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.
The paper classifies affine minimal translation surfaces and finds their properties.
The isotropic 3-space \mathbb{I}^{3} is a real affine 3-space endowed with the metric dx^{2}+dy^{2}. In this paper we describe Weingarten and linear Weingarten affine translation surfaces in \mathbb{I}^{3}. Further we classify the affine translation surfaces in \mathbb{I}^{3} that satisfy certain equations in terms of …
Classifies surfaces translating under specific curvature flows.
We prove some estimates of the volumes of the sets of translation surfaces of unit area having several independent small saddle connections in a rank one affine submanifold.
In this paper, we extend the notion of affine translation surfaces introduced by Liu and Yu (Proc. Japan Acad. Ser. A Math. Sci. 89, 111--113, 2013) in a Euclidean space R^{3} to higher dimensional ambient spaces. We provide that an affine translation hypersurface of constant Gauss-Kronocker curvature K_{0} in R^{n+1} …
The study of which mapping class group elements can be realized as affine automorphisms of dilation surfaces.
To every half-translation surface, we associate a saddle connection graph, which is a subgraph of the arc graph. We prove that every isomorphism between two saddle connection graphs is induced by an affine homeomorphism between the underlying half-translation surfaces. We also investigate the automorphism group of the …
Book explores infinite translation surfaces, challenging traditional geometry.
Study decomposes geometric surfaces, finding special curves.
This is a review with examples concerning the concepts of affine (in particular, constant and linear) vector fields and fundamental vector fields on a manifold. The affine, linear and constant vector fields on a manifold are shown to be in a bijective correspondence with the fundamental vector fields on it of respectiv…
New unsupervised image translation method detects changes without labeled data.
Transforms classical connections using pushforwards and gauge transformations.
In [DM] it was asked whether all flat holomorphic Cartan geometries (G,H) on a complex torus are translation invariant. We answer this affimatively under the assumption that the complex Lie group G is affine. More precisely, we show that every holomorphic Cartan geometry of type (G,H), with G a complex affine Lie group…
We complete the classification of rank two affine manifolds in the moduli space of translation surfaces in genus three. Combined with a recent result of Mirzakhani and Wright, this completes the classification of higher rank affine manifolds in genus three.
To any connected and simply connected nilpotent Lie group N, one can associate its group of affine transformations Aff(N). In this paper, we study simply transitive actions of a given nilpotent Lie group G on another nilpotent Lie group N, via such affine transformations. We succeed in translating the existence questio…
Algorithm constructs surfaces with specific Veech groups in lattice strata.
Researchers compute gap distributions for saddle connection directions on specific translation surfaces.
We give a new characterization of flat affine manifolds in terms of an action of the Lie algebra of classical infinitesimal affine transformations on the bundle of linear frames. We characterize flat affine symplectic Lie groups using symplectic étale affine representations and as a consequence of this, we show that a …
Affine deformations of convex cones on projective surfaces.
Study on Monge-Ampère equations with polynomial growth rates.
For a half-translation surface (S,q), the associated saddle connection complex A(S,q) is the simplicial complex where vertices are the saddle connections on (S,q), with simplices spanned by sets of pairwise disjoint saddle connections. This complex can be naturally regarded as an induced subcomplex of the arc complex. …
New algorithm for computing Veech groups from translation surfaces.
Study shows periodic points of Prym eigenforms in specific genera.
We classify the affine connections on compact orientable surfaces for which the pseudogroup of local isometries acts transitively. We prove that such a connection is either torsion-free and flat, the Levi-Civita connection of a Riemannian metric of constant curvature or the quotient of a translation-invariant connectio…
Constructs subvarieties in translation surface strata using combinatorial input.
We characterize subgroups of the mapping class group that stabilize a Teichmueller disk in terms of ellipses and strips that are immersed in the associated translation surface. In particular, we show that the space of immersed ellipses/strips that meet at least three cone points is naturally a (non-manifold) 2-dimensio…
New method constructs translationally equivariant hyperbolic affine spheres.
The affine-additive group is hyperbolic with a non-vanishing 4-capacity.
We investigate specific examples of locally-defined real vector-fields on strata of translation surfaces. Integrating SL(2,R)-loci of Veech surfaces along these vector-fields yield interesting new examples of horocyle-invariant ergodic measures. These measures are supported on closed immersed manifolds with boundary th…
In this paper, we investigate the parametric version and non-parametric version of rigidity theorem of spacelike translating solitons in pseudo-Euclidean space . Firstly, we classify -dimensional complete spacelike translating solitons in by affine technique and classical…
We construct noncompact solutions to the affine normal flow of hypersurfaces, and show that all ancient solutions must be either ellipsoids (shrinking solitons) or paraboloids (translating solitons). We also provide a new proof of the existence of a hyperbolic affine sphere asymptotic to the boundary of a convex cone c…
In this paper we determine the at least -dimensional affine reductive homogeneous manifolds for an at most -dimensional simple Lie group or an at most -dimensional semi-simple Lie group. Those reductive spaces among them which admit a sharply transitive differentiable section yield local almost differentiable …
We prove that the focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the source, while the focal surface is translation invariant. This is done by const…
The paper shows measures equidistribute on affine submanifolds with a rate.
We prove that holomorphic normal projective connections on compact complex surfaces are flat. We show that a holomorphic torsion-free affine connection on a compact complex surface is locally modelled on a translations-invariant affine connection on $\C^2$, except if is a generic connection on a princ…
Develops a theory for mth order p-affine capacity for convex bodies containing the origin.
The study classifies holomorphic projective connections on complex threefolds.
We give a gauge invariant characterisation of the elliptic affine sphere equation and the closely related Tzitzéica equation as reductions of real forms of $SL(3, \C)$ anti--self--dual Yang--Mills equations by two translations, or equivalently as a special case of the Hitchin equation. We use the Loftin--Yau--Zaslow co…
Paper proves certain closed affine manifolds without invariant lines don't exist.
We study the boundary of an affine invariant submanifold of a stratum of translation surfaces in a partial compactification consisting of all finite area Abelian differentials over nodal Riemann surfaces, modulo zero area components. The main result is a formula for the tangent space to the boundary. We also prove fini…
Identifies holonomy of affine surfaces via meromorphic connections.
We show that closed aspherical manifolds supporting an affine structure, whose holonomy map is injective and contains a pure translation, must have vanishing simplicial volume. This provides some further evidence for the veracity of the Auslander Conjecture. Along the way, we provide a simple cohomological criterion fo…
We give a characterization of flat affine connections on manifolds by means of a natural affine representation of the universal covering of the Lie group of diffeomorphisms preserving the connection. From the infinitesimal point of view, this representation is determined by the 1-connection form and the fundamental for…
This paper is a continuation of the previous paper of the author[M]. We show that an affine deformation space of a hyperbolic surface of type (g,b) can be parametrized by Margulis invariants and affine twist parameters with a certain decomposition of the surface, which are associated with the Fenchel-Nielsen coordinate…
In this article we study Weinstein structures endowed with a Lefschetz fibration in terms of the Legendrian front projection. First we provide a systematic recipe for translating from a Weinstein Lefschetz bifibration to a Legendrian handlebody. Then we present several applications of this technique to symplectic topol…
Affine deformations of convex cones yield special spacetime structures.
The object of this paper is to study GL(2,R) orbit closures in hyperelliptic components of strata of abelian differentials. The main result is that all higher rank affine invariant submanifolds in hyperelliptic components are branched covering constructions, i.e. every translation surface in the affine invariant subman…
Defines foliation criterion for dense isoperiodic leaves in rank 1 affine orbifolds.