arXiv research

Unique maximal curve systems found for up to 5 punctures.

Sparse curves on surfaces grow at a specific intermediate rate.

Study on loops on non-orientable surfaces, determining cardinality and order.

Minimal energy local systems on curves are compact components of character varieties.

Superintegrable systems on curved manifolds found to have Hessian structures.

In this paper we study the convergence behavior of grafting rays to the Thurston boundary of Teichmuller space. When the grafting is done along a weighted system of simple closed curves or along a maximal uniquely ergodic lamination this behavior is the same as for Teichmuller geodesics and lines of minima. We also sho…

The paper classifies maximal translation surfaces in Lorentz-Minkowski space.

Maximal causal curves for Lipschitz metrics are either lightlike or timelike.

Curved Frobenius manifolds link to Hessian metrics in geometry.

Torically maximal curves (known also as simple Harnack curves) are real algebraic curves in the projective plane such that their logarithmic Gauß map is totally real. In this paper we show that hyperplanes in projective spaces are the only torically maximal hypersurfaces of higher dimensions.

Consider a family of K3 surfaces over a hyperbolic curve (i.e. Riemann surface). Their second cohomology groups form a local system, and we show that its top Lyapunov exponent is a rational number. One proof uses the Kuga-Satake construction, which reduces the question to Hodge structures of weight 1. A second proof us…

New solutions to SU(n+1) Toda system found on compact Riemann surfaces with cone singularities.

For one-dimensional systems of conservation laws admitting two additional conservation laws we assign a ruled surface of codimension two in projective space. We call two such systems dual if the corresponding ruled surfaces are dual. We show that a Hamiltonian system is autodual, its ruled surface sits in some quadric,…

Reformulated sigma models for complex Grassmannians using Gross-Neveu formalism.

Study extremals on Lie groups with asymmetric polyhedral Finsler structures using Pontryagin's Maximal Principle.

We classify closed, simply-connected, non-negatively curved 6-manifolds of almost maximal symmetry rank up to equivariant diffeomorphism.

Monodromy and vanishing cycles computed for ample linear systems on simply connected surfaces.

Przytycki has shown that the size $\mathcal{N}_{k}(S)$ of a maximal collection of simple closed curves that pairwise intersect at most $k$ times on a topological surface $S$ grows at most as a polynomial in $|χ(S)|$ of degree $k^{2}+k+1$. In this paper, we narrow Przytycki's bounds by showing that $$ \mathcal{N}_{k}(S)…

Interpolates curves using maximal and minimal surfaces in different spaces.

Researchers solve a Plateau problem for maximal surfaces in pseudo-hyperbolic spaces.

The paper explores fully affine maximal curves and their properties.

We give a different formulation for describing maximal surfaces in Lorentz-Minkowski space, $\mathbb{L}^3$, using the identification of $\mathbb L^3$ with $\mathbb C\times \mathbb R$. Further we give a different proof for the singular Björling problem for the case of closed real analytic null curve. As an application, …

We show that for n>2 the following equivalence problems are essentially the same: the equivalence problem for Lagrangians of order n with one dependent and one independent variable considered up to a contact transformation, a multiplication by a nonzero constant, and modulo divergence; the equivalence problem for the s…

The symmetry-rank of a riemannian manifold is by definition the rank of its isometry group. We determine precisely which smooth closed manifolds admit a positively curved metric with maximal symmetry-rank.

Classifies special quartic curves up to equivalence.

Investor optimizes utility in a market with endogenous pricing.

The paper studies billiards in symmetric tables and finds a measure bound for maximizing orbits.

Maximizing energy on flexible curves yields regular or convex polygons.

Optimizes profit in targeted marketing across multiple markets with varying marketing expenditures.

New findings on hypersurfaces in Euclidean space that are both maximal and minimal.

This work studies learning curves for revenue maximization algorithms.

Consider an immersed Legendrian surface in the five dimensional complex projective space equipped with the standard homogeneous contact structure. We introduce a class of fourth order projective Legendrian deformation called \emph{$\,Ψ$-deformation}, and give a differential geometric characterization of surfaces admitt…

Generalizes symmetries of curved manifolds.

Study of trigonal curves in abelian differentials with specific divisor properties.

Extends Euler's problem to Lorentz-Minkowski plane.

We prove that Ricci flows with almost maximal extinction time must be nearly round, provided that they have positive isotropic curvature when crossed with $\mathbb{R}^{2}$. As an application, we show that positively curved metrics on $S^{3}$ and $RP^{3}$ with almost maximal width must be nearly round.

We prove a blow-up criterion in terms of an $L_2$-bound of the curvature for solutions to the curve diffusion flow if the maximal time of existence is finite. In our setting, we consider an evolving family of curves driven by curve diffusion flow, which has free boundary points supported on a line. The evolving curve h…

In this note we study the distribution of real inflection points among the ovals of a real non-singular hyperbolic curve of even degree. Using Hilbert's method we show that for any integers $d$ and $r$ such that $4\leq r \leq 2d^2-2d$, there is a non-singular hyperbolic curve of degree $2d$ in $\mathbb R^2$ with exactl…

Paper estimates optimal ROC curve arc length and AUC, improving classification performance.

Let $\mathcal{M}_{0}^n$ be the class of closed, simply-connected, non-negatively curved Riemannian manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if $M\in \mathcal{M}_{0}^n$, then $M$ is equivariantly diffeomorphic to the free linear quotient by a torus of a product of spheres…

The paper uses Tannakian reconstruction to understand hyperbolic log-orbi curves.

We present a new mixture model-based discriminant analysis approach for functional data using a specific hidden process regression model. The approach allows for fitting flexible curve-models to each class of complex-shaped curves presenting regime changes. The model parameters are learned by maximizing the observed-da…

This study examines abnormal geodesics in 2D-Zermelo navigation problems, revealing their role in separating time minimal and maximal curves.

The curvature of web curves is studied in 3D manifolds.

This paper introduces a novel mixture model-based approach for simultaneous clustering and optimal segmentation of functional data which are curves presenting regime changes. The proposed model consists in a finite mixture of piecewise polynomial regression models. Each piecewise polynomial regression model is associat…

This paper studies gradient flows in asymmetric metric spaces and proves existence results.

There is a well known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns-Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least three and irreducible topological spherical bui…

Maximizes filling systems on surfaces with given boundary components.