In this paper we give a construction of Lagrangian torus fibration for Fermat type quintic \cy hypersurfaces via the method of gradient flow. We also compute the monodromy of the expected special Lagrangian torus fibration and discuss structures of singular fibers.
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48 results for “Fermat quintic”
Neural nets approximate Ricci flat metrics for Calabi-Yau manifolds.
problem Lack of analytical Ricci flat metrics for Calabi-Yau threefolds.
method Employed neural network approximations for several Calabi-Yau manifolds of dimensions two and three.
result Measures of Ricci flatness improved by three orders of magnitude after training.
Algorithm computes eigenvalues and eigenforms on Calabi-Yau threefolds.
problem Computing spectra of Laplace-de Rham operator on Calabi-Yau manifolds.
method Numerical algorithm using approximate Calabi-Yau metric.
result Computed eigenvalues and eigenforms for the Fermat quintic threefold.
Proves resurgent nature of a series solution to deformed Painlevé I equation.
problem Analyzing the resurgent nature of a series solution to the deformed Painlevé I equation.
method Proves resurgent nature through formal -power series solution and Borel summability.
result Borel transform defines a global multivalued holomorphic function on a Fermat quintic surface.
The -dimensional link of a weighted homogeneous hypersurface on the round -sphere in has a nontrivial null Sasakian structure which is contact Calabi-Yau, in many cases. It admits a canonical co-closed -structure induced by the Calabi-Yau -orbifold basic geometry. We disti…
We identify a set of "energy" functionals on the space of metrics in a given Kaehler class on a Calabi-Yau manifold, which are bounded below and minimized uniquely on the Ricci-flat metric in that class. Using these functionals, we recast the problem of numerically solving the Einstein equation as an optimization probl…
Fermat constants fail to fully identify Clairaut constants for certain geodesics on a surface of revolution.
problem Identifying Clairaut constants from Fermat constants for specific geodesics.
method Analytical proof for a specific class of geodesics on a surface of revolution.
result Fermat constants do not fully determine Clairaut constants for some geodesics, except for a standard sphere.
Determined the balanced cone of a specific geometric space.
problem Determining the balanced cone of the small resolution of the quintic conifold.
method Intersection number method to explicitly determine the cone.
result Explicitly determined the balanced cone of the small resolution of the quintic conifold.
Calculates Laplacian spectra on Calabi-Yau hypersurfaces.
problem Computing the spectrum of the Laplacian on complex manifolds.
method Numerical computation of eigenvalues and eigenmodes for line bundles.
result Agreement with exact results for and a torus, first numerical results for Fermat quintic.
Paper studies weighted Fermat-Frechet problem for simplex edge lengths.
problem Finding optimal edge lengths for simplex deformations.
method Isometric embedding techniques for -Space.
result New variational method to solve weighted Fermat-Frechet problem.
Study shows connections between Jacobian torsors and Fermat curves.
problem Understanding torsors of Jacobian of universal Fermat curves.
method Analyzes torsors of Jacobian of universal family of degree- Fermat curves.
result Every torsor is a connected component of the Picard scheme.
We analyze convergence of Fermat distances and their application in clustering.
problem Understanding convergence properties of Fermat distances on Riemannian manifolds.
method Geometric and statistical arguments in percolation theory, leveraging novel arguments for non-uniform densities and curved domains.
result Discrete, sample-based Fermat distances converge to their continuum analogues with a precise rate dependent on intrinsic dimensionality.
We compute some value of the harmonic volume for the Fermat sextic. Using this computation, we prove that some special algebraic cycle in the Jacobian variety of the Fermat sextic is not algebraically equivalent to zero.
This is the first in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we study theories of supercommutative algebras for which infinitely differentiable functions can be evaluated on elements. Such…
In this review, we collect several results for conformally standard stationary spacetimes (SxR,g) obtained in terms of a Finsler metric of Randers type on the orbit manifold S that we call Fermat metric. This metric is obtained by applying the relativistic Fermat principle and it turns out that it encodes all the causa…
The paper investigates unique solutions for Fermat-Torricelli problem in specific norms.
problem Finding unique solutions for the Fermat-Torricelli problem in normed planes.
method Formulating and proving a uniqueness criterion for the problem.
result A criterion for the uniqueness of solutions in norms defined by regular polygons (lambda planes).
A new method finds a subconscious point on curved surfaces.
problem Finding a point on variable curvature surfaces that minimizes distances.
method New variational method based on geodesic arc length.
result Solution involves a 'subconscious' point with positive curvature.
By calculating the Fermat limit of certain q-Fermat functions, we get explicit surgery formulae for the second and third Ohtsuki invariants for homology 3-spheres. The surgery formula of the second Ohtsuki invariant λ_2, combined with an argument using the general theory of finite type invariants of homology 3-spheres,…
Fermat-Torricelli points help assess investment risks by smoothing series data.
problem Analyzing investment risks in series with large variance, nonlinear trends, or non-normal distributions.
method Construct Fermat-Torricelli points to reduce random component influence.
result Smoothing series by Fermat-Torricelli points reduces risk assessment errors.
Quantum K-theory of quintic 3-fold conjectured with non-polynomial coefficients.
problem Reconstructing quantum K-theory for quintic 3-fold.
method Formulated explicit conjecture for small J-function and its q-difference equation.
result Coefficients of q-difference equations are non-polynomial functions of Gopakumar-Vafa invariants.
A new classifier uses Fermat distance for semi-supervised learning in high dimensions.
problem Semi-supervised classification with limited labeled data in high-dimensional settings.
method Proposes weighted k-NN and MDS classifiers using Fermat distance.
result The weighted k-NN classifier is minimax optimal and outperforms other methods.
We use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. We apply this construction to the tropical curves obtained from the 2875 lines on the quintic Calabi-Yau threefold. Each admissible tropical curve gives a Lagrang…
Researchers create special fibrations on Calabi-Yau hypersurfaces.
problem Understanding special Lagrangian fibrations on Calabi-Yau hypersurfaces.
method Produced special Lagrangian T^n-fibrations on Calabi-Yau hypersurfaces in the Fermat family.
result Demonstrated fibrations on generic regions of Calabi-Yau hypersurfaces in the large complex structure limit.
Constructs independent bases for cubic curve families using Hessian structures.
problem Finding independent bases for cubic curve families.
method Uses a Hessian structure to define a cost function for constructing bases.
result Constructs valuatively independent bases for .
In this article we construct Lagrangian torus fibrations for general quintic \cy hypersurfaces near the large complex limit and their mirror manifolds using gradient flow method. Then we prove the Strominger-Yau-Zaslow mirror conjecture for this class of \cy manifolds in symplectic category.
In this paper we first study some global properties of the energy functional on a non-reversible Finsler manifold. In particular we present a fully detailed proof of the Palais--Smale condition under the completeness of the Finsler metric. Moreover we define a Finsler metric of Randers type, which we call Fermat metric…
We consider the discriminant locus of the Fermat cubic under the twistor fibration . We show that it has a conformal symmetry group of order and use this to identify its topology.
Capturing Smile Dynamics with the Quintic Volatility Model: SPX, Skew-Stickiness Ratio and VIXq-fin.MF
Model captures SPX and VIX volatility surfaces and skew-stickiness ratio.
problem Capturing volatility dynamics in financial markets.
method Two-factor Quintic Ornstein-Uhlenbeck (OU) model with polynomial volatility.
result Model accurately represents SPX and VIX volatility surfaces and SSR.
Generalizes Fermat's principle for wave propagation in cone structures.
problem Wave propagation in complex media with discontinuities and anisotropy.
method Generalizes Fermat's principle to smooth interfaces separating two cone structures representing wave propagation in various media.
result Conditions for critical points of arrival time functional, generalizing Snell's law and reflection.
Topological Mirror Symmetrymath.AG
This paper focuses on a topological version on the Strominger-Yau-Zaslow mirror symmetry conjecture. Roughly put, the SYZ conjecture suggests that mirror pairs of Calabi-Yau manifolds are related by the existence of dual special Lagrangian torus fibrations. We explore this conjecture without reference to the special La…
The paper explores selecting the parameter α for Fermat distance to balance geometry and noise.
problem Choosing the optimal parameter α for Fermat distance to navigate geometry and noise.
method Theoretical and simulation studies to determine the best α value.
result An optimal α value is identified to balance geometry and noise.
Method combines LD and Fermat Distance for neural network uncertainty.
problem Measuring uncertainty in neural network predictions.
method Statistical Depth (LD) combined with Fermat Distance.
result Effective uncertainty estimation without impacting original model performance.
Develops arithmetic PDE geometry using Fermat quotients.
problem Creating an arithmetic PDE analogue of Riemannian geometry.
method Using Fermat quotients and Frobenius elements in the absolute Galois group of a -adic field.
result Existence and uniqueness of geodesics and connections proved.
The paper proves K-stability of special Gushel-Mukai manifolds.
problem Proving K-stability of special Gushel-Mukai manifolds.
method Analyzing the structure of Gushel-Mukai manifolds and their K-stability.
result General special Gushel-Mukai n-folds are K-stable for 3 ≤ n ≤ 6.
Study on minimal surfaces and their Gauss maps intersecting a specific hypersurface.
problem Understanding intersections of complete minimal surfaces and a Fermat hypersurface.
method Established modified defect relations for the Gauss map of a complete minimal surface.
result Finite total curvature of a complete minimal surface if it intersects a specific hypersurface.
Survey of Fermat principle in general relativity and beyond.
problem Mathematical challenges in variational formulation of Fermat principle in Lorentzian geometry.
method Proof in smooth lightlike curves, analysis of null condition, alternative frameworks, multiplicity results.
result Space of lightlike curves does not admit a smooth manifold structure due to cone nature of null condition.
A complete description of the global monodromy of a Lefschetz fibration arising from the Fermat surface of degree 4 is given. As a by-product we get a positive relation among right hand Dehn twists in the mapping class group of a closed orientable surface of genus 3.
We compute the Chen-Ruan orbifold cohomology ring of the Batyrev mirror orbifold of a smooth quintic hypersurface in 4-dimensional projective space. We identify the obstruction bundle for this example by using the Riemann bilinear relations for periods. We outline a general method of computing the Chen-Ruan ring for Ca…
In this paper we prove several multiplicity results of -periodic light rays in conformally stationary spacetimes using the Fermat metric and the extensions of the classical theorems of Gromoll-Meyer and Bangert-Hingston to Finsler manifolds. Moreover, we exhibit some stationary spacetimes with a finite number of …
In this work, a version of Fermat's principle for causal curves with the same energy in time orientable Finsler spacetimes is proved. We calculate the secondvariation of the {\it time arrival functional} along a geodesic in terms of the index form associated with the Finsler spacetime Lagrangian. Then the character of …
A polynomial curve of degree 5, , is a helix, if and only if both $||α^'||$ and $||α^'\wedge α^{''}||$ are polynomial functions.
Geometric proof shows primes of form 3k+1 are norms of Eisenstein integers.
problem Geometric proof of primes of form 3k+1 being norms of Eisenstein integers.
method Geometric proof using Penner's λ-length and norms of Eisenstein integers.
result Every prime of the form is the norm of an Eisenstein integer.
We study a geometry associated with rank 3 distributions in dimension 8, whose symbol algebra is constant and has a simple Lie algebra sp(3,R) as Tanaka prolongation. We restrict our considerations to only those distributions that are defined in terms of a systems of ODEs of the form $\dot{z}_{ij}=\frac{\partial^2 f(\d…
In this paper we investigate the properties of series of vacua in the string theory landscape. In particular, we study minima to the flux potential in type IIB compactifications on the mirror quintic. Using geometric transitions, we embed its one dimensional complex structure moduli space in that of another Calabi-Yau …
The tools of presymplectic geometry are used to study light rays trajectories in anisotropic media.
We improve density-based distances using normalizing flows and score matching.
problem Inaccurate density estimates and poor convergence in graph-based methods for high-dimensional spaces.
method Learn densities with normalizing flows and refine geodesics with a score model.
result Improved density-based distances that scale to high dimensions and improve numerical stability.
We study log canonical thresholds on quartic threefolds, quintic fourfolds, and double spaces. As an application, we show that they have a Kaehler-Einstein metric if they are general.
This paper provides the technical details of gradient flow construction and related problems, which are essential for our construction of Lagrangian torus fibrations for Calabi-Yau hypersurfaces.