Paper proves minimizing movements match smooth droplet flow in 3D.
problem Consistency of minimizing movements with smooth mean curvature flow.
method Proved minimizing movements coincide with smooth droplet flow.
result Minimizing movements and smooth mean curvature flow are consistent in 3D.
Study on droplet flow on uneven surfaces, proving existence and properties.
problem Understanding droplet movement on irregular surfaces.
method Existence of smooth flow and 1/2-Hölder continuous minimizing movement solutions.
result Properties of minimizing movements including comparison principles and uniform boundedness.
Two oppositely charged droplets of (say) water in e.g. oil or air will tend to drift together under the influence of their charges. As they make contact, one might expect them to coalesce and form one large droplet, and this indeed happens when the charge difference is sufficiently small. However, Ristenpart et al disc…
New Heintze-Karcher inequality helps understand droplet shapes.
problem Characterize the shape of droplets inside smooth containers.
method Obtained a new form of the Heintze-Karcher inequality for mean convex hypersurfaces with boundary on curved substrates.
result New mathematical inequality aids in understanding droplet shapes.
Study phase transition in liquid crystal droplets using mathematical analysis.
problem Mathematical analysis of phase transition between isotropic and nematic states of liquid crystals.
method Rigorous mathematical analysis using the Ericksen model and Γ-convergence theory.
result Γ-limit provides geometric description and anchoring conditions for liquid crystal orientations.
Motivation: Single cell transcriptome sequencing (scRNA-Seq) has become a revolutionary tool to study cellular and molecular processes at single cell resolution. Among existing technologies, the recently developed droplet-based platform enables efficient parallel processing of thousands of single cells with direct coun…
Experimental fractal landscape dynamics observed in emulsions.
problem Understanding anomalous motions in soft glassy materials.
method Quantitative analysis of oil droplet trajectories in dense emulsions.
result Experimental fractal geometry matches computational model of soft glassy dynamics.
A theory of exceptional extreme events, characterized by their abnormal sizes compared with the rest of the distribution, is presented. Such outliers, called "dragon-kings", have been reported in the distribution of financial drawdowns, city-size distributions (e.g., Paris in France and London in the UK), in material f…
This paper solves the normalizability crisis in sequential inference by introducing bounded information geometry.
problem Structural failure in standard sequential inference architectures when dealing with extreme outliers.
method Non-parametric field actions and bounded information geometry to truncate infinite tails of spatial distributions.
result Empirical benchmarks across three domains show robust estimation without infinite-tailed distributional assumptions.
The study computes Bergman kernels and point process asymptotics on Kähler manifolds.
problem Computing asymptotics of Bergman kernels and point process distributions on Kähler manifolds.
method Equivariant and partial Bergman kernels, determinantal point processes, asymptotic analysis.
result The distribution of linear statistics converges to a centered normal variable with specific variances.
One of the most pressing questions in climate science is that of the effect of anthropogenic aerosol on the Earth's energy balance. Aerosols provide the `seeds' on which cloud droplets form, and changes in the amount of aerosol available to a cloud can change its brightness and other physical properties such as optical…
Study on stability of 3D sessile drops, identifying degenerate kernel.
problem Linear stability of three-dimensional sessile drops with a free contact line.
method Derived constrained second variation, formulated Jacobi problem, combined geometric and Fourier analysis.
result Kernel of the constrained Jacobi operator is exactly the space of horizontal translations under pressure-volume nondegeneracy.
Motivation: With the development of droplet based systems, massive single cell transcriptome data has become available, which enables analysis of cellular and molecular processes at single cell resolution and is instrumental to understanding many biological processes. While state-of-the-art clustering methods have been…
MOAI evaluates indoor airflow's impact on COVID-19 transmission.
problem Understanding indoor airflow's role in COVID-19 transmission.
method Developed a privacy-preserving app and model to evaluate risk exposure.
result Quantified factors contributing to higher or lower contamination in settings.
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.
The paper examines Ricci flows with closed and smooth tangent flows, proving uniqueness and characterizing ancient flows.
problem Characterizing and understanding Ricci flows with closed and smooth tangent flows.
method Analyzing ancient and finite-time singularity Ricci flows to prove uniqueness and characterizations.
result The tangent flow is unique and characterizes ancient and finite-time singularity flows.
Proves uniqueness of geometric flow in various Riemannian manifolds.
problem Proving uniqueness of geometric flow in general Riemannian manifolds.
method Two backward uniqueness theorems for extrinsic geometric flow.
result Backward uniqueness of extrinsic geometric flow in general ambient manifolds.
Study of twisted Calabi flow connecting J-flow and Calabi flow on Kähler manifolds.
problem Existence and convergence of twisted Calabi flow on compact Kähler manifolds.
method Analysis of a family of twisted Calabi flows connecting J-flow and Calabi flow, showing long-time existence and convergence to cscK metrics.
result Long-time existence and convergence of twisted Calabi flow to cscK metrics, implying openness of continuity method.
Investigate scalar curvature under geometric flows
problem Behavior of scalar curvature under geometric flows
method Three specific cases: Ricci flow, Kähler-Ricci flow, Laplacian flow
result Long-time existence of flows
Streets and Tian introduced pluriclosed flow and symplectic curvature flow in recent years. Here we construct a curvature flow to unify these two flows. We show the short time existence of our flow and exhibit an obstruction to long time existence.
We consider four extended Ricci flow systems---that is, Ricci flow coupled with other geometric flows---and prove dynamical stability of certain classes of stationary solutions of these flows. The systems include Ricci flow coupled with harmonic map flow (studied abstractly and in the context of Ricci flow on warped pr…
The article calculates the F-convergence rate for Ricci flows with closed and smooth tangent flows.
problem Analyzing the convergence rate of Ricci flows with specific tangent flows.
method Calculating the F-convergence rate for Ricci flows with closed and smooth tangent flows. result A Ricci flow with closed and smooth tangent flow is ∣logλ∣−θ close to its tangent flow in the F-sense. Paper introduces Tensor Gauge Flow Models for better data encoding.
problem Lack of expressive flow dynamics in existing Generative Flow Models.
method Incorporates higher-order Tensor Gauge Fields into the Flow Equation.
result Tensor Gauge Flow Models achieve improved generative performance.
Study K-R flow on Hirzebruch surfaces, showing tangent flows are K-R flows with orbifold singularities.
problem Finite time singularities in Kähler-Ricci flow on Hirzebruch surfaces.
method Analyze tangent flows based at singular points.
result Tangent flows are K-R flows with orbifold singularities.
Ancient curve shortening flows have entropy and curvature bounds equivalent.
problem Bounding entropy and total curvature for ancient curve shortening flows.
method Equivalence of entropy and total curvature conditions for ancient curve shortening flows.
result Entropy and total curvature bounds are equivalent for ancient curve shortening flows.
The study disproves rotating ancient flows in 4D space.
problem The existence of rotating ancient flows in R4. method Analysis of ancient noncollapsed flows in R4. result Nonexistence of rotating ancient flows among ancient noncollapsed flows in R4. Simplifies residual flows to make flow-based modeling more practical.
problem Extremely high computational cost of residual flows limits their applicability.
method Introduces Quasi-Autoregressive (QuAR) approach to residual flows.
result Significantly reduces compute time and memory requirements for flow-based modeling.
Existence of translating solutions shown for curve diffusion flow.
problem Existence of translating solutions for curve diffusion flow.
method Higher order curve shortening flow approach.
result Properly immersed translating solutions exist.
Modeling bone microarchitecture adaptation using geometric flows.
problem Bone microarchitecture adaptation modeling.
method Advection and mean curvature flow model with a sphere as a test case.
result Closed-form solution for sphere under advection and mean curvature flow.
Variational inference relies on flexible approximate posterior distributions. Normalizing flows provide a general recipe to construct flexible variational posteriors. We introduce Sylvester normalizing flows, which can be seen as a generalization of planar flows. Sylvester normalizing flows remove the well-known single…
The Hodge star mean curvature flow on a 3-dimension Riemannian or pseudo-Riemannian manifold, the geometric Airy flow on a Riemannian manifold, the Schrodingier flow on Hermitian manifolds, and the shape operator curve flow on submanifolds are natural non-linear dispersive curve flows in geometric analysis. A curve flo…
In many fields of science, high-dimensional integration is required. Numerical methods have been developed to evaluate these complex integrals. We introduce the code i-flow, a python package that performs high-dimensional numerical integration utilizing normalizing flows. Normalizing flows are machine-learned, bijectiv…
The study examines mass drop and multiplicity in mean curvature flow.
problem Analyzing mass drop and multiplicity in mean curvature flow.
method Defined Brakke flow with variational inequality, proved mass drop conditions.
result Mass drop and multiplicity one conjecture are equivalent for Brakke flows.
We explore the harmonic-Ricci flow---that is, Ricci flow coupled with harmonic map flow---both as it arises naturally in certain principal bundle constructions related to Ricci flow and as a geometric flow in its own right. We demonstrate that one natural geometric context for the flow is a special case of the locally …
Using the conformally invariant Cotton tensor, we define a geometric flow, the "Cotton flow", which is exclusive to three dimensions. This flow tends to evolve the initial metrics into conformally flat ones, and is somewhat orthogonal to the Yamabe flow, the latter being a flow within a conformal class. We define an en…
The paper studies mean curvature flow in a Ricci flow background with extended Ricci flow.
problem Analyzing mean curvature flow in a Ricci flow background.
method Computing variational properties and deriving evolution equations for mean curvature and second fundamental form.
result Established a Huisken's monotonicity-type formula for mean curvature solitons in an extended Ricci flow.
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.
New derivation of Type IIA flow metrics.
problem Flow of metrics in Type IIA theory.
method Adapted to Laplacian flow, uses projected Levi-Civita connection.
result New derivation of flow equations.
Survey of geometric flows from unified string theories.
problem None explicitly stated, but related to understanding geometric flows in string theories.
method Survey of geometric flows in various geometries (complex, almost-complex, symplectic) motivated by string theories.
result Intermediate flows between Ricci and Kähler-Ricci flows, often coupled to additional fields.
SurVAE Flows combine VAEs and flows using surjective transformations.
problem Combining the strengths of VAEs and flows to model complex densities.
method Modular framework of composable deterministic and stochastic transformations.
result Exact likelihood computation and lower bound on likelihood.
Study describes global existence and convergence of flows on surfaces and fibrations.
problem Global existence and convergence of flows on surfaces and fibrations.
method Complete description of Ricci-Yang-Mills flow and pluriclosed flow on Tk bundles over Riemann surfaces. result Equivalence of solutions to generalized Ricci flow and pluriclosed flow with symmetry.
Gauge Flow Models use a learnable Gauge Field in Generative Flow Models.
problem Improving generative model performance.
method Integrates a learnable Gauge Field into Flow ODEs.
result Gauge Flow Models outperform traditional Flow Models in Flow Matching experiments.
By the method of discrete Morse flows, we construct an energy reducing multiple-valued function flow. The flow we get is Holder continuous with respect to the L-2 norm. We also give another way of constructing flows in some special cases, where the flow we get behaves like ordinary heat flow.
Higher Gauge Flow Models integrate higher geometry and symmetries into Generative Flow Models.
problem Improving generative models' performance.
method Integrates L∞-algebra into Generative Flow Models, leveraging higher geometry and symmetries. result Substantial performance improvements on Gaussian Mixture Model datasets.
A new geometric flow K-flow on 3-manifolds shrinks or preserves homogeneous spheres.
problem Analyzing the behavior of Thurston's model geometries under the K-flow. method Defining and studying the K-flow on 3-dimensional Riemannian manifolds, using a DeTurck-type argument for short-time existence. result The K-flow shrinks or preserves homogeneous spheres, showing short-time existence. New perspective on G2-structures flow from DeTurck Laplacian.
problem Understanding G2-structures and their flows.
method Introducing a new flow (DeTurck Laplacian flow) for G2-structures.
result DeTurck Laplacian flow is a flow of G2-structures.
Study G2-flows reducing to complex geometry flows, focusing on G2-anomaly and G2-Laplacian coflow.
problem Investigate flows of G2-structures in relation to complex geometry. method Analyze G2-Laplacian coflow and G2-anomaly flow, compare their properties. result Compare G2-anomaly flow to G2-Laplacian coflow, investigate short-time existence and fixed points. Smooth flows with surgery approximate weak mean curvature flows with spherical and neck-pinch singularities.
problem Approximating weak mean curvature flows with singularities using smooth flows.
method Combining Choi-Haslhofer-Hershkovits and Choi-Haslhofer-Hershkovits-White work on canonical neighbourhoods and barriers to flows with surgery.
result Smooth flows with surgery can approximate weak mean curvature flows with spherical and neck-pinch singularities.