Study Moishezon twistor spaces using quartic hypersurfaces and Del Pezzo fibrations.
problem Classify Moishezon twistor spaces with specific half-anti-canonical systems.
method Utilize pluri-half-anti-canonical maps and quartic hypersurfaces to investigate twistor spaces.
result Complete classification of Moishezon twistor spaces with half-anti-canonical systems as pencils.
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.
Researchers calculate Morse index and nullity for two specific minimal hypersurfaces.
problem Calculating Morse index and nullity for homogeneous minimal hypersurfaces with g=4 or 6. method Analyzing two specific homogeneous minimal hypersurfaces in Sn with g=4. result Obtained irrational eigenvalues in Laplace spectra for the two hypersurfaces.
New classification of complex hypersurfaces in 3D.
problem Classifying simply-transitive Levi non-degenerate hypersurfaces in C3. method Novel Lie algebraic approach, new coordinate-free formula for quartic tensor.
result Unique non-tubular model with geometric relations to planar equi-affine geometry.
In this paper we investigate Moishezon twistor spaces which have a structure of double covering over a very simple rational threefold. These spaces can be regarded as a direct generalization of the twistor spaces studied by Poon and Kreussler-Kurke to the case of arbitrary signature. In particular, the branch divisor o…
Study shows K-moduli spaces connect quartic surfaces to K3 surfaces, verifying predictions and classifying degenerations.
problem Understanding the moduli spaces of quartic K3 surfaces and their birational models.
method Interpolates between GIT and Baily-Borel moduli spaces, describes wall crossings, and classifies degenerations.
result Verifies Laza-O'Grady's prediction and classifies Gorenstein canonical Fano degenerations of \(\mathbb{P}^3\).
Classifies special quartic curves up to equivalence.
problem Classifying maximal quartic curves up to equivalence.
method Analyzing intersections of quartic polynomials and their level sets.
result Quartic generalised projective special real manifolds have non-regular boundary behavior.
We study complex spatial quartic surfaces with simple singularities up to equisingular deformations; as a first step, give a complete equisingular deformation classification of the so-called non-special simple quartic surfaces.
Segre quartic surfaces linked to minitwistor spaces with Einstein-Weyl structures.
problem Understanding the relationship between Segre quartic surfaces and minitwistor spaces.
method Using Penrose correspondence and detailed investigation of dual varieties.
result Determined the degrees and structure of components of dual varieties.
Minimal surfaces with isothermal parameters admitting Bézier representation were studied by Cosin and Monterde. They showed that, up to an affine transformation, the Enneper surface is the only bi-cubic isothermal minimal surface. Here we study bi-quartic isothermal minimal surfaces and establish the general form of th…
Study on null curves in Klein's quartic moduli space.
problem Existence of minimal surfaces with specific planar ends.
method Methods developed by Robert Bryant.
result Minimal surfaces with 9 embedded planar ends do not exist.
Classifies branched Willmore spheres using conformal Gauss maps.
problem Classifying branched Willmore spheres.
method Analyzing the asymptotic expansion of the conformal Gauss map at branched points.
result Full classification of branched Willmore spheres.
Mathematicians embed a Klein's quartic cover in hyperbolic space.
problem Embedding a Klein's quartic cover in R3. method Constructing an infinite {3,7}-surface in H3 by gluing prisms and antiprisms. result No proof of embedding in R3, but exploration of possible embedding in H3. The study characterizes surfaces with specific harmonic properties in pseudo-conformal geometry.
problem Characterizing surfaces with harmonic properties in pseudo-conformal geometry.
method Investigating sphere congruences, quasi-umbilical surfaces, and constant mean curvature surfaces.
result Generically, Bryant's quartic differential is divergence free if and only if the surface is superconformal or orthogonal to a harmonic congruence of spheres.
Study on Finsler spaces with specific metric changes and reversible geodesics.
problem Characterizing Finsler spaces with reversible geodesics.
method Analyzing a Finsler space with a Randers change of Quartic metric and deriving conditions for reversible geodesics.
result The Finsler metric F induces a generalized weighted quasi-distance on the space.
Researchers compute monodromy groups of surface families over quartic curves.
problem Computing monodromy groups of surface families over smooth quartic curves.
method Analyzing cyclic branched covers of P2 over smooth quartic curves, computing monodromy groups for del Pezzo and K3 surfaces. result Obtained monodromy groups for del Pezzo and K3 surfaces, including Weyl group $W\left(E_{7}
ight)$ and arithmetic lattice $U\left(h_{L_{-}}
ight)$.
We study the discriminant of a degree 4 extension given by a deformed bidouble cover, i.e., by equations z^2= u + a w, w^2= v + bz. We first show that the discriminant surface is a quartic which is cuspidal on a twisted cubic, i.e.,is the discriminant of the general equation of degree 3. We then take a(u,v), b(u,v) and…
Klein quartic maximizes the first positive Laplacian eigenvalue's multiplicity to 8.
problem Maximizing the first positive eigenvalue's multiplicity of the Laplacian.
method Analyzing hyperbolic surfaces of genus 3 and 2, proving the Klein quartic's maximality.
result Klein quartic maximizes the first positive Laplacian eigenvalue's multiplicity to 8.
We give an algebro-geometric construction of some of the non-arithmetic ball quotients constructed by the author, Parker and Paupert. The new construction reveals a relationship between the corresponding orbifold fundamental groups and the automorphism group of the Klein quartic, and also with groups constructed by Bar…
Study Einstein-Weyl spaces from Segre quartic surfaces, finding unique geodesics and deformations.
problem Characterize Einstein-Weyl spaces associated with Segre quartic surfaces.
method Explicit construction and analysis of minitwistor spaces, focusing on singularities and geodesics.
result Found unique closed geodesics on Einstein-Weyl spaces, showing deformations and non-compactifications.
We prove some value of the harmonic volume for the Klein quartic C is nonzero modulo ${1/2}\{mathbb Z}$, using special values of the generalized hypergeometric function 3F2. This result tells us the algebraic cycle C−C− is not algebraically equivalent to zero in the Jacobian variety J(C).
Paper constructs an infinite 3-7 surface in 3D space.
problem Can an infinite 3-7 surface be embedded in Euclidean space?
method Constructs an immersed, but not embedded, infinite 3-7 surface in R^3.
result Realizes an infinite 3-7 surface as a cover of Klein's quartic.
Holomorphic maps between configuration spaces are classified, resolving quartic and elliptic curve problems.
problem Classifying holomorphic maps between configuration spaces.
method Using braid groups, elliptic curves, and complex analysis, the authors classify maps and families of elliptic curves.
result The only non-trivial, non-identity holomorphic maps are the resolving quartic map and a map from elliptic curves.
Study of zero-divisors in sedenions via determinant factorization.
problem Characterizing zero-divisors in the sedenion algebra.
method Factorization of determinant of left multiplication, reduction to quaternionic normal form, block computation.
result Quartic polynomial factorization of determinant, geometric model of zero-divisor locus.
Study calculates volumes of Fano K-moduli spaces in various dimensions.
problem Computing volumes of Fano K-moduli spaces in different dimensions.
method Computed volumes of Fano K-moduli spaces of Quartic del Pezzo and log Fano hyperplane arrangements in dimensions one and two.
result Relates computed volumes to Weil-Petersson volumes, extending the Weil-Petersson metric to the log case.
It is shown that the sum of class numbers of orders in totally complex quartic fields with no real quadratic subfield obeys an asymptotic law similar to the prime numbers, as the bound on the regulators tends to infinity. Here only orders which are maximal at a given set of primes containing an even number of elements …
Study on homological Dehn functions of groups of type FP2.
problem Understanding the homological Dehn functions of groups of type FP2. method Proved foundational results, studied homological Dehn functions of Leary's groups, and provided methods to obtain groups with specific homological Dehn functions.
result Found groups of type FP2 with quartic homological Dehn function and unsolvable word problem. Compactifies a component by studying metric degeneration.
problem Compactify the SO(2,3)-Hitchin component.
method Study metric degeneration on a surface in pseudo-hyperbolic space.
result Establish closure in projectivized geodesic currents space.
A Laguerre geometric local characterization is given of L-minimal surfaces and Laguerre deformations (T-transforms) of L-minimal isothermic surfaces in terms of the holomorphicity of a quartic and a quadratic differential. This is used to prove that, via their Laguerre Gauss maps, the T-transforms of L-minimal isotherm…
Improved bound on the product of first Laplacian eigenvalue and area for genus three surfaces.
problem Bounding the product of the first eigenvalue of the Laplacian and the area for compact surfaces of genus three.
method Improved the bound established by Yang and Yau, using numerical computations for the hyperbolic Klein quartic surface.
result Showed that the product of the first eigenvalue of the Laplacian and the area is bounded above by approximately 21.668π.
Sharp curvature pinching for mean curvature flow in spheres proved.
problem Proving sharp curvature pinching for mean curvature flow in spheres.
method Using blow-up arguments, codimension and cylindrical estimates, and rescaling.
result Smooth convergence to a totally geodesic limit in infinite time.
Study on Einstein metrics on specific homogeneous spaces.
problem Existence of invariant Einstein metrics on aligned homogeneous spaces.
method Analysis of G_1xG_2-invariant Einstein metrics on G_1/K x G_2/K for compact Lie groups.
result Existence of Einstein metrics is equivalent to a real root of a quartic polynomial.
We prove that any weakly triholomorphic map from a compact hyperkähler surface to an algebraic K3 surface defined by a homogeneous polynomial of degree 4 in CP3 has only isolated singularities.
In this paper we deal with the classical question of existence of polynomial in momenta integrals for geodesic flows on the 2-torus. For the quasi-linear system on coefficients of the polynomial integral we consider the region (so called elliptic regions) where there are complex-conjugate eigenvalues. We show that for …
Automatically explores geometric loci of curves using software networking.
problem Exploring hyperbolisms and geometric loci of plane curves.
method Parametric equations, Groebner bases, and elimination for deriving polynomial equations.
result Derives new constructions of lemniscates and other geometric loci.
Paper extends known α-minimizing hypercones using polynomial analysis.
problem Limitations in the class of known α-minimizing hypercones. method Sub-calibration methods and analysis of cubic and quartic polynomials.
result Significantly extended class of α-minimizing hypercones. This article proposes a new method for the estimation of the parameters of a simple linear regression model which accounts for the role of co-moments in non-Gaussian distributions being based on the minimization of a quartic loss function. Although the proposed method is very general, we examine its application to fina…
We investigate the action of the automorphism group of a closed Riemann surface on its set of theta characteristics (or spin structures). We give criteria for when an automorphism fixes all spin structures, or when it fixes just one. The case of hyperelliptic curves and of the Klein quartic are discussed in detail.
Let B be a reducible reduced plane curve. We introduce a new point of view to study the topology of $(\PP^2, {\mathcal {B}})$ via Galois covers and Alexander polynomials. We show its effectiveness through examples of Zariski N-plets for conic and conic-quartic configurations.
Study of large group actions on surfaces, focusing on Hurwitz and handlebody groups.
problem Characterizing and understanding group actions on surfaces, especially maximal handlebody and Hurwitz groups.
method Analyzing various group actions, comparing Hurwitz and handlebody groups, and examining bounding actions.
result Relationship between Hurwitz groups and maximal handlebody groups, and insights into geometric bounding actions.
The paper tackles learning mixtures of two multinomial logits, showing identifiability and presenting an algorithm.
problem Learning an arbitrary mixture of two multinomial logits.
method Reduction to solving a system of univariate quartic equations, followed by an algorithm using polynomial and linear samples.
result Identifiability of the mixture models may only fail on an algebraic variety of negligible measure.
We construct minimal laminations with prescribed singularities on a line segment using perturbation techniques and PDE methods. In addition to the singular set, the rate of curvature blowup is also prescribable in our construction, and we show that all curvature blowup rates between quadratic and quartic arise. Our res…
In this paper we expose on the dual 1-jet space J^{1*}(R,M^4) the distinguished (d-) Riemannian geometry (in the sense of d-connection, d-torsions, d-curvatures and some gravitational-like and electromagnetic-like geometrical models) for the (t,x)-conformal deformed Berwald-Moor Hamiltonian metric of order four.
In this paper, we prove that a quartic polynomial solution of the eikonal equation ∣∇xf∣2=16x6 in Rn is either an isoparametric polynomial or congruent to a polynomial f=(∑i=1nxi2)2−8(∑i=1kxi2)(∑i=k+1nxi2), k=0,1,.˙.,[2n].
We connect the algebraic geometry and representation theory associated to Freudenthal's magic square. We give unified geometric descriptions of several classes of orbit closures, describing their hyperplane sections and desingularizations, and interpreting them in terms of composition algebras. In particular, we show h…
We provide a computer-assisted proof of the holomorphy of the quartic and the octic meromorphic differentials arising in the main Theorem 4.11 of our paper 'The Classification of Branched Willmore spheres in the 3-Sphere and the 4-Sphere' (arXiv:1706.01405), using the free mathematical software Sage.
Study para-Kähler-Einstein metrics and their non-integrable twistor distributions.
problem Characterize para-Kähler-Einstein metrics and their associated non-integrable twistor distributions.
method Use Cartan's method of equivalence and analyze the anti-self-dual Weyl tensor.
result Establish a correspondence between the anti-self-dual Weyl tensor and the Cartan quartic of the twistor distribution.
A new ODE model explains gradient descent dynamics near edge of stability.
problem Understanding gradient-based training over non-convex landscapes.
method Rod Flow, a new ODE approximation of GD dynamics.
result Rod Flow accurately predicts critical sharpness threshold and self-stabilization in quartic potentials.