The paper proves a theorem about higher-dimensional expansion and its topological implications.
problem Higher-dimensional expansion properties and their topological consequences.
method Detailed proof of Gromov's Topological Overlap Theorem using cellular cochains and simplicial complexes.
result The theorem states that if a complex has strong higher-dimensional expansion properties, it has a topological overlap property.
RSHT algorithm simplifies complex shapes to points.
problem Simplifying complex shapes to points in higher dimensions.
method Combines simplicial collapses and expansions.
result Reduces triangulated d-manifolds to points using RSHT.
Study of higher-dimensional Willmore energies via minimal submanifold asymptotics.
problem Understanding conformally invariant generalizations of Willmore energy.
method Derives and studies a new energy functional for submanifolds, connects it to minimal submanifold asymptotics in Poincare-Einstein spaces.
result Explicitly identifies the energy for four-dimensional submanifolds and studies its variational properties.
The paper analyzes covariance of embedded manifolds to extract curvature information.
problem Understanding curvature of submanifolds embedded in higher-dimensional spaces.
method Covariance analysis of point sets on embedded Riemannian manifolds, focusing on volume and curvature.
result Eigenvalue decompositions of covariance matrices have asymptotic expansions containing curvature information.
The study examines numerical aspects of Karhunen-Loève expansions for stochastic processes.
problem Constructing Karhunen-Loève expansions for second-order stochastic processes.
method Spectral decomposition of covariance operator via Fredholm integral equation, discretization, singular value decomposition of weight-scaled sample matrix.
result Consistent solutions for model-based and data-driven KLE construction, characterized by convergence of SVD-based eigenvalue estimates and KL coefficients distributions.
The paper studies partition functions of point processes on Kähler manifolds, generalizing geometric functionals and relating to QHE.
problem Analyzing partition functions of determinantal point processes on Kähler manifolds.
method Using geometric functionals and TYZ expansion coefficients of the Bergman kernel.
result The coefficients of the partition function expansion are geometric functionals on Kähler metrics.
The paper introduces a new method to create stable ergodic actions on higher-dimensional manifolds.
problem Stable ergodicity of group actions on smooth manifolds restricted to one-dimensional cases.
method Geometric method using quasi-conformal blender for constructing stable local dynamics.
result Every closed manifold admits stably ergodic finitely generated group actions by diffeomorphisms of class C1+α. Algorithm calculates L2-Euler characteristic for complex spaces.
problem Computing the twisted L2-Euler characteristic for complex spaces. method Uses Oki's matrix expansion algorithm to indirectly evaluate the Dieudonné determinant.
result Truncated algorithm produces good results in various complex spaces.
We calculate a_4 term in heat kernel expansion for noncommutative tori.
problem Calculating the term a_4 in the heat kernel expansion for noncommutative tori.
method Local expression calculation and functional relations derivation.
result Validated the calculated expressions through functional relations and partial differential system.
The paper finds power series for Bach-flat metrics from spacetimes, including Einstein and constant curvature cases.
problem Extracting asymptotically anti-de Sitter Einstein 4-metrics from Bach-flat spacetimes.
method Using conformally compact Riemannian setting and formal power series, the paper finds expansions about conformal infinity.
result The mass is part of the free data at conformal infinity, leading to Einstein metrics.
The paper connects tensor models to crystallization theory to study manifold properties.
problem Understanding topological and geometrical properties of tensor models.
method Using crystallization theory to analyze colored tensor models and their PL-manifold representations.
result The G-degree of PL-manifolds is finite-to-one in any dimension, and classification theorems are obtained for specific dimensions.
Defines a transgression functor for higher-dimensional Courant algebroids.
problem None explicitly stated; focuses on definition and properties.
method Definition of transgression functor for Courant algebroids.
result Established a connection between Courant algebroids and Lie algebroids.
An observable for nonabelian, higher-dimensional forms is introduced, its properties are discussed and its expectation value in BF theory is described. This is shown to produce potential and genuine invariants of higher-dimensional knots.
A novel method visualizes higher-dimensional spaces using hyperbolic geometry.
problem Challenges in visualizing higher-dimensional spaces.
method Interactive visualization of higher-dimensional grids based on hyperbolic geometry.
result Our method shows the whole higher-dimensional space at once and avoids disadvantages of previous methods.
The (abelian bosonic) heterotic string effective action, equations of motion and Bianchi identity at order alpha prime in ten dimensions, are shown to be equivalent to a higher dimensional action, its derived equations of motion and Bianchi identity. The two actions are the same up to the gauge fields: the latter are a…
New higher-dimensional Schwarz and Scherk surfaces discovered.
problem Constructing complete embedded periodic minimal hypersurfaces.
method Higher dimensional generalizations of Schwarz's and Scherk's surfaces.
result Complete embedded periodic minimal hypersurfaces constructed in Rn. Paper extends eigenvalue inequality to higher dimensions.
problem Eigenvalue inequality for Steklov eigenvalues in higher dimensions.
method Extended Hersch-Payne-Schiffer trick to higher dimensional manifolds.
result Generalized inequality for Steklov eigenvalues.
Study higher dimensional Reidemeister torsion for twist knots surgeries.
problem Asymptotic behavior of Reidemeister torsion for exceptional surgeries.
method Analyzing graph manifolds and twist knot groups, determining limits of coefficients.
result Explicit set of limits for leading coefficients in higher dimensional Reidemeister torsion.
We provide a generalization of Bianchi's Bäcklund transformation from 2-dimensional quadrics to higher dimensional quadrics. The starting point of our investigation is the higher dimensional (infinitesimal) version of Bianchi's main four theorems on the theory of deformations of quadrics and Bianchi's treatment of the …
This paper concerns some stability properties of higher dimensional catenoids in $\rr^{n+1}$ with n≥3. We prove that higher dimensional catenoids have index one. We use δ-stablity for minimal hypersurfaces and show that the catenoid is n2-stable and a complete n2-stable minimal hypersurface is a …
Researchers find explicit Bäcklund transforms for specific quadrics.
problem Isometric deformations of diagonal higher dimensional quadrics without center.
method Explicitly found Bäcklund transforms using the Bianchi Permutability Theorem and 3-moving Möbius configuration.
result Explicit solutions can be iterated with arbitrary constants.
Transformed quadrics from 2D to higher dimensions.
problem Generalizing quadric transformations to higher dimensions.
method Bianchi's Hazzidakis transformation method.
result Generalization to higher dimensional quadrics.
Higher-dimensional contact manifolds lack certain properties.
problem Characterizing properties of higher-dimensional contact manifolds.
method Symplectic handle attachment and alternative proof techniques.
result They do not arise as nonseparating weak contact-type hypersurfaces in closed symplectic manifolds.
We introduce the foliated anti-self dual equation for higher dimensional smooth manifolds with codimension-4 Riemannian foliations. Several fundamental results are established, towards the defining of a Donaldson type invariant for such foliations.
We give a definition of higher dimensional iterated integrals based on integration over membranes. We prove basic properties of this definition and formulate a conjecture which extends Chen's de Rham Theorem for iterated integrals to the membrane case.
The paper connects geodesic flows and higher-dimensional Reidemeister torsion for hyperbolic orbifolds.
problem Understanding the relationship between geodesic flows and higher-dimensional Reidemeister torsion.
method Using the integral expression of the Ruelle zeta function and the Selberg zeta function.
result The absolute value at zero of the Ruelle zeta function equals the higher-dimensional Reidemeister torsion.
Constructs Gabor frames for curved manifolds to detect boundaries.
problem Signal analysis on curved manifolds with boundaries.
method Higher-dimensional Gabor frames for local linearizations.
result Detection of higher-dimensional boundaries in curved manifolds.
The classical Cohn-Vossen theorem states that two isometric compact convex surfaces in R3 are congruent. In this short note, we generalize the classical Cohn-Vossen Theorem to higher dimensional surfaces in space form Nn+1(K) for n≥2.
The construction (by Kapranov) of the space of infinitesimal paths on a manifold is extended to include higher dimensional infinitesimal objects, encoding contractions of infinitesimal loops. This full infinitesimal groupoid is shown to have the algebra of polyvector fields as its non-linear cohomology.
Multisections generalize Heegaard splittings and trisections to higher dimensions.
problem Decomposing higher-dimensional manifolds into 1-handlebodies with specific intersection properties.
method Considering multisections of higher-dimensional smooth or PL closed orientable manifolds, and associating diagrams to them.
result Multisection diagrams uniquely determine manifolds in all dimensions up to 6.
We analyze higher-dimensional sliding puzzles, finding solvability patterns.
problem Solvability of higher-dimensional cubical sliding puzzles.
method Study of puzzle graphs and token movement constraints.
result Characterization of solvability regimes from stuck to fully solvable.
Modeling high-dimensional surfaces for toxicity testing using tensor product basis functions.
problem Characterizing complex high-dimensional surfaces from high-throughput toxicity testing data.
method Developed a novel Bayesian additive adaptive basis tensor product model.
result Model accurately predicts dose-responses for untested chemicals.
The paper studies higher dimensional isoperimetry and divergence in groups using combinatorial methods.
problem Estimating filling functions and divergence in finitely generated groups.
method Combinatorial study of universal covers of compact simplicial complexes, focusing on round and unfolded simplicial spheres.
result The problem of estimating higher dimensional divergence can be restricted to round spheres.
This is a survey of higher-dimensional Kleinian groups, i.e., discrete isometry groups of the hyperbolic n-space for n greater than 3. Our main emphasis is on the topological and geometric aspects of higher-dimensional Kleinian groups and their contrast with the discrete groups of isometry of the hyperbolic 3-space.
Rescaling expansiveness proven for k*-expansive vector fields.
problem Proving rescaling expansiveness for k*-expansive vector fields.
method Introducing and exploring singular-expansive flows.
result Rescaling expansiveness established for k*-expansive vector fields.
The paper proves the existence of infinitely many minimal hypersurfaces in higher-dimensional manifolds.
problem Finding minimal hypersurfaces in higher-dimensional closed manifolds.
method Generic metrics and Baire sense arguments.
result Infinitely many singular minimal hypersurfaces are found in closed manifolds with optimal regularity.
Paper proves contact structures overtwisted if small plastikstufe with toric core exists.
problem Contact structures and overtwistedness in higher-dimensional contact manifolds.
method Proves overtwistedness via existence of a small plastikstufe with toric core.
result Contact structures are overtwisted if and only if a small plastikstufe with toric core exists.
The paper characterizes Eguchi-Hanson space and its higher-dimensional analogs using Lichnerowicz Laplacian.
problem Characterizing complete Ricci-flat ALE orbifolds.
method Analytical proof using Lichnerowicz Laplacian and dimension constraints.
result Uniqueness of Eguchi-Hanson space and its higher-dimensional analogs among Ricci-flat Kähler ALE orbifolds.
Study on higher-dimensional black holes, focusing on retractions and scalar quasibound states.
problem Examining the physics of a five-dimensional non-extremal Reissner-Nordström black hole.
method Analyzing the line element and scalar field perturbations using polynomial conditions of Heun functions.
result Obtained analytical expressions for quasibound state frequencies and discussed system stability.
Computational techniques calculate dimensions of complex structures.
problem Calculating dimensions of complex structures on manifolds.
method Developed computational techniques to calculate Kodaira dimension and Dolbeault harmonic forms.
result Computed dimensions of left-invariant almost complex structures.
Study shows bounded cohomology vanishes for higher dimensional sphere diffeomorphisms.
problem Vanishing of bounded cohomology for higher dimensional sphere diffeomorphism groups.
method Proved vanishing of bounded cohomology with real coefficients for n≥4 and 1≤r≤∞. result Vanishing of bounded cohomology for higher dimensional spheres.
Develops method to compute Chern-Simons potentials from higher-dimensional Pontryagin densities.
problem Computing Chern-Simons potentials from higher-dimensional Pontryagin densities.
method Systematic approach using a generic affine connection with non-vanishing torsion and non-metricity.
result Algorithm and code for determining Chern-Simons potential from Pontryagin density in arbitrary even dimensions.
We study the problem of prescribing the Paneitz curvature on higher dimensional spheres. Particular attention is paid to the blow-up points, i.e. the critical points at infinity of the corresponding variational problem. Using topological tools and a careful analysis of the gradient flow lines in the neighborhood of suc…
The paper derives expansions for Green's operators and resolvents using Hadamard methods.
problem Analyzing normally hyperbolic operators and their Green's functions.
method Hadamard expansions for powers of Green's operators and resolvents.
result Derives expansions involving Hadamard coefficients for advanced/retarded Green's operators.
Paper introduces a new operator and solves equations on higher-dimensional almost Kähler manifolds.
problem Investigate ∂ˉ-problem and generalized Monge-Ampère equation on almost Kähler manifolds. method Introduce DJ+ operator and use it to solve equations. result Established a uniqueness up to a constant and local existence theorem for the generalized Monge-Ampère equation.
Maximally hyperbolic solutions contain future neighborhoods of intersecting hypersurfaces.
problem Maximally globally hyperbolic solutions of higher-dimensional vacuum Einstein equations.
method Analyzing intersections of characteristic hypersurfaces.
result Contains a future neighborhood of intersecting hypersurfaces.
Formula for mass in higher-dimensional graphs proves mass theorems.
problem Proving mass theorems for higher-dimensional graphs.
method Explicit formula for Gauss-Bonnet-Chern mass, applied to asymptotically flat graphical manifolds.
result Proves positive mass theorem and Penrose inequality for graphs with flat normal bundle.
Method learns PDE dynamics via evolving latent manifold using Ricci flow.
problem Learning dynamics in time, especially PDEs, with low-dimensional representations.
method Parameterizes latent manifold, simulates Ricci flow physics-informedly, matching manifold quantities.
result Ricci flow facilitates learning for out-of-distribution data and adversarial robustness.