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 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 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.
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 systoles in translation surfaces, finding maxima and saddle connections.
problem Characterize maxima of systole function on translation surfaces.
method Define systole, characterize maxima, provide examples, study saddle connections.
result Found local but nonglobal maxima on strata of genus at least three.
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…
Translation surfaces' saddle connections equidistribute over time.
problem Equidistribution of saddle connections on translation surfaces.
method Using ergodic theory and directional flow, proving the existence of weak limits and their equivalence to Lebesgue measure.
result The weak limit of measures on saddle connections converges to the Lebesgue measure as the connection length increases.
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.
First-order methods avoid saddle points for most initializations.
problem Avoiding saddle points in optimization problems.
method First-order methods, including gradient descent and variants, analyzed using dynamical systems and the Stable Manifold Theorem.
result First-order methods avoid saddle points for almost all initializations.
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.
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.
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.
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.
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 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…
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 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.
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…
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. 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.
The paper calculates gap distributions for translation surfaces, focusing on the double heptagon.
problem Calculating gap distributions for translation surfaces.
method Describes a procedure to find winning holonomy vectors and applies it to the double heptagon.
result Explicitly computed gap distribution for the regular double heptagon translation surface.
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.
The Schoen-Yau theorem is extended to saddle maps.
problem Univalentness of saddle maps between discs.
method Proof of Schoen-Yau theorem for saddle maps.
result Analog of Schoen-Yau theorem proven for saddle maps.
Deep ReLU networks escape from the origin via saddle points with a low-rank bias.
problem Understanding the dynamics of gradient descent in deep ReLU networks.
method Analysis of escape directions and singular values of weight matrices.
result The first singular value of the ℓ-th layer weight matrix is at least ℓ41 larger than any other singular value. DLNs dynamics change with variance, leading to saddle-to-saddle training phases.
problem Understanding the dynamics of DLNs with varying initialization variance.
method Analyzing the phase transition of DLNs' dynamics as variance changes.
result Gradient descent visits a sequence of saddles, reaching a sparse global minimum.
Study flows with isolated non-saddle sets and their region of influence.
problem Understanding the complexity of isolated non-saddle sets in flows.
method Cohomological analysis and construction of flows with prescribed structures.
result Construct flows with specific structures for their region of influence.
Study geometric properties of loss functions to understand neural network performance.
problem Understanding the geometric properties of high-dimensional loss functions to improve neural network performance.
method Combine concepts from high-dimensional probability and differential geometry to study curvature properties in lower-dimensional loss representations.
result Mean curvature in the original loss space determines if saddle points appear as minima, maxima, or flat regions.
Study of holonomy behavior along differential rays on Riemann surfaces.
problem Analyzing the asymptotic behavior of holonomy along differential rays on Riemann surfaces.
method Investigates the asymptotic formulas for holonomy eigenvalues and growth rates using local fourth roots of the differential.
result Explicit asymptotic formulas for holonomy eigenvalues and their growth rates are derived.
Decentralized learning for GLMs with feature distribution and network connectivity.
problem Optimizing generalized linear models in a decentralized network with feature partitioning.
method Chambolle--Pock primal--dual algorithm applied to an equivalent saddle-point formulation.
result Convergence rates for empirical risk minimization under Lipschitz and square root Lipschitz assumptions.
Paper analyzes algorithms for nonstationary saddle-point optimization problems.
problem Nonstationary saddle-point optimization problems in game theory, reinforcement learning, and machine learning.
method Proposes extragradient and Frank-Wolfe algorithms for online and bandit settings.
result Establishes sub-linear regret bounds for the proposed algorithms.
Algorithm classifies saddle-focus singularities in Hamiltonian systems.
problem Classifying nondegenerate saddle-focus singularities in integrable Hamiltonian systems.
method Developed an algorithm based on semi-local equivalence to represent singularities as almost direct products.
result Obtained complete lists of saddle-focus singularities of complexities 1, 2, and 3.
Proximal methods avoid local minima in weakly convex problems.
problem Weakly convex optimization problems with strict saddle properties.
method Proximal methods on nonsmooth functions with strict saddle guarantees.
result Proximal methods converge to local minimizers only, when initialized randomly.