I use harmonic maps and minimal surfaces to study quadratic equations in groups.
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arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
168,657 papers · 148 categories
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48 results for “quadratic geometry”
The quadratic linking degreemath.AG
Introduces quadratic linking degree in algebraic geometry.
problem No specific problem stated; focuses on new concept.
method Motivic homotopy theory and Witt group.
result Explicit computation of quadratic linking degree.
Develops second order infinitesimal structures on Teichmüller space.
problem Understand the infinitesimal structures of Teichmüller space.
method Formulated second order infinitesimal structures over Teichmüller space.
result Affirmative answers to two folklore problems on Teichmüller space.
We compare the flat geometry associated to a quadratic differential with the hyperbolic geometry associated to the underlying Riemann surface. We show that if a curve is contained in a thick subsurface, then its hyperbolic length is comparable to its flat length times the flat size of the subsurface.
Investigates smoothness of specific algebra structures.
problem Smoothness of bi-quadratic algebras on three generators.
method Analyzes differential smoothness with PBW basis.
result Characterizes conditions for smoothness.
Weyl Tensors, Strongly Regular Graphs, Multiplicative Characters, and a Quadratic Matrix Equationmath.GR
Solutions to a quadratic matrix equation are linked to strongly regular graphs and multiplicative characters.
problem Solving a specific quadratic matrix equation in Riemannian geometry.
method Constructing nonzero solutions using group rings and multiplicative characters of finite fields.
result Solutions relate to strongly regular graphs and multiplicative characters of finite fields.
The paper studies optimal transport in linear quadratic systems and derives interpolation inequalities.
problem Optimal transport problem in Linear Quadratic optimal control systems.
method Well-posedness of the Monge problem, regularity of optimal transport map, displacement interpolation of measures.
result Derivation of general interpolation inequalities for entropy functionals.
This paper deals with the notion of quadratic differential in spherical CR geometry (or more generally on strictly pseudoconvex CR manifolds). We get to this notion by studying a splitting of Rumin complex and discuss its first features such as trajectories and length. We also define several differential operators on q…
On Surfaces in R^n via Gauss Map, Caustics, Duality and Pseudo Euclidean Geometry of Quadratic Formsmath.DG
Unified view of surfaces in R^n using Gauss map, caustics, and quadratic forms.
problem Understanding smooth surfaces in R^n via various geometric perspectives.
method Combining evolute, curvature ellipse, Gauss map, and pseudo-Euclidean geometry of quadratic forms.
result Intersection of caustic with normal space of a surface yields polar dual of curvature ellipse.
New derivation shows spacetime interval is quadratic without light.
problem Deriving spacetime geometry from fundamental principles.
method Formalizing axioms of smoothness, homogeneity, isotropy, and determinism.
result Invariant spacetime interval is quadratic, independent of light.
The paper studies new curvature properties in Finsler geometry.
problem Properties of projectively equivalent Finsler metrics and their curvature structures.
method Introducing new characterizations of quadratic curvature properties in Finsler manifolds.
result Novel insights into curvature behavior under generalized projective sprays.
We construct a tangent bundle exponential map and locally autoparallel coordinates for geometries based on a general connection on the tangent bundle of a manifold. As concrete application we use these new coordinates for Finslerian geometries and obtain Finslerian geodesic coordinates. They generalise normal coordinat…
Markov's theorem classifies the worst irrational numbers with respect to rational approximation and the indefinite binary quadratic forms whose values for integer arguments stay farthest away from zero. The main purpose of this paper is to present a new proof of Markov's theorem using hyperbolic geometry. The main ingr…
New aesthetic curves in equiaffine geometry include the quadratic and logarithmic spiral.
problem Designing aesthetic shapes in equiaffine geometry.
method Introducing a new symmetry (ESA) to characterize planar curves.
result The new class of curves includes the quadratic curve and logarithmic spiral.
New metrics defined in Finsler geometry with specific properties.
problem Understanding the properties of Finsler metrics and their subclasses.
method Introducing the generalized Berwald projective Weyl metric and proving properties of the class of generalized Douglas metrics.
result All metrics with vanishing Landsberg curvature are of R-quadratic type.
Algebraic Number Starscapesmath.NT
The paper explores the geometry of algebraic numbers and their roots.
problem Understanding the geometry of algebraic numbers and their roots.
method Computer visualization and geometric analysis of polynomial roots.
result Geometric insights into the embedding properties of polynomial root mappings.
The paper classifies a specific type of quadratic variety with a small codimension.
problem Classifying nondegenerate smooth projective varieties of dimension defined by quadratic equations.
method Using the Hartshorne conjecture on complete intersections and classification techniques.
result The paper classifies varieties with and proves they are complete intersections.
Relativity Without Light: Homogeneity, Isotropy, and Determinism Force Quadratic Spacetime Metricsmath-ph
Develops a foundational argument for Lorentzian or Euclidean spacetime geometry without light or electromagnetic phenomena.
problem Relativity without light
method Formalizing physical principles as axioms about an invariant interval function
result Invariant interval functions are powers of nondegenerate quadratic forms
Study connects curvature to graph theory and reveals differences.
problem Exploring differences between Quadratic Orthogonal Bisectional Curvature and Real Bisectional Curvature.
method Real (1,1)--forms and Weitzenböck curvature operator used to represent graph Dirichlet energy.
result Curvature differences illuminated between Quadratic Orthogonal Bisectional Curvature and Real Bisectional Curvature.
We study computational and statistical consequences of problem geometry in stochastic and online optimization. By focusing on constraint set and gradient geometry, we characterize the problem families for which stochastic- and adaptive-gradient methods are (minimax) optimal and, conversely, when nonlinear updates -- su…
Optimizes quadratic bandits with tight Hessian-dependent sample complexity bounds.
problem Understanding optimal sample complexity for quadratic functions.
method Introduces energy allocation and optimal energy spectrum to prove tight lower bounds. Solves for Hessian-independent optimal algorithm.
result Proves optimal Hessian-dependent sample complexities and existence of a universally optimal algorithm.
Survey of recent progress in gravitational instantons
problem Classification of hyperkähler and Hermitian gravitational instantons
method Survey of recent progress in gravitational instantons
result Survey of recent progress in gravitational instantons
Paper proves stable minimal surfaces in 3D are flat.
problem Understanding stable minimal surfaces in 3D.
method Analyzes quadratic area growth and stability conditions.
result Stable minimal Plateau surfaces in 3D are flat.
Study on regularity of optimal transport maps on convex domains with quadratic cost.
problem Regularity of optimal transport maps between convex domains with quadratic cost.
method Analysis of -densities and boundary conditions, monotonicity formula for optimal transport maps.
result Proves -regularity for nondegenerate -densities and -regularity for boundary.
A meromorphic quadratic differential on a punctured Riemann surface induces horizontal and vertical measured foliations with pole-singularities. In a neighborhood of a pole such a foliation comprises foliated strips and half-planes, and its leaf-space determines a metric graph. We introduce the notion of an asymptotic …
The split version of the Freudenthal-Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras [3, 17, 18]. The geometries appearing in the second row are Severi-Brauer varieties [20]. We provide an easy uniform axiomatization of these geometries and related ones,…
Geodesic flows with specific integrals are linked to special 4-webs.
problem Characterizing geodesic flows with commuting quadratic integrals.
method Characterization through geodesic 4-webs and geometric hypothesis.
result Metrics of Stäckel type for geodesic flows with specific integrals.
Research classifies quadratic forms over various fields.
problem Classifying conjugacy classes in PSL(2, K).
method Synthetic approach, focusing on Lie groups and their actions.
result Provides a complete description of PSL(2, Z)-classes of integral binary quadratic forms.
New non-quadratic hypersurfaces found for higher dimensions.
problem Bernstein problem for affine maximal type hypersurfaces in higher dimensions.
method Constructing non-quadratic hypersurfaces for N>=3, θ\in(1/2,(N-1)/N).
result Found non-quadratic affine maximal type hypersurfaces for N>=3.
Effective field theories with explicit Lorentz violation are intimately linked to Riemann-Finsler geometry. The quadratic single-fermion restriction of the Standard-Model Extension provides a rich source of pseudo-Riemann-Finsler spacetimes and Riemann-Finsler spaces. An example is presented that is constructed from a …
We generalize Conway's approach to integral binary quadratic forms on Q to study integral binary hermitian forms on quadratic imaginary extensions of Q. In Conway's case, an indefinite form that doesn't represent 0 determines a line ("river") in the spine T associated with SL(2,Z) in the hyperbolic plane. In our genera…
3d Super Hyperbolic Geometrymath.GT
Extends Wigner's representation to study super hyperbolic geometry.
problem Understanding geometry in super hyperbolic three-space.
method Extended Wigner's representation of the Lorentz group to OSp_C(1|2) and applied to Minkowski (3,1|4)-dimensional super space.
result Proof of divergence of the volume of a typical ideal tetrahedron in super hyperbolic three-space.
This study connects financial volatility to quantum mechanics on hyperbolic manifolds.
problem Deriving a geometric interpretation of financial volatility.
method Mapping financial pricing to quantum Hamiltonians via transformations.
result Financial volatility is a diffusion process on a hyperbolic manifold.
In this paper, we present a unified study of the moduli space of tropical curves and Outer space which we link via period maps to the moduli space of tropical abelian varieties and the space of positive definite quadratic forms. Our work is a first step towards exhibiting Outer space and the space of positive definite …
Survey explores geometric aspects of policy optimization in control systems.
problem Understanding the geometric relationships between control design and optimization.
method Geometric perspective on policy optimization, focusing on parameterization and topology.
result Implications of policy geometry on stability and performance of local search algorithms.
CWGD measures gradient diversity weighted by curvature, improving SGD convergence.
problem Gradient noise in high-curvature directions is underestimated by standard methods.
method CWGD weights gradient diversity by the inverse square root of the Hessian.
result CWGD-Cosine reduces optimization error by up to 20% compared to standard cosine annealing.
We show that the Dehn function of the handlebody group is exponential in any genus . On the other hand, we show that the handlebody group of genus is cubical, biautomatic, and therefore has a quadratic Dehn function.
We study modular fibers of elliptic differentials, which are roughly spaces of torus-coverings over a fixed base torus. For genus 2 torus covers with fixed degree we show, that the modular fibers F_d(1,1) are itself connected torus covers with Veech group SL_2(Z). Using results of Eskin, Masur and Schmoll we calculate …
This is a paper based on a talk given at the conference on Conformal Geometry which held at Roscoff in France in the 2008 summer. We study some aspects of the equation arising from the problem of the existence on a given closed Riemannian manifold of dimension at leat 4, of a conformal metric with constant curvat…
Lecture notes on non-Kähler complex threefolds, focusing on conifold transitions.
problem Understanding non-Kähler complex threefolds and conifold transitions.
method Review of basics, description of topological features, survey of recent developments.
result Survey of recent developments on the geometrization of conifold transitions.
The correspondence between Riemann-Finsler geometries and effective field theories with spin-independent Lorentz violation is explored. We obtain the general quadratic action for effective scalar field theories in any spacetime dimension with Lorentz-violating operators of arbitrary mass dimension. Classical relativist…
We study the asymptotic geometry of Teichmueller geodesic rays. We show that when the transverse measures to the vertical foliations of the quadratic differentials determining two different rays are topologically equivalent, but are not absolutely continuous with respect to each other, then the rays diverge in Teichmue…
We formulate natural conformally invariant conditions on a 4-manifold for the existence of a metric whose Schouten tensor satisfies a quadratic inequality. This inequality implies that the eigenvalues of the Ricci tensor are positively pinched.
We use meromorphic quadratic differentials with higher order poles to parametrize the Teichmüller space of crowned hyperbolic surfaces. Such a surface is obtained on uniformizing a compact Riemann surface with marked points on its boundary components, and has non-compact ends with boundary cusps. This extends Wolf's pa…
Deformations of singularities of meromorphic -connections and meromorphic quadratic differentialsmath.CA
The paper studies deformations and confluences of singularities in meromorphic connections and quadratic differentials.
problem Understanding singularities and deformations in meromorphic connections and quadratic differentials.
method Local formal invariants and jets of meromorphic quadratic differentials, universal isomonodromic deformation, unfolded Stokes phenomenon, horizontal and vertical foliations.
result Establishes a correspondence between local formal invariants and jets of meromorphic quadratic differentials, describing parameter spaces and moduli spaces.
Extends classification of hyperbolic Dehn fillings in quadratic fields.
problem Classifying hyperbolic Dehn fillings with specific cusp shapes.
method Analyzing manifolds with cusp shapes in the same quadratic field.
result Complete classification of hyperbolic Dehn fillings under mild assumptions.
We study the global convergence of generative adversarial imitation learning for linear quadratic regulators, which is posed as minimax optimization. To address the challenges arising from non-convex-concave geometry, we analyze the alternating gradient algorithm and establish its Q-linear rate of convergence to a uniq…
Classifies all toric Kahler surfaces with twistor 2-forms.
problem Classifying smooth 4-dimensional Kahler geometries with specific properties.
method Complete classification through algebraic and geometric analysis.
result Found six geometrically distinct families of toric Kahler surfaces.