Related papers: Dispersion points and rational curves
Analytic curves are classified w.r.t. their symmetry under a regular and separately analytic Lie group action on an analytic manifold. We show that an analytic curve is either exponential or splits into countably many analytic immersive…
We prove two general decomposition theorems for fixed-point invariants: one for the Lefschetz number and one for the Reidemeister trace. These theorems imply the familiar additivity results for these invariants. Moreover, the proofs of…
We describe a method to show a plane quartic over a number field has no rational points. The method can be adapted to show that a curve does not have divisors of degree 1 or 2 and can be generalized to arbitrary smooth projective curves.…
In the presence of a positive, compactly supported measure on an affine algebraic curve, we relate the density of polynomials in Lebesgue $L^2$-space to the existence of analytic bounded point evaluations. Analogues to the complex plane…
We give necessary and sufficient topological conditions for a simple closed curve on a real rational surface to be approximable by smooth rational curves. We also study approximation by smooth rational curves with given complex…
We initiate the study of inflection curves of rational vector fields on the Riemann sphere. For a rational vector field $v_R=-R(z)\frac{\partial}{\partial z}, \qquad R(z)=\frac{Q(z)}{P(z)} $ we define its affine regular inflection locus by…
This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarnik type theorems for submanifolds of R^n. These problems have attracted a lot of interest since Kleinbock and…
We look at the elliptic curve E(q), where q is a fixed rational number. A point (p,r) on E(q) is called a rational point if both p and r are rational numbers. We introduce the concept of conjugate points and show that not both can be…
We study the subsets of metric spaces that are negligible for the infimal length of connecting curves; such sets are called metrically removable. In particular, we show that every totally disconnected set with finite Hausdorff measure of…
Two discrete dynamical systems are discussed and analyzed whose trajectories encode significant explicit information about a number of problems in combinatorial probability, including graphical enumeration on Riemann surfaces and random…
By considering mirror symmetry applied to conformal field theories corresponding to strings propagating in quintic hypersurfaces in projective 4-space, Candelas, de la Ossa, Green and Parkes calculated the ``number of rational curves on the…
The motivating problem addressed by this paper is to describe those non-degenerate sets of points $Z$ in $\mathbb P^3$ whose general projection to a general plane is a complete intersection of curves in that plane. One large class of such…
This article surveys the mathematics of the cut and project method as applied to point sets, called here {\em model sets}. It covers the geometric, arithmetic, and analytical sides of this theory as well as diffraction and the connection…
We study intersections of projective convex sets in the sense of Steinitz. In a projective space, an intersection of a nonempty family of convex sets splits into multiple connected components each of which is a convex set. Hence, such an…
Many questions about triangles and quadrilaterals with rational sides, diagonals and areas can be reduced to solving certain Diophantine equations. We look at a number of such questions including the question of approximating arbitrary…
We investigate the relationship between measurable differentiable structures on doubling metric measure spaces and derivations. We prove: [1] a decomposition theorem for the module of derivations into free modules; [2] the existence of a…
We show that the points that converge to infinity under iteration of the exponential map form a connected subset of the complex plane.
We prove that the focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the…
We construct an irreducible rational curve of degree 10 in $CP^2$ which has 12 triple points and a union of three rational quartics with 19 triple points. This gives counter-examples to a conjecture by Dimca, Harbourne, and Sticlaru. We…
A system of plane curves defined by prescribing n points of multiplicity m in general position is regular if n > (2m)^2. The proof uses computation of limits of linear systems acquiring fixed divisors, an interesting problem in itself.