Related papers: Constructing Strebel differentials via Belyi maps …

Translation surfaces with poles correspond to meromorphic differentials on compact Riemann surfaces. They appear in compactifications of strata of the moduli space of Abelian differentials and in the study of stability conditions. Such…

We construct a trace map on the chiral homology of chiral Weyl algebra for any smooth Riemann surface. Our trace map can be viewed as a chiral version of the deformed HKR quasi-isomorphism. This also provides a mathematical rigorous…

In terms of appropriate extended moduli spaces, we develop a finite-dimensional construction of the self-duality and related moduli spaces over a closed Riemann surface as stratified holomorphic symplectic spaces by singular…

Compactifications of moduli spaces of (1,p)-polarized abelian surfaces with level structures of canonical type have been described in great detail by Hulek, Kahn and Weintraub. The aim of this paper is to determine some invariants of smooth…

We construct spectral triples on C*-algebraic extensions of unital C*-algebras by stable ideals satisfying a certain Toeplitz type property using given spectral triples on the quotient and ideal. Our construction behaves well with respect…

We develop a systematic framework for constructing spherical harmonics on the two-dimensional unit sphere as superpositions of Gaussian beams whose poles form well-separated point configurations. The distributional and analytic properties…

The space of Lam\'e functions of order m is isomorphic to the space of pairs (elliptic curve, Abelian differential) where the differential has a single zero of order 2m at the origin and m double poles with vanishing residues. We describe…

In this paper, we address the following question: What does a typical compact Riemann surface of large genus look like geometrically? We do so by constructing compact Riemann surfaces from oriented 3-regular graphs. The set for such Riemann…

The diffraction of a scalar plane wave from a doubly-periodic surface on which either the Dirichlet or Neumann boundary condition is imposed is studied by means of a rigorous numerical solution of the Rayleigh equation for the amplitudes of…

We construct and study a natural homeomorphism between the moduli space of polynomial cubic differentials of degree d on the complex plane and the space of projective equivalence classes of oriented convex polygons with d+3 vertices. This…

We construct rational elliptic surfaces of index two by explicitly constructing their associated Halphen pencils in the projective plane $\mathbb{P}^2$. For each of the types of singular fibers that occur we construct at least one example…

Let $S$ be a Riemann surface of type $(p,n)$ with $3p-3+n>0$. Let $\omega$ be a pseudo-Anosov map of $S$ that is obtained from Dehn twists along two families $\{A,B\}$ of simple closed geodesics that fill $S$. Then $\omega$ can be realized…

Let Sol be the three-dimensional solvable Lie group equipped with its standard left-invariant Riemannian metric. We give a precise description of the cut locus of the identity, and a maximal domain in the Lie algebra on which the Riemannian…

In this paper, we study the translation surfaces corresponding to meromorphic differentials on compact Riemann surfaces. We compute the number of connected components of the corresponding strata of the moduli space. We show that in genus…

We introduce dessins d'enfants from the various existing points of view: As topological covering spaces, as surfaces with triangulations, and as algebraic curves with functions ramified over three points. We prove Belyi's theorem that such…

Brane tilings, sometimes called dimer models, are a class of bipartite graphs on a torus which encode the gauge theory data of four-dimensional SCFTs dual to D3-branes probing toric Calabi--Yau threefolds. An efficient way of encoding this…

The local invariants of a meromorphic quadratic differential on a compact Riemann surface are the orders of zeros and poles, and the residues at the poles of even orders. The main result of this paper is that with few exceptions, every…

In this paper, we construct polynomial growth harmonic maps from once-punctured Riemann surfaces of any finite genus to any even-sided, regular, ideal polygon in the hyperbolic plane. We also establish their uniqueness within a class of…

We make a systematic investigation of quadrature properties for quadrics, namely integration of holomorphic functions over planar domains bounded by second degree curves. A full understanding requires extending traditional settings by…

For compact Riemann surfaces, the collar theorem and Bers' partition theorem are major tools for working with simple closed geodesics. The main goal of this paper is to prove similar theorems for hyperbolic cone-surfaces. Hyperbolic…