 4-form of a G2-structure φ. The modified flow is now parabolic in the direction of closed forms upto diffeomorphisms. We then prove short time existence and uniqueness of solutions to the modified f…
We derive and study supergravity BPS flow equations for M5 or D3 branes wrapping a Riemann surface. They take the form of novel geometric flows intrinsically defined on the surface. Their dual field-theoretic interpretation suggests the existence of solutions interpolating between an arbitrary metric in the UV and the …