Related papers: Tessellations of rational complex functions and th…

We analyze a random lozenge tiling model of a large regular hexagon, whose underlying weight structure is periodic of period $2$ in both the horizontal and vertical directions. This is a determinantal point process whose correlation kernel…

We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudo-Riemannian manifold $(M,\rg)$. In other words, we establish a canonical isomorphism between the spaces of…

In [B.Gruenbaum, G.C. Shephard, Spherical tilings with transitivity properties, in: The geometric vein, Springer, New York, 1981, pp. 65-98], they proved "for every spherical normal tiling by congruent tiles, if it is isohedral, then the…

In these lectures we first review the important properties of the Riemann $\zeta$-function that are necessary to understand the nature and importance of the Riemann hypothesis (RH). In particular this first part describes the analytic…

This paper is the second part of a two-part study of an elementary functorial construction of tesselated surfaces from finite groups. This elementary construction was discussed in the first part and generally results in a large collection…

Let W -> X be a real smooth projective 3-fold fibred by rational curves. J. Koll\'ar proved that, if W(R) is orientable, then a connected component N of W(R) is essentially either a Seifert fibred manifold or a connected sum of lens spaces.…

I present two independent proofs of the Riemann Hypothesis considered by many the greatest unsolved problem in mathematics. I find that the admissible domain of complex zeros of the Riemann Zeta Function is the critical line. The methods…

Rational maps on the Riemann sphere occupy a distinguished niche in the general theory of smooth dynamical systems. First, rational maps are complex-analytic, so a broad spectrum of techniques can contribute to their study (quasiconformal…

We find a system of two polynomial equations in two unknowns, whose solution allows to give an explicit expression of the conformal representation of a simply connected three sheeted compact Riemann surface onto the extended complex plane.…

We introduce a family of variational functionals for spinor fields on a compact Riemann surface $M$ that can be used to find close-to-conformal immersions of $M$ into $\mathbb{R}^3$ in a prescribed regular homotopy class. Numerical…

We study here slopes of periodicity of tilings. A tiling is of slope if it is periodic along direction but has no other direction of periodicity. We characterize in this paper the set of slopes we can achieve with tilings, and prove they…

We study the set $R$ of nonplanar rational curves of degree $d<q+2$ on a smooth Hermitian surface $X$ of degree $q+1$ defined over an algebraically closed field of characteristic $p>0$, where $q$ is a power of $p$. We prove that $R$ is the…

We recall the theory of linear discrete Riemann surfaces and show how to use it in order to interpret a surface embedded in R^3 as a discrete Riemann surface and compute its basis of holomorphic forms on it. We present numerical examples,…

We consider a class of planar tree-level four-point functions in N=4 SYM in a special kinematic regime: one BMN operator with two scalar excitations and three half-BPS operators are put onto a line in configuration space; additionally, for…

The individual terms of the series representing the Riemann zeta function are examined geometrically from their accumulated plot in the complex plane. Symmetry is identified and determined mathematically for comparison with more traditional…

In this work we consider an equation for the Riemann zeta-function in the critical half-strip. With the help of this equation we prove that finding non-trivial zeros of the Riemann zeta-function outside the critical line would be equivalent…

The moduli spaces of compact and connected Riemann surfaces has been a central topic in modern mathematics in recent years. Thus their homological dimensions become important invariants. Motivated by the emergence mathematical counterparts…

Given a compact Riemann surface $\bar{X}$ of genus $g$ and a point $q$ on $\bar{X}$, we consider $X:=\bar{X}\setminus\{q\}$ with a basepoint $p\in X$. The extension of mixed Hodge structures, given by the weights -1 and -2, of the mixed…

Classically, an indecomposable class $R$ in the cone of effective curves on a K3 surface $X$ is representable by a smooth rational curve if and only if $R^2=-2$. We prove a higher-dimensional generalization conjectured by Hassett and…

Riemann's minimal surfaces are a complete, embeddable, one-parameter family of minimal surfaces with translational symmetry along one direction. It's infinite number of planar ends are joined together by an array of necks, closely matching…