The paper characterizes coverings over the projective plane with minimal defect.
problem Characterizing minimal defect branched coverings over the projective plane.
method Characterization through properties of decomposable and indecomposable coverings.
result Extended family of realizations and generalized results on primitive permutation groups.
Simply-connected surfaces of general type for n≥5.
problem Topological structures of Galois covers of surfaces of minimal degree.
method Investigation of Galois covers of surfaces of minimal degree in complex projective space.
result Galois covers of surfaces of minimal degree are simply-connected for n≥5.
Researchers compute covering type of all closed surfaces.
problem Measuring the complexity of closed surfaces using covering type.
method Using the concept of covering type introduced by Karoubi and Weibel, the researchers computed the minimum number of vertices in simplicial complexes homotopy equivalent to closed surfaces.
result Results completely settle a problem posed by Karoubi and Weibel, and provide insights into the relationship between surface topology and minimal triangulations.
Minimal example found for two finite CW-complexes sharing a common covering.
problem Finding the minimum number of 2-cells in two finite CW-complexes that share a common covering.
method Constructing an almost minimal example with two 2-cells in each complex.
result Minimal example with two 2-cells in each complex.
Minimal number of caps covering an n-sphere is n+2.
problem Covering an n-sphere with minimal number of caps.
method Analyzing short closed sets and caps, proving theorems about their intersections and coverings.
result The minimal number of short closed sets (caps) covering an n-sphere is n+2.
Study horocycle orbits in Z-covers of hyperbolic surfaces.
problem Classify horocycle orbit closures in Z-covers of compact hyperbolic surfaces. method Careful analysis of distance minimizing geodesic rays in the cover.
result All non-maximal horocycle orbit closures have integer Hausdorff dimension.
Survey of universal portfolio techniques for minimizing investment regret.
problem Minimizing investment regret in algorithmic trading.
method Explains various universal portfolio techniques and their proofs.
result Coverage of fundamental concepts and algorithms in regret minimization.
We show that the conjectural cusped complex hyperbolic 2-orbifolds of minimal volume are the two smallest arithmetic complex hyperbolic 2-orbifolds. We then show that every arithmetic cusped complex hyperbolic 2-manifold of minimal volume covers one of these two orbifolds. We also give all minimal volume manifolds that…
Study of fundamental groups of 3D small covers using Morse theory.
problem Understanding the fundamental groups of 3D small covers.
method Morse-theoretic approach to get explicit, balanced presentations of fundamental groups.
result Explicit, balanced presentations of fundamental groups with minimal generators and minimal Heegaard splittings.
The study connects lamination and orbit closures in hyperbolic manifolds.
problem Understanding the geometric and dynamical properties of horocycle orbit closures in Z-covers of compact hyperbolic manifolds. method Exposes connections between distance minimizing laminations and horospherical orbit closures in Z-covers of compact hyperbolic manifolds. Provides novel constructions and explicit descriptions. result Even slight perturbations to hyperbolic metrics can drastically change horocycle orbit closures.
Estimates open sets for fibrations, leading to volume vanishing results.
problem Estimating open sets for fibrations.
method Straightforward estimate for open sets with fundamental group constraints.
result Vanishing results for simplicial volume and minimal volume entropy for certain mapping tori.
Study on minimal foliations in 3D manifolds with specific conditions.
problem Characterizing minimal foliations in 3D manifolds.
method Analyzing Anosov foliations and their intersections.
result Necessary and sufficient conditions for orbit foliation of Anosov flows.
In this article we determine, for an infinite family of maps on the plane, the topology of the surface on which the minimal regular covering occurs. This infinite family includes all Archimedean maps.
Small covers were introduced by Davis and Januszkiewicz in 1991. We introduce the notion of equilibrium triangulations for small covers. We study equilibrium and vertex minimal Z22-equivariant triangulations of 2-dimensional small covers. We discuss vertex minimal equilibrium triangulations of $\mathbb{R…
This paper uses results on the classification of minimal triangulations of 3-manifolds to produce additional results, using covering spaces. Using previous work on minimal triangulations of lens spaces, it is shown that the lens space L(4k,2k−1) and the generalised quaternionic space S3/Q4k have complexity $k,…
The paper proves properties of minimal surfaces and Heegaard splittings in 3-manifolds.
problem Characterizing minimal surfaces and Heegaard splittings in 3-manifolds.
method Analyzes strongly irreducible Heegaard splittings and proves properties of minimal surfaces.
result Strongly irreducible Heegaard splittings are either isotopic to minimal surfaces of index at most one or to non-orientable minimal surfaces with a vertical handle.
For Finsler metrics (no reversibility assumed) on closed orientable surfaces of genus greater than one, we study the dynamics of minimal rays and minimal geodesics in the universal cover. We prove in particular, that for almost all asymptotic directions the minimal rays with these directions laminate the universal cove…
Estimates the growth of Morse index for free boundary minimal hypersurfaces.
problem Estimating the Morse index of free boundary minimal hypersurfaces.
method Adapting Song's method to the free boundary case.
result The Morse index grows linearly with the sum of Betti numbers.
Hyperbolic surface bundles over the circle have minimal entropy close to 1/g.
problem Understanding the minimal entropy of hyperbolic surface bundles over the circle.
method Proved the minimal entropy behavior of hyperbolic surface bundles as branched double covers of the 3-sphere.
result Minimal entropy of hyperbolic surface bundles over the circle behaves like 1/g.
Let M be a smooth 4-manifold which admits a relatively minimal hyperelliptic genus h Lefschetz fibration over the 2-sphere. If all of the vanishing cycles for this fibration are nonseparating curves, then we show that M is a 2-fold cover of a 2-sphere bundle over the 2-sphere, branched over an embedded surface. If the …
Minimal action on universal circle for foliations on 3-manifolds.
problem Action of fundamental group on universal circle of foliations.
method Analyzes uniform foliations on 3-manifolds, proving minimality and transitivity.
result Action is minimal and transitive on pairs of different points.
Unique minimal model for LCK manifolds proved.
problem Characterizing unique minimal models for LCK manifolds.
method Proving bimeromorphic maps are holomorphic.
result LCK manifolds have a unique minimal model.
Paper shows a surface with positive genus under a tetrahedron, less than cone's area.
problem Existence of area-minimizing surfaces with positive genus spanning a tetrahedron.
method Triple cover approach, using BV functions and Caccioppoli sets.
result Found a surface of positive genus with area less than a conic surface.
Study on Gauss map of anisotropic minimal surfaces with Morse index estimates.
problem Estimating the Morse index of anisotropic minimal surfaces.
method Local analysis of Gauss map, conformal geometric techniques applied to the Gauss map.
result Upper and lower estimates for the Morse index of anisotropic minimal surfaces.
New algorithm finds minimal causal models for complex latent variables.
problem Learning causal structure in presence of latent variables and measurement dependencies.
method Graph theoretic edge clique cover problem, non-parametric algorithm.
result Minimality in minimal causal models implies specific properties.
Minimal constructions of meanders and hyperelliptic pillowcase covers help in understanding ratio-optimizing pseudo-Anosovs.
problem Understanding ratio-optimizing pseudo-Anosovs in moduli spaces of quadratic differentials.
method Minimal constructions of meanders and hyperelliptic pillowcase covers.
result Existence of ratio-optimizing pseudo-Anosovs deep in the Johnson filtration.
The paper proves the existence of certain minimal surfaces in specific manifolds.
problem Existence of minimal surfaces with specific properties in Riemannian manifolds.
method Proof of existence using Morse index, bumpy metrics, and cyclic coverings.
result Connected, immersed Morse index one, closed minimal hypersurfaces with unbounded volumes.
A smooth five-dimensional s-cobordism becomes a smooth product if stabilized by a finite number n of S2xS2x[0,1]'s. We show that for amenable fundamental groups, the minimal n is subextensive in covers, i.e., n(cover)/index(cover) has limit 0. We focus on the notion of sweepout width, which is a bridge between 4-di…
Constructs minimal laminations with controlled leaf topologies.
problem Creating minimal laminations with specific surface topologies.
method Using towers of finite coverings and developing a relative version of residual finiteness.
result Established finite covers with control on the second systole.
New minimal surfaces found via covering spaces for complex boundaries.
problem Finding minimal surfaces for complex boundary configurations.
method Constructing minimal surfaces via covering spaces for proper subgroups of the fundamental group.
result Existence of a proper normal subgroup N0 such that the area of the minimal surface ΣN0 is the infimum of all other minimal surfaces. Study eigenvalues of knot covers to bound knot properties.
problem Bounding knot genera, dealternating number, and alternation number.
method Minimal positive eigenvalues of double branched covers over spanning surfaces.
result Eigenvalue bounds provide estimates for knot properties.
Simple analysis for fast rates in empirical minimization with concave losses and convex regularization.
problem Fast rates in empirical minimization with concave losses and convex regularization.
method Simple analysis using covering number and concentration inequality.
result First result of fast rates with high probability for exponential concave empirical risk minimization.
This paper uses Brin and Thickstun's theory of end reductions of non-compact 3-manifolds to study groups of covering translations of irreducible contractible open 3-manifolds W which are not homeomorphic to R^3. We associate to W an object S(W) called the simplicial complex of minimal R^2-irreducible end reductions of …
New orbits found in Lagrangian systems on surfaces.
problem Finding action minimizing periodic orbits in Tonelli Lagrangian systems.
method Analyzing minimal boundaries and using graph theorems.
result Existence of action minimizing simple periodic orbits.
The paper studies entropy in branched covers of 3-manifolds.
problem Entropy of pseudo-Anosov monodromies in fibered 3-manifolds.
method Analyzes the minimal entropy of genus g surface bundles over the circle as 2-fold branched covers of 3-manifolds.
result There exist infinitely many 3-manifolds with comparable minimal entropy to 1/g.
The study examines stable minimal surfaces in higher dimensions and provides bounds on their properties.
problem Properties of stable minimal surfaces in higher codimension.
method Structural analysis of holomorphic vector bundles and geometric inequalities.
result Explicit bounds on the systole for stable minimal tori and surfaces.
This paper classifies periodic weaves and their universal cover, extending Tait's conjectures.
problem Classifying periodic weaves and their universal cover in thickened surfaces.
method Introducing hyperbolic periodic weaves, extending Tait's conjectures, and using a generalized Kauffman bracket polynomial.
result Tait's conjectures are extended to minimal reduced alternating weaving motifs.
We give a sharp upper bound for the area of a minimal two-sphere in a three-manifold (M,g) with positive scalar curvature. If equality holds, we show that the universal cover of (M,g) is isometric to a cylinder.
The elastic trefoil is the twice covered circle, a key finding in knot elasticity.
problem Characterizing the elastic behavior of knotted loops of springy wire.
method Minimizing bending energy and ropelength to penalize self-intersection.
result The elastic trefoil is the twice covered circle, not the round circle.
For most metrics, many minimal hypersurfaces densely cover a manifold.
problem Finding many minimal hypersurfaces in generic metrics.
method Analyzing Riemannian metrics on closed manifolds.
result The union of all closed, smooth, embedded minimal hypersurfaces is dense for generic metrics.
Minimal genus surfaces solve homology problems in finite complexes.
problem Finding surface representatives of homology classes with minimal genus.
method Minimizing genus and Euler characteristic, analyzing surgeries and homotopy.
result Minimizers are homotopic to cellwise coverings, but problem is undecidable in general.
Algorithm calculates genus of embedded graphs on surfaces.
problem Finding minimal and maximal genus for graph embeddings.
method Constructing special branched coverings of the 2-sphere.
result Algorithm calculates orientable genus and maximal genus of graphs.
Improved cover detection in music datasets with novel triplet loss.
problem Challenging task of automatically detecting covers in audio datasets.
method Convolutional neural network mapping melodic features to embeddings, training to minimize cover distance and maximize non-cover distance.
result New prototypical triplet loss improves accuracy for large datasets and live songs.
Study minimal annuli in a slab, estimating their area.
problem Estimating the area of minimal annuli in a slab.
method Organized minimal annuli based on winding number, deduced convexity of length function, compared to catenoid waist area.
result Deduced convexity of length function and estimated area of minimal annuli.
In this paper we study the covering numbers of the space of convex and uniformly bounded functions in multi-dimension. We find optimal upper and lower bounds for the ε-covering number of $\C([a, b]^d, B)$, in the Lp-metric, 1≤p<∞, in terms of the relevant constants, where d≥1, $a < b \in \mathb…
The paper constructs non-abelian covers for knots with non-trivial Alexander polynomials.
problem Finding minimal non-cyclic covers of knots with non-trivial Alexander polynomials.
method Using covering space theory and Alexander stratifications, the paper constructs finite non-abelian representations and bounds the order of the group.
result The first Betti number of the cover can be arbitrarily large, and it contains non-peripheral homology.
This paper shows semi-equivelar toroidal maps are vertex-transitive covers.
problem Understanding the relationship between semi-equivelar and vertex-transitive toroidal maps.
method Proving semi-equivelar toroidal maps are quotients of vertex-transitive toroidal maps.
result Each semi-equivelar toroidal map has a finite vertex-transitive cover.
Improved lower bound for geodesics on manifolds.
problem Finding a lower bound for the number of minimal geodesics on Riemannian manifolds.
method Refined Bangert's method using the stable norm unit ball on the first homology.
result Quadratic lower bound for the number of minimal geodesics.