arXiv research

The paper extends deformation theory for curves of fixed degree in graded manifolds.

The paper studies submanifolds of fixed degree with constraints on variations.

Study conic line arrangements of degree 7, finding their topology and connected components.

Complex manifold describes solvable Pell-Abel equations with fixed degrees.

We provide linear lower bounds for $f_ρ(L)$, the smallest integer so that every curve on a fixed hyperbolic surface $(S,ρ)$ of length at most $L$ lifts to a simple curve on a cover of degree at most $f_ρ(L)$. This bound is independent of hyperbolic structure $ρ$, and improves on a recent bound of Gupta-Kapovich. When $…

We consider a regular embedded network composed by two curves, one of them closed, in a convex domain $Ω$. The two curves meet only in one point, forming angle of $120$ degrees. The non-closed curve has a fixed end point on $\partialΩ$. We study the evolution by curvature of this network. We show that the maximal exist…

Proves cup product homomorphism for bounded cohomology on negatively curved manifolds.

Research examines curves of degree 8 with specific singularities.

In this paper, we prove that the tangent bundle of the moduli space $\cSU_C(r,d)$ of stable bundles of rank $r>2$ and of fixed determinant of degree $d$ (such that $(r,d)=1$), on a smooth projective curve $C$ is always stable, in the sense of Mumford-Takemoto. This verifies a well-known conjecture, and is related to a …

Moduli spaces of semi-stable real and quaternionic vector bundles of a fixed topological type admit a presentation as Lagrangian quotients, and can be embedded into the symplectic quotient corresponding to the moduli variety of semi-stable holomorphic vector bundles of fixed rank and degree on a smooth complex projecti…

This paper is motivated by the real symplectic isotopy problem : does there exists a nonsingular real pseudoholomorphic curve not isotopic in the projective plane to any real algebraic curve of the same degree? Here, we focus our study on symmetric real curves on the projective plane. We give a classification of real s…

New inequality for odd-degree flexible curves using surface doubling.

The study constructs balanced and rigid curves on specific types of hypersurfaces and complete intersections.

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)…

Consider genus $g$ curves that admit degree $d$ covers to elliptic curves only branched at one point with a fixed ramification type. The locus of such covers forms a one parameter family $Y$ that naturally maps into the moduli space of stable genus $g$ curves $\bar{\mathcal M}_{g}$. We study the geometry of $Y$, and pr…

The paper adapts differential signatures to algebraic curves under group actions.

New hexagonal circular 3-webs with reducible curves classified.

New algebraic theory classifies symplectic curves in complex projective space.

Classifies real rational knots and curves in a specific quadric space.

We study the degree of polynomial representations of knots. We obtain the lexicographic degree for two-bridge torus knots and generalized twist knots. The proof uses the braid theoretical method developed by Orevkov to study real plane curves, combined with previous results from [KP10] and [BKP14]. We also give a sharp…

We show that $\mathbb{Q}$-Fano varieties of fixed dimension with anti-canonical degrees and alpha-invariants bounded from below form a bounded family. As a corollary, K-semistable $\mathbb{Q}$-Fano varieties of fixed dimension with anti-canonical degrees bounded from below form a bounded family.

Curves with constant torsion can be deformed arbitrarily.

Classifies curves up to symplectic isotopy.

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…

The degree of certain holomorphic 2-spheres is bounded.

A limaçon-like curve, allowing 2π-transition with monotone curvature between concentric curvature elements, is presented. The curve is 4th degree algebraic, 4th degree rational, and shares other common features with Pascal's limaçon.

The space of all immersed closed curves of rotation degree 0 in the plane modulo reparametrizations has the same homotopy groups as the circle times the 2-sphere.

The study finds lower bounds for the warping degree of a knot projection.

Given a null-homologous knot $K$ in a rational homology 3-sphere $M$, and the standard infinite cyclic covering $\tilde{X}$ of $(M,K)$, we define an invariant of triples of curves in $\tilde{X}$, by means of equivariant triple intersections of surfaces. We prove that this invariant provides a map $φ$ on $\Al^{\otimes 3…

We use invariants of Hendricks and Manolescu coming from involutive Heegaard Floer theory to find constraints on possible configurations of singular points of a rational cuspidal curve of odd degree in the projective plane. We show that the results do not carry over to rational cuspidal curves of even degree.

Study shows double descent curve in high-dimensional linear regression with random projections.

This paper constructs PH spline curves with prescribed arc lengths.

Formula derived for Gromov-Witten invariants of smooth curves.

We apply Murasugi-Tristram inequality to real algebraic curves of odd degree on $RP^2$ with a deep nest, i.e. a nest of the depth $k-1$ where $2k+1$ is the degree. For such curves, the ingredients of the Murasugi-Tristram inequality can be computed (or estimated) inductively using the computations for iterated torus li…

The paper gives topological as well as rigid isotopy classification of smooth irreducible algebraic curves in the real projective 3-space for the case when the degree of the curve is at most six and its genus is at most one.

Margalit and Schleimer constructed nontrivial roots of the Dehn twist about a nonseparating curve. We prove that the conjugacy classes of roots of the Dehn twist about a nonseparating curve correspond to the conjugacy classes of periodic maps with certain conditions. Futhermore, we give data set which determine the con…

Study shows connections between Jacobian torsors and Fermat curves.

This paper explores constantly curved holomorphic 2-spheres in complex Grassmannian and confirms their rarity.

Model shows loss curve with two distinct exponents due to sparse activations.

Paper proves unique energy-minimizing curves in constrained spaces.

We give bounds on the gap functions of the singularities of a cuspidal plane curve of arbitrary genus, generalising recent work of Borodzik and Livingston. We apply these inequalities to unicuspidal curves whose singularity has one Puiseux pair: we prove two identities tying the parameters of the singularity, the genus…

Study extends graph approach to elastic curves with fixed ends.

The paper establishes a correspondence between Higgs torsors and connections on curves.

Consider a flat bundle over a complex curve. We prove a conjecture of Fei Yu that the sum of the top k Lyapunov exponents of the flat bundle is always greater or equal to the degree of any rank k holomorphic subbundle. We generalize the original context from Teichmueller curves to any local system over a curve with non…

A new FFT-based method for fast rigid alignment of 2D closed curves.

Consider a Riemann surface $X$ of genus $g \geq 2$ equipped with an antiholomorphic involution $τ$. This induces a natural involution on the moduli space $M(r,d)$ of semistable Higgs bundles of rank $r$ and degree $d$. If $D$ is a divisor such that $τ(D) = D$, this restricts to an involution on the moduli space $M(r,D)…

Study Lyapunov exponents on curves, relating them to holomorphic bundles and moduli spaces.

Rough and Hodge Laplacians eigenvalues approach zero with fixed volume.