Related papers: Numerical approximations to extremal metrics on to…
The paper develops a continiuty method for solutions of the Abreu equation, which include extremal metrics on toric surfaces. Results are obtained, assuming a hypothesis (the "M-condition") on the solutions.
This paper consists of real and complex affine techniques for studying the Abreu equation on toric surfaces. In particular, an interior estimate for Ricci tensor is given.
We develop an algorithm computing the transcendental lattice and the Mordell--Weil group of an extremal elliptic surface. As an example, we compute the lattices of four exponentially large series of surfaces
This paper completes a programme to determine which toric surfaces admit Kahler metrics of constant scalar curvature/
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to…
The paper is devoted to some extremal problems, related to convex polygons in the Euclidean plane and their perimeters. We present a number of results that have simple formulations, but rather intricate proofs. Related and still unsolved…
In this article a new upper bounds for the multiple trigonometrical integrals are found. The method of the work based on a new method of estimation for the areas of algebraic surfaces.
We give a qualitative description of extremals for Morrey's inequality. Our theory is based on exploiting the invariances of this inequality, studying the equation satisfied by extremals and the observation that extremals are optimal for a…
In a previous work of the authors, a result to algorithmically compute the topology types of the level curves of an algebraic surface, is given. From this result, here we derive applications based on level curves to determine some…
The paper considers estimates for some sums and products of functions of prime numbers. Several assertions on this topic have been proven. We also study extremal estimates for strongly additive and strongly multiplicative arithmetic…
The aim of this paper is to highlight recent progress in using conic optimization methods to study geometric packing problems. We will look at four geometric packing problems of different kinds: two on the unit sphere -- the kissing number…
This work describes numerical methods that are useful in many areas: examples include statistical modelling (bioinformatics, computational biology), theoretical physics, and even pure mathematics. The methods are primarily useful for the…
Let $X$ be a compact Riemann surface of genus $\geq 2$ of constant negative curvature -1. An extremal disk is an embedded (resp. covering) disk of maximal (resp. minimal) radius. A surface containing an extremal disk is an {\em extremal…
Extremal compact hyperbolic surfaces contain a packing of discs of the largest possible radius permitted by the topology of the surface. It is well known that arithmetic conditions on the uniformizing group are necessary for the existence…
This article focuses on the study of toric algebraic statistical models which correspond to toric Del Pezzo surfaces with Du Val singularities. A closed-form for the Maximum Likelihood Estimate of algebraic statistical models which…
We derive a sharp cusp count for finite volume complex hyperbolic surfaces which admit smooth toroidal compactifications. We use this result, and the techniques developed in [DiC12], to study the geometry of cusped complex hyperbolic…
We give two examples of surfaces with canonical map of degree 4 onto a canonical surface.
We discuss conformal metrics of curvature 1 on tori and on the sphere, with four conic singularities whose angles are multiples of pi/2. Besides some general results we study in detail the family of such symmetric metrics on the sphere,…
We present a notion of a random toric surface modeled on a notion of a random graph. We then study some threshold phenomena related to the smoothness of the resulting surfaces.
This paper surveys some selected topics in the theory of conformal metrics and their connections to complex analysis, partial differential equations and conformal differential geometry.