Related papers: Geodesic Webs of Hypersurfaces
We investigate the property of boundary rigidity for the projective structures associated to torsion-free affine connections on connected analytic manifolds with boundary. We show that these structures are generically boundary rigid,…
We study the local analytic classification of affine structures with logarithmic pole on complex surfaces. With this result in hand, we can get the local classification of the logarithmic parallelizable d-webs, d $\ge$ 3.
It is demonstrated that hypersurfaces with a flat centroaffine metric are governed by a system of nonlinear PDEs known as the equations of associativity of 2-dimensional topological field theory.
The first examples of complete projective connections are uncovered: normal projective connections on surfaces whose geodesics are all closed and embedded are complete, as are normal projective connections induced from complete affine…
The aim of this work is to study the foliations on the complex projective plane with flat \textsc{Legendre} transform (dual web). We establish some effective criteria for the flatness of the dual $d$-web of a homogeneous foliation of degree…
We give a Chern-Weil map for the Gel'fand-Fuks characteristic classes of Haefliger-singular foliations, those foliations defined by smooth Haefliger structures with dense regular set. Our characteristic map constructs, out of singular…
In order to understand the structure of the cohomologies involved in the study of projectively equivariant quantizations, we introduce a notion of affine representation of a Lie algebra.We show how it is related to linear representations…
The $\text{PSL}(4,\mathbb{R})$ Hitchin component of a closed surface group $\pi_1(S)$ consists of holonomies of properly convex foliated projective structures on the unit tangent bundle of $S$. We prove that the leaves of the…
We study the symmetries and geodesics of an infinite translation surface which arises as a limit of translation surfaces built from regular polygons, studied by Veech. We find the affine symmetry group of this infinite translation surface,…
We identify the moduli space of complex affine surfaces with the moduli space of regular meromorphic connections on Riemann surfaces and show that it satisfies a corresponding universal property. As a consequence, we identify the tangent…
The topological vertex formalism for 5d $\mathcal{N}=1$ gauge theories is not only a convenient tool to compute the instanton partition function of these theories, but it is also accompanied by a nice algebraic structure that reveals…
Geodesic nets on Riemannian manifolds form a natural class of stationary objects generalizing geodesics. Yet almost nothing is known about their classification or general properties even when the ambient Riemannian manifold is the Euclidean…
We prove that a one-dimensional foliation with generic singularities on a projective space, exhibiting a Lie group transverse structure in the complement of some codimension one algebraic subset is logarithmic, i.e., it is the intersection…
We study geometric structures on the complement of a toric mirror arrangement associated with a root system. Inspired by those special hypergeometric functions found by Heckman-Opdam, as well as the work of Couwenberg-Heckman-Looijenga on…
The goal of this study is to provide a method for computing the following: Given a network of curves in 3d (satisfying a condition at the intersection points), compute efficiently a smooth surface such that the curves are geodesics on it.…
We study affine maps between affine manifolds. Even when the fibers are compact and diffeomorphic, two of them can inherit different affine structures from the source space. This leads to a fixed linear holonomy deformation theory of the…
We construct families of smooth affine surfaces with pairwise non isomorphic A 1-cylinders but whose A 2-cylinders are all isomorphic. These arise as complements of cuspidal hyperplane sections of smooth projective cubic surfaces.
Machine learning problems have an intrinsic geometric structure as central objects including a neural network's weight space and the loss function associated with a particular task can be viewed as encoding the intrinsic geometry of a given…
An affine hypersurface is said to admit a pointwise symmetry, if there exists a subgroup of the automorphism group of the tangent space, which preserves (pointwise) the affine metric h, the difference tensor K and the affine shape operator…
In this work, we investigate the behaviour of the covering gonality of a very general hypersurface in a product of projective spaces. Inspired by the work of Bastianelli, Ciliberto, Flamini and Suppino in [BCFS19] which addresses the case…