Affine equivalence of half-translation surfaces via saddle connection graphs.
problem Understanding affine equivalence of half-translation surfaces.
method Association of saddle connection graphs and investigation of their automorphism groups.
result Every isomorphism between saddle connection graphs is induced by an affine homeomorphism between the underlying half-translation surfaces.
Study shows saddle connection graph's geometry and quasi-isometry properties.
problem Characterize the geometry and quasi-isometry of saddle connection graphs.
method Proved 4-hyperbolicity and uniform quasi-isometry to a tree, used generalised unicorn paths.
result Saddle connection graph is not quasi-isometrically rigid and its boundary is straight foliations.
The altenating knots, links and twists projected on the S_2 sphere are identified with the phase Space of a Hamiltonian dynamic system of one degree of freedom. The saddles of the system correspond to the crossing points, the edges, to the stable and unstable manifolds, connecting the saddles. Each facxe is then orient…
The study embeds graphs on translation surfaces, proving essential-systolic embeddings and estimating surface genera.
problem Embedding graphs on translation surfaces with specific properties.
method Proving essential-systolic embeddings and estimating surface genera.
result Finite graphs admit essential-systolic embeddings on translation surfaces with estimated genera.
Research describes all possible gradient vector fields on a sphere with up to ten singular points.
problem Characterizing gradient vector fields on a sphere with limited singular points.
method Using a graph to represent one-dimensional stable manifolds, specifying singularities and connections.
result Identified all topological structures of codimension one gradient vector fields on a sphere with up to ten singular points.
Study finds saddle connections on random surfaces follow Poisson distribution.
problem Distribution of saddle connections on random translation surfaces.
method Analysis of saddle connections on surfaces of large genus.
result Number of saddle connections in given lengths converges to Poisson distribution.
Saddle connection complexes are rigid under affine equivalence.
problem Characterizing the rigidity of saddle connection complexes.
method Proving simplicial isomorphisms between saddle connection complexes are induced by affine diffeomorphisms.
result Saddle connection complexes are complete invariants of affine equivalence classes of half-translation surfaces.
Study saddle connections on hyperelliptic surfaces, finding growth rates.
problem Count saddle connections on hyperelliptic surfaces without interior intersections.
method Used horocycle renormalization to prove lower bound growth rate.
result Found saddle connections satisfy L(logL)d−2 growth rate. We extend asymptotic formulas for saddle connections on translation surfaces.
problem Counting saddle connections on translation surfaces with large genus.
method Recursive formulas and asymptotic analysis for all strata and multiplicities.
result Asymptotics for all saddle connections on translation surfaces of growing genus.
Study precise rates of horizontal gap shrinkage on generic translation surfaces.
problem Understanding precise decay rates of horizontal gaps in translation surfaces.
method Analyzing saddle connections and their angles on translation surfaces.
result Obtained precise decay rates for the difference in angle between almost horizontal saddle connections.
Golden L surface has unbounded bunching of saddle connections
problem Unbounded bunching of saddle connections on translation surfaces
method Translation surface with golden ratio
result Every positive integer K has a ball containing at least K saddle connection periods
Classifies Morse flows on 3-sphere with specific saddle connections.
problem Classifying Morse-Smale flows on a 3-sphere with specific saddle connections.
method Used generalized Heegaard diagrams (Pr-diagrams) to classify flows.
result Found all possible, up to homeomorphism, ways to embed two circles in a 2-sphere with no more than 10 points of transversal intersection.
Bounds on saddle connections on flat spheres with conical singularities.
problem Counting saddle connections on flat spheres with conical singularities.
method Geometry of immersed disks and explicit upper bounds.
result Explicit upper bounds on the number and lengths of saddle connections.
The study shows how to measure translation surfaces with short saddle connections.
problem Measuring the probability of surfaces with short saddle connections.
method Using the multi-scale compactification of strata and algebraicity results.
result Proves strong regularity for invariant measures on translation surfaces.
Study systolic geometry of translation surfaces and origamis.
problem Investigate systolic ratios of translation surfaces and origamis.
method Analyze systoles and saddle connections, develop algorithm, compute ratios.
result Compute maximal systolic ratio of origamis in H(1,1) up to 67 squares. Translation surfaces with poles correspond to meromorphic differentials on compact Riemann surfaces. They appear in compactifications of strata of the moduli space of Abelian differentials and in the study of stability conditions. Such structures have different geometrical and dynamical properties than usual translatio…
Gradient-like flows on certain manifolds restrict saddle Morse indices to 1 or n-1.
problem Restricting Morse indices of saddles in gradient-like flows.
method Analyzing invariant manifolds and their intersections for gradient-like flows.
result Morse indices of saddles are either 1 or n-1, no other indices possible.
Researchers compute gap distributions for saddle connection directions on specific translation surfaces.
problem Computing gap distributions for saddle connection directions on translation surfaces.
method Translation to dynamical question of return times to a transversal under the horocycle flow.
result Gap distributions have support at 0 and quadratic tail decay.
Flat surfaces that correspond to k-differentials on compact Riemann surfaces are of finite area provided there is no pole of order k or higher. We denote by \textit{flat surfaces with poles of higher order} those surfaces with flat structures defined by a k-differential with at least one pole of order at least $k…
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.
Motivated by the study of billiards in polygons, we prove fine results for the distribution of gaps of directions of saddle connections on translation surfaces. As an application we prove that for almost every holomorphic differential ω on a Riemann surface of genus g≥2 the smallest gap between saddle connecti…
The paper introduces horizon saddle connections to study dilation surfaces.
problem Understanding the geometric and dynamical properties of dilation surfaces.
method Introducing horizon saddle connections and quasi-Hopf surfaces.
result Existence of horizon saddle connections restricts the Veech group of dilation surfaces.
Horizon saddle connections imply dense hyperbolic geodesics on dilation surfaces.
problem Characterize dilation surfaces with dense hyperbolic geodesics.
method Analyzing saddle connections and directional flow properties.
result Dilation surfaces with horizon saddle connections have dense hyperbolic geodesics.
Researchers create new triply periodic minimal surfaces by gluing saddle towers.
problem Creating triply periodic minimal surfaces without symmetry constraints.
method Gluing Karcher-Scherk saddle towers with phase differences and balancing under vertical interaction.
result Expands known triply periodic minimal surfaces into new 5-parameter families.
Computes constants for specific geometric structures.
problem Calculating constants for specific geometric structures.
method Analyzes saddle connections and Prym eigenforms.
result Computed Siegel-Veech constants for real quadratic orders.
Fix a translation surface X, and consider the measures on X coming from averaging the uniform measures on all the saddle connections of length at most R. Then as R→∞, the weak limit of these measures exists and is equal to the Lebesgue measure on X. We also show that any weak limit of a subsequence of …
Researchers calculate complexity of billiard paths in regular polygons.
problem Calculating the complexity of billiard paths in regular polygons.
method Counting saddle connections on lattice surfaces, focusing on combinatorial length.
result They answered a question about billiard language complexity in regular polygons.
For a translation surface, we define the systole to be the length of the shortest saddle connection. We give a characterization of the maxima of the systole function on a stratum, and give a family of examples providing local but nonglobal maxima on each stratum of genus at least three. We further study the relation be…
The study bounds the complexity of meromorphic differentials' directions.
problem Understanding the descriptive complexity of meromorphic differentials.
method Geometric lemma and topological analysis of saddle connections.
result Sharp upper bound on the Cantor-Bendixson rank of meromorphic differentials.
Revisit Fenn's table theorem from a differential-topological perspective.
problem Prove zero-existence theorem on a cylinder and horizontal square-table theorem under Fenn's boundary conditions.
method Differential-topological approach.
result Prove horizontal square-table theorem under more general boundary conditions.
Alternative proof for non-existence of complete curves in differential strata.
problem Non-existence of complete algebraic curves in strata of holomorphic differentials.
method Using positivity of divisor classes on moduli spaces of curves.
result Alternative proof confirming Gendron's result on non-existence.
Given k>=2, we construct a (2k-2)-parameter family of properly embedded minimal surfaces in H^2 x R invariant by a vertical translation T, called Saddle Towers, which have total intrinsic curvature 4 pi(1-k), genus zero and 2k vertical Scherk-type ends in the quotient by T. As limits of those Saddle Towers, we obtain J…
Study on connection points on double regular polygons, providing coordinates and proving non-connection points.
problem Identifying connection points on double regular polygons.
method Examined coordinates in trace field, provided constructive proof for prime n. result For n=7, conjectured all remaining points are connection points; for n≥7 prime, provided explicit separatrix. Configurations of rigid collections of saddle connections are connected component invariants for strata of the moduli space of quadratic differentials. They have been classified for strata of Abelian differentials by Eskin, Masur and Zorich. Similar work for strata of quadratic differentials has been done in Masur and …
We establish that first-order methods avoid saddle points for almost all initializations. Our results apply to a wide variety of first-order methods, including gradient descent, block coordinate descent, mirror descent and variants thereof. The connecting thread is that such algorithms can be studied from a dynamical s…
The paper proves geometric properties of square tables and saddle surfaces.
problem The mathematical table problem from a geometric-topological perspective.
method Geometric-topological proofs on cylinder, saddle surfaces, and level sets of Fenn graphs.
result Zero-existence theorem on a cylinder, proving Fenn's square-table theorem under different boundary conditions.
The paper classifies surfaces formed by quadrilateral gluings.
problem Classifying topological surfaces formed by quadrilateral gluings.
method Review of graphs embedded into surfaces, algorithms based on labeling schemes of fundamental polygons.
result Computing numbers of possible gluings for classification.
We explicitly compute the limiting gap distribution for slopes of saddle connections on the flat surface associated to the regular octagon with opposite sides identified. This is the first such computation where the Veech group of the translation surface has multiple cusps. We also show how to parametrize a Poincaré se…
We construct most symmetric Saddle towers in Heisenberg space i.e. periodic minimal surfaces that can be seen as the desingularization of vertical planes intersecting equiangularly. The key point is the construction of a suitable barrier to ensure the convergence of a family of bounded minimal disks. Such a barrier is …
Graph-based framework for provably robust adversarial training.
problem Adversarial robustness of machine learning models.
method Formulates adversarial robustness as loss minimization with a Lipschitz constraint, using graph-based discretization and primal-dual algorithms.
result Establishes a connection between elliptic operators and adversarial learning, and proves fundamental lower bounds on adversarial sensitivity.
SGD learns neural networks with a complexity measure called leap.
problem Time complexity of SGD learning on neural networks.
method Introduced a complexity measure called leap, proved conjecture for Gaussian data, and showed saddle-to-saddle dynamics.
result Proved a conjecture about the time complexity of learning functions with low-dimensional support.
Study describes bifurcations of gradient flows on 2-sphere with holes.
problem Analyzing gradient flows on a 2-sphere with up to six singular points.
method Using separatrix diagrams to specify saddle-node and saddle connections.
result Identified all possible topological structures of bifurcations.
The alternating knots, links and twists projected on the S2 sphere were identified with the phase space of a Hamiltonian dynamic system of one degree of freedom. The saddles of the system correspond to the crossings, the edges correspond to the stable and unstable manifolds connecting the saddles. Each face is then …
The study of Platonic solids' unfoldings leads to high genus Teichmüller curves.
problem Understanding the topology and geometry of Teichmüller curves from Platonic solids.
method Computing Teichmüller curves using lattice surfaces and algorithmic approaches.
result The Teichmüller curve of the unfolded dodecahedron has genus 131 with specific singularities and cusps.
Study calculates slope gaps on polygon surfaces, finding non-unimodal distributions.
problem Understanding the distribution of slope gaps on polygon surfaces.
method Explicit computation of slope gap distributions for 2n-gons, providing bounds on non-differentiability points.
result Slope gap distributions are not always unimodal, answering a question by Athreya.
A new Lagrangian method for graph neural networks accelerates state computation.
problem Efficiently computing states in graph neural networks for complex data.
method Lagrangian optimization for state convergence in graph neural networks.
result The proposed method accelerates state computation without iterative phases.
Proposes a new method for GNNs that avoids iterative node state convergence.
problem Iterative computation of node states in GNNs is inefficient and requires many epochs.
method Constrained optimization in the Lagrangian framework to learn transition function and node states simultaneously.
result The proposed method compares favorably with existing models on various benchmarks.
A distributed optimization method solves saddle point problems with strong concavity and convexity.
problem Solving saddle point problems with distributed and heterogeneous data.
method GT-GDA, a distributed first-order method using gradient tracking and consensus over coupling matrices.
result GT-GDA converges linearly to the unique saddle point solution under specific conditions.