Related papers: Schottky uniformization and Bloch-Wigner dilogarit…

We develop linear discretization of complex analysis, originally introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We prove convergence of discrete period matrices and discrete Abelian integrals to their continuous…

Using recent work of Bourgain-Dyatlov we show that for any convex co-compact hyperbolic surface Strichartz estimates for the Schr\"odinger equation hold with an arbitrarily small loss of regularity.

We prove explicit bounds on canonical Green functions of Riemann surfaces obtained as compactifications of quotients of the upper half-plane by Fuchsian groups.

It is well known that the collection of uniformizations of a closed Riemann surface $S$ is partially ordered; the lowest ones are the Schottky unformizations, that is, tuples $(\Omega,\Gamma,P:\Omega \to S)$, where $\Gamma$ is a Schottky…

We unify and advance a host of works on iterated function systems of holomorphic self-maps of hyperbolic Riemann surfaces. Our foremost result is a generalisation to left iterated function systems of an unpublished and little known theorem…

The intuitive notion of the Gromov invariant for maps from a Riemann surface to a Grassmannian is shown to agree with the definition in \cite{BDW}. Also, an induction on the genus is proved, which extends the results of \cite{BDW} to a…

We describe Zhu recursion for a vertex operator algebra (VOA) on a general genus Riemann surface in the Schottky uniformization where $n$-point correlation functions are written as linear combinations of $(n-1)$-point functions with…

We study the Dirac and the Laplacian operators on orientable Riemann surfaces of arbitrary genus g. In particular we compute their determinants with twisted boundary conditions along the b-cycles. All the ingredients of the final results…

In this paper we develop a general approach to dimer models analogous to Krichever's scheme in the theory of integrable systems. We start with a Riemann surface and the simplest generic meromorphic functions on it and demonstrate how to…

We interprete results of Markman on monodromy operators as a universality statement for descendent integrals over moduli spaces of stable sheaves on $K3$ surfaces. This yields effective methods to reduce these descendent integrals to…

We give a general framework for uniform, constant-time one-and two-dimensional scalar multiplication algorithms for elliptic curves and Jacobians of genus 2 curves that operate by projecting to the x-line or Kummer surface, where we can…

An Ansatz for the Poincar\'e metric on compact Riemann surfaces is proposed. This implies that the Liouville equation reduces to an equation resembling a non chiral analogous of the higher genus relationships (KP equation) arising in the…

We give a new proof of Faltings's p-adic Eichler-Shimura decomposition of the modular curves via BGG methods and the Hodge-Tate period map. The key property is the relation between the Tate module and the Faltings extension, which was…

In this paper a class of oscillatory integrals is interpreted as a limit of Lebesgue integrals with Gaussian regularizers. The convergence of the regularized integrals is shown with an improved version of iterative integration by parts that…

In this paper we prove new explicit formulas for Faltings' $\delta$-invariant of an arbitrary hyperelliptic Riemann surface. This has several applications: For example we obtain an explicit lower bound for $\delta$ depending only on the…

We introduce the notion of Schr\"odinger integral operators and prove sharp local and global regularity results for these (including propagators for the quantum mechanical harmonic oscillator). Furthermore we introduce general classes of…

We give a simple proof of the sharp decay of the Fourier-transform of surface-carried measures of two-dimensional generic surfaces. The estimates are applied to prove Strichartz and resolvent estimates for elliptic operators whose…

We study Witt groups of smooth curves and surfaces over algebraically closed fields of characteristic not two. In both dimensions, we determine both the classical Witt group and Balmer's shifted Witt groups. In the case of curves, the…

Bloch and Okounkov's correlation function on the infinite wedge space has connections to Gromov-Witten theory, Hilbert schemes, symmetric groups, and certain character functions of $\hgl_\infty$-modules of level one. Recent works have…

The paper presents an analog of the old result by the author and V. Voevodsky, according to which a Riemann surface admits a conformal structure, defined by an equilateral triangulation, if and only if the corresponding algebraic curve can…