Expanding on subgroup dimension, this paper covers new dimensions and compactness.
problem Understanding subgroup dimensions and their properties.
method Generalization of dimension datum, study of disconnected subgroups, and isospectral sets.
result New insights into subgroup dimensions and compactness of isospectral sets.
The paper explores rational functions with 3 branching points on the Riemann sphere.
problem Existence of rational functions with specific branching points.
method Utilizes complex analysis to establish properties of rational functions.
result Identifies new types of exceptional branching data.
Given a smooth bounded domain Ø⊆R2, we consider the equation $\D v = 2 v_x \wedge v_y$ in Ø, where v:Ø→R3. We prescribe Dirichlet boundary datum, and consider the case in which this datum converges to zero. An asymptotic study of the corresponding Euler functional is performed, analyzing multiple…
The paper classifies isoparametric hypersurfaces in Randers space forms.
problem Classifying isoparametric hypersurfaces in Randers space forms.
method Anisotropic submanifolds and isoparametric hypersurfaces in Randers space forms (N,F) with (h,W) were studied.
result The isoparametric hypersurfaces in a Randers space form (N,F) are the same as in Riemannian space, despite different isoparametric functions.
Study of tori of revolution under Willmore flow converges to Clifford Torus.
problem Long-time behavior and convergence of Willmore flow for tori of revolution.
method Gradient flow of Willmore energy for tori of revolution, analyzing energy threshold and convergence to Clifford Torus.
result Convergence of Willmore flow to Clifford Torus for initial energy below 8π.
Navigation in Lorentz Finsler geometry induces isoparametric hypersurfaces.
problem Defining and analyzing isoparametric hypersurfaces in Lorentz Finsler geometry.
method Using a navigation process with a Finsler metric and a tangent vector field, isoparametric functions and hypersurfaces are defined and analyzed.
result Local correspondences between isoparametric functions and hypersurfaces are established.
Ancient solutions found for a specific flow on symplectic half-flat structures.
problem Existence of solutions for a particular geometric flow.
method Analyzes Type IIA flow on symplectic half-flat SU(3)-structures.
result Existence of ancient, immortal, and eternal solutions under suitable conditions.
Proves existence of multi-phase flows from arbitrary initial data.
problem Non-uniqueness issue in Brakke flows.
method Global existence proof for multi-phase mean curvature flow.
result Validates explicit identity for evolving grain volumes.
We prove that the Calabi-Yau equation on the Kodaira-Thurston manifold has a unique solution for every S1-invariant initial datum.
A Bayesian treatment of latent directed graph structure for non-iid data is provided where each child datum is sampled with a directed conditional dependence on a single unknown parent datum. The latent graph structure is assumed to lie in the family of directed out-tree graphs which leads to efficient Bayesian inferen…
To a branched cover between closed, connected and orientable surfaces one associates a "branch datum", which consists of the two surfaces, the total degree d, and the partitions of d given by the collections of local degrees over the branching points. This datum must satisfy the Riemann-Hurwitz formula. A "candidate su…
Global existence and convergence of heat flow for p-harmonic maps.
problem Global existence and convergence of heat flow for p-harmonic maps between manifolds.
method Analysis of heat flow equations for p-harmonic maps.
result Global existence and convergence of heat flow for p-harmonic maps under certain conditions.
Proposes a new prior for deep generative models to capture latent properties.
problem Complex non-linear relationships between data and latent properties.
method Factorial mixture prior with Gaussian mixture models for quantization.
result Empirically evaluated method for learning discrete properties in unsupervised or semi-supervised settings.
Rational maps structure theorem with geometric decomposition and realizability proof.
problem Realizability of rational maps branch data.
method Geometric decomposition of pullback metric into footballs and application to realizability.
result Realizability of branch data for rational maps when k>l+1. The paper addresses errors in online selective conformal prediction and proposes new strategies to ensure valid inference.
problem Online selective conformal prediction's exchangeability issues and false coverage rate control problems.
method Evaluation and correction of existing calibration selection strategies, proposing new ones that preserve exchangeability.
result Novel calibration selection strategies ensure both selection-conditional coverage and FCR control.
Three types of branched covers are identified.
problem Identifying the number of distinct branched covers for a given branch datum.
method Analyzing the compatibility conditions and using Grothendieck's dessins d'enfant.
result There are exactly three distinct types of branched covers.
Proves existence and uniqueness of curvature motion for regular networks.
problem Existence and uniqueness of motion by curvature for regular networks.
method Proves existence and uniqueness using $W^{2- rac{2}{p}}_p$ initial data and investigates regularization effects.
result Proves existence and uniqueness of motion by curvature for regular networks.
The study examines surface branch data on spheres with specific properties and computes the number of realizations.
problem Examining surface branch data on spheres with specific properties and computing the number of realizations.
method Analyzing surface branch data on spheres with three branching points and two partitions of degree d, computing the number of realizations based on arithmetic properties of the entries of the third partition.
result In the only case where n is 0, the entries have a common divisor, supporting conjectures by Edmonds-Kulkarny-Stong and Zieve.
Study shows diffused interface flows to single diffused balls over time.
problem Volume-preserving mean curvature flow in Euclidean space.
method Diffused interface version, exponential convergence proof.
result Exponential convergence to single diffused balls.
We introduce a first order flow of G2-structures and construct its explicit solution in case of a cone over S3×S3. Also we prove for this situation that starting from certain initial datum the flow deforms corresponding to G2-structure metric to a conic metric up to homotheties.
We prove that a strictly stable minimal Ch2 intrinsic graph G is locally area-minimizing, i.e. given any Ch1 graph S with the same boundary, Area(G)<Area(S) unless G=S. As a consequence we show the existence and the uniqueness of C∞ minimal graphs with prescribed small boundary datum…
The paper studies non-Kähler LVMB manifolds and their metrics.
problem Existence and properties of special metrics on LVMB manifolds.
method Analyzes LVMB manifolds with holomorphic bundle structures over toric bases, using combinatorial data.
result Constructs a new example of a balanced metric on an LVMB manifold.
Paper solves the Hurwitz existence problem using fiber products.
problem Determining when a combinatorial map datum corresponds to a holomorphic map.
method Using fiber products of holomorphic maps between Riemann surfaces.
result Proves non-realizability of many branch data and constructs new data.
Unique solution found for quaternionic Monge-Ampère equation on specific HKT manifolds.
problem Solving the quaternionic Monge-Ampère equation on HKT manifolds with an HKT foliation.
method Study of quaternionic Monge-Ampère equation on HKT manifolds with specific foliation properties.
result Unique solution for the equation for every basic datum.
Noise stabilizes solutions to transport equations, preventing blow-up.
problem Proving global existence and uniqueness of solutions to stochastic transport equations.
method Characteristics-based techniques exploiting the geometric structure of transport equations.
result Noise prevents blow-up in deterministic solutions and ensures global existence and uniqueness of solutions.
Study traveling waves in hyperbolic space for Fisher-KPP equations.
problem Understanding wave behavior in hyperbolic space for Fisher-KPP equations.
method Analyzes the Cauchy problem in hyperbolic space for heat equation with Fisher-KPP forcing term.
result Proves new results on the dichotomy of solution propagation or vanishing based on diffusion and reaction strength.
We define the notion of a loop Hodge structure -- an infinite dimensional generalization of a Hodge structure -- and prove that a suitable variation of this object over a complex manifold is equivalent to the datum of a harmonic bundle. Hence one can study harmonic bundles using classical tools of Hodge theory, especia…
Global existence of Willmore flow with boundary via Li-Yau inequality.
problem Global existence of Willmore flow with boundary conditions.
method Extending Li-Yau inequality to surfaces with boundary and using geometric measure theory.
result Global existence of Willmore flow with Dirichlet boundary data below a specific energy threshold.
Agrachev's problem on circle turns is solved for various topologies.
problem How many times must a circle be turned to allow deformation with non-degenerate Frenet frame?
method Introduced decorated turn data to retain a nontrivial turn-counting problem. Analyzed different topologies and dimensions.
result For Cn curve topology, k(2)=1, k(3)=2, and k(n)=1 for n≥4. Spherical Fenchel obstruction in all dimensions n≥4. The paper ensures positivity of solutions to stochastic equations with positive initial data.
problem Ensuring positivity of solutions to stochastic equations with positive initial data.
method Providing sufficient conditions on coefficients for positivity of mild solutions.
result Sufficient conditions for positivity of solutions to stochastic equations.
We review some constructions and properties of complex manifolds admitting pluriclosed and balanced metrics. We prove that for a 6-dimensional solvmanifold endowed with an invariant complex structure J having holomorphically trivial canonical bundle the pluriclosed flow has a long time solution for every invariant init…
Study proves existence of a specific type of flow in geometry.
problem Existence of canonical multi-phase free boundary Brakke flows.
method Global-in-time existence established using Brakke flow and uniform density ratio assumption.
result Existence of the flow with no positive mass on the free boundary for some short time.
Given a singular Riemannian foliation on a compact Riemannian manifold, we study the mean curvature flow equation with a regular leaf as initial datum. We prove that if the leaves are compact and the mean curvature vector field is basic, then any finite time singularity is a singular leaf, and the singularity is of typ…
Solves Plateau's Problem in Heisenberg group for graphs.
problem Plateau's Problem in the Heisenberg group for intrinsic graphs.
method Geometric construction and calibration argument.
result Solves Plateau's Problem under smallness conditions.
Proposes a method to quantify uncertainty in deterministic image classifiers.
problem Uncertainty in deterministic image classifiers.
method Introduces Wellington Posterior for inductive transfer from scenes.
result Validates Wellington Posterior using various methods.
In this work we study the decomposability property of branched coverings of degree d odd, over the projective plane, where the covering surface has Euler characteristic ≤0. The latter condition is equivalent to say that the defect of the covering is greater than d. We show that, given a datum $\mathscr{D}=\{D…
For many tasks and data types, there are natural transformations to which the data should be invariant or insensitive. For instance, in visual recognition, natural images should be insensitive to rotation and translation. This requirement and its implications have been important in many machine learning applications, a…
Study on existence and properties of continuous solutions to complex Hessian equations.
problem Existence and properties of continuous solutions to complex Hessian equations.
method Established new capacity estimates and weak stability estimates for the m-Hessian measure. result Existence of continuous solutions to the complex Hessian equation under certain conditions.
Infinite fractal tree solves shortest connection problem.
problem Finding the shortest connection for a fractal set.
method Constructing an infinite planar self-similar binary tree.
result The tree is the unique solution to the Steiner problem.
Study continuity and Hölder estimates for solutions on Stein spaces.
problem Continuity and Hölder estimates for solutions to degenerate complex Monge-Ampère equations.
method Prove continuity up to the boundary and local Hölder estimates on the regular locus.
result Local Hölder estimates on the regular locus for solutions to degenerate complex Monge-Ampère equations.
A new proof of an extension theorem with bounded generators.
problem Extension theorems in complex analysis.
method Skoda-type L2 division theorem with bounded generators. result The new division theorem allows α to be 1 in the norm of the datum. In this paper we investigate the following existence problem for rational functions: for a given collection Π of partitions of a number n to define whether there exists a rational function f of degree n for which Π is the branch datum. An important particular case when the answer to this problem is known is t…
New proof shows diffusion models implicitly estimate intrinsic dimensionality.
problem Estimating intrinsic dimensionality of data from diffusion models.
method Formal proof of FLIPD under realistic assumptions.
result FLIPD's correctness proven under realistic conditions.
Study on curvature equation in Heisenberg group with convex boundary.
problem Existence of solutions to prescribed mean curvature equation in sub-Finsler Heisenberg group.
method Finsler approximation scheme to prove existence of Lipschitz solutions.
result Existence of a Lipschitz solution for the Dirichlet problem.
We study the geometry of type II supergravity compactifications in terms of an oriented vector bundle E, endowed with a bundle metric of split signature and further datum. The geometric structure is associated with a so-called generalised G-structure and characterised by an E-spinor ρ, which we can regard as a …
The study characterizes and rules out collapsing in convex ancient mean curvature flow.
problem Characterizing and ruling out collapsing in convex ancient mean curvature flow.
method Characterization and counterexamples.
result Collapsing occurs if and only if the flow is asymptotic to at least one Grim hyperplane.
Study on MCF of foliations, focusing on singular cases.
problem Investigate mean curvature flow of foliations with singularities.
method Generalizes previous results on MCF of foliations, focusing on singular cases.
result Finite time singularities are singular leaves of type I under bounded curvature conditions.
Abstract framework for two meromorphic forms on punctured surfaces.
problem Developing a framework for two meromorphic forms on punctured Riemann surfaces.
method Abstract framework with Teichmüller regularity, degeneration detection, and pushability.
result Existence of a surface carrying two meromorphic differentials realizing any prescribed restricted pair.