On some covering problems in geometry

We present a method to obtain upper bounds on covering numbers. As applications of this method, we reprove and generalize results of Rogers on economically covering Euclidean $n$-space with translates of a convex body, or more generally, any measurable set. We obtain a bound for the density of covering the $n$-sphere by rotated copies of a spherically convex set (or, any measurable set). Using the same method, we sharpen an estimate by Artstein--Avidan and Slomka on covering a bounded set by translates of another. The main novelty of our method is that it is not probabilistic. The key idea, which makes our proofs rather simple and uniform through different settings, is an algorithmic result of Lov\'asz and Stein.Comment: 9 pages. IMPORTANT CHANGE: In previous versions of the paper, the illumination problem was also considered, and I presented a construction of a body close to the Euclidean ball with high illumination number. Now, I removed this part from this manuscript and made it a separate paper, 'A Spiky Ball'. It can be found at http://arxiv.org/abs/1510.0078

Some Curvature Problems in Semi-Riemannian Geometry

In this survey article we review several results on the curvature of semi-Riemannian metrics which are motivated by the positive mass theorem. The main themes are estimates of the Riemann tensor of an asymptotically flat manifold and the construction of Lorentzian metrics which satisfy the dominant energy condition.Comment: 25 pages, LaTeX, 4 figure

Combinatorial problems in finite geometry and lacunary polynomials

We describe some combinatorial problems in finite projective planes and indicate how R\'edei's theory of lacunary polynomials can be applied to them

Statistical mechanics approach to some problems in conformal geometry

A weak law of large numbers is established for a sequence of systems of N classical point particles with logarithmic pair potential in \bbR^n, or \bbS^n, n\in \bbN, which are distributed according to the configurational microcanonical measure $\delta(E-H)$, or rather some regularization thereof, where H is the configurational Hamiltonian and E the configurational energy. When $N\to\infty$ with non-extensive energy scaling E=N^2 \vareps, the particle positions become i.i.d. according to a self-consistent Boltzmann distribution, respectively a superposition of such distributions. The self-consistency condition in n dimensions is some nonlinear elliptic PDE of order n (pseudo-PDE if n is odd) with an exponential nonlinearity. When n=2, this PDE is known in statistical mechanics as Poisson-Boltzmann equation, with applications to point vortices, 2D Coulomb and magnetized plasmas and gravitational systems. It is then also known in conformal differential geometry, where it is the central equation in Nirenberg's problem of prescribed Gaussian curvature. For constant Gauss curvature it becomes Liouville's equation, which also appears in two-dimensional so-called quantum Liouville gravity. The PDE for n=4 is Paneitz' equation, and while it is not known in statistical mechanics, it originated from a study of the conformal invariance of Maxwell's electromagnetism and has made its appearance in some recent model of four-dimensional quantum gravity. In differential geometry, the Paneitz equation and its higher order n generalizations have applications in the conformal geometry of n-manifolds, but no physical applications yet for general n. Interestingly, though, all the Paneitz equations have an interpretation in terms of statistical mechanics.Comment: 17 pages. To appear in Physica

A new geometric approach to problems in birational geometry

A classical set of birational invariants of a variety are its spaces of pluricanonical forms and some of their canonically defined subspaces. Each of these vector spaces admits a typical metric structure which is also birationally invariant. These vector spaces so metrized will be referred to as the pseudonormed spaces of the original varieties. A fundamental question is the following: given two mildly singular projective varieties with some of the first variety's pseudonormed spaces being isometric to the corresponding ones of the second variety's, can one construct a birational map between them which induces these isometries? In this work a positive answer to this question is given for varieties of general type. This can be thought of as a theorem of Torelli type for birational equivalence.Comment: 13 pages, to appear in PNA

Homogeneous variational problems: a minicourse

A Finsler geometry may be understood as a homogeneous variational problem, where the Finsler function is the Lagrangian. The extremals in Finsler geometry are curves, but in more general variational problems we might consider extremal submanifolds of dimension $m$. In this minicourse we discuss these problems from a geometric point of view.Comment: This paper is a written-up version of the major part of a minicourse given at the sixth Bilateral Workshop on Differential Geometry and its Applications, held in Ostrava in May 201