arXiv research

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

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

Formula counts rational curves with a specific singular point in projective space.

In this paper we classify, up to rigid isotopy, non-singular real rational curves of degrees less than or equal to 6 in a quadric homeomorphic to the 3-sphere. We also study their connections with rigid isotopy classes of real rational knots in $\mathbb{RP}^3$.

These are notes on some algebraic geometry of complex projective curves, together with an application to studying the contact curves in CP^3 and the null curves in the complex quadric Q^3 in CP^4, related by the well-known Klein correspondence. Most of this note consists of recounting the classical background. The main…

In this paper, we analyze the diversity of term structure functions (e.g., yield curves, swap curves, credit curves) constructed in a process which complies with some admissible properties: arbitrage-freeness, ability to fit market quotes and a certain degree of smooth- ness. When present values of building instruments…

Classifies low energy maps from curved surfaces into spheres.

Research examines curves of degree 8 with specific singularities.

Classifies low-energy harmonic maps from curved surfaces to spheres.

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.

Low-degree method fails to predict robust subspace recovery problem.

Survey on using low-degree polynomials to assess statistical tasks complexity.

In this paper, we adapt the differential signature construction to the equivalence problem for complex plane algebraic curves under the actions of the projective group and its subgroups. Given an action of a group $G$, a signature map assigns to a plane algebraic curve another plane algebraic curve (a signature curve) …

New findings on computational limits for estimating hidden structures.

Paper connects free-energy and low-degree hardness in high-dimensional statistics.

New hexagonal circular 3-webs with reducible curves classified.

New algebraic theory classifies symplectic curves in complex projective space.

Study disproves conjecture about low-degree polynomials in hypothesis testing.

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…

Learn low-degree functions with few random queries.

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…

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 degree of certain holomorphic 2-spheres is bounded.

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.

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.

We consider a length functional for $C^1$ curves of fixed degree in graded manifolds equipped with a Riemannian metric. The first variation of this length functional can be computed only if the curve can be deformed in a suitable sense, and this condition is expressed via a differential equation along the curve. In the…

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.

Sublinear algorithms detect cliques in graphs with high probability.

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.

New work shows FP potential monotonicity equals low-degree polynomial estimators limits.

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…

These notes survey and explore an emerging method, which we call the low-degree method, for predicting and understanding statistical-versus-computational tradeoffs in high-dimensional inference problems. In short, the method posits that a certain quantity -- the second moment of the low-degree likelihood ratio -- gives…

Study rational homology of moduli space via Morse functions, proving stability phenomena.

Detects dense subhypergraphs in random hypergraphs using low-degree polynomials.

New method explains computational barriers in high-dimensional statistical models.

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

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…

GCNs favor high-degree nodes, leading to biased performance; a new method mitigates this.

We discuss hypercomplex and hyperkähler structures obtained from higher degree curves in complex spaces fibring over ${\mathbb{P}}^1$.

Study pseudo-hyperkähler geometry of curves in hyperkähler twistor spaces.