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We investigate the elliptic integrable model introduced by Deguchi and Martin, which is an elliptic extension of the Perk-Schultz model. We introduce and study a class of partition functions of the elliptic model by using the…

In this paper, we construct holomorphic families of Riemann surfaces from Veech groups and characterize their sections by some points of corresponding flat surfaces. The construction gives us concrete solutions for some Diophantine…

A cyclic cover of the complex projective line branched at four appropriate points has a natural structure of a square-tiled surface. We describe the combinatorics of such a square-tiled surface, the geometry of the corresponding…

We introduce a notion of resultant of two meromorphic functions on a compact Riemann surface and demonstrate its usefulness in several respects. For example, we exhibit several integral formulas for the resultant, relate it to potential…

We study the generalization of the Willmore functional for surfaces in the three-Heisenberg group. Its construction is based on the spectral theory of the Dirac operator coming to the Weierstrass representation of surfaces (see…

In this note we clarify that the Rcci flow can be used to give an independent proof of the uniformization theorem of Riemann surfaces.

We prove that the integral of a certain Riesz-type kernel over $(n-1)$-rectifiable sets in $\mathbb{R}^n$ is constant, from which a formula for surface measure immediately follows. Geometric interpretations are given, and the solution to a…

Let $k$ be an algebraic closure of a finite field of odd characteristic. We prove that for any rank two graded Higgs bundle with maximal Higgs field over a generic hyperbolic curve $X_1$ defined over $k$, there exists a lifting $X$ of the…

We introduce a class of iterated integrals, defined through a set of linearly independent integration kernels on elliptic curves. As a direct generalisation of multiple polylogarithms, we construct our set of integration kernels ensuring…

The purpose of these notes is to show that the methods introduced by Bauer and Furuta in order to refine the Seiberg-Witten invariants of smooth 4-dimensional manifolds can also be used to obtain stable homotopy classes from Riemann…

We consider smooth locally Hamiltonian flows on compact surfaces of genus $g\geq 2$ to prove their deviation of Birkhoff integrals for smooth observables. Our work generalizes results of Forni and Bufetov which prove the existence of…

We prove a simultaneous generalization of the classical Riemann-Hurwitz and Plucker formulas, addressing the total inflection of a morphism from a (smooth, projective) curve to an arbitrary (smooth, projective) higher-dimensional variety.…

In this paper we continue our study on the canonical metrics on the Teichm\"uller and the moduli space of Riemman surfaces. We first prove the equivalence of the Bergman metric and the Carath\'eodory metric to the K\"ahler-Einstein metric,…

We study various properties of the gradients of solutions to harmonic functions on Lipschitz surfaces. We improve an exponential bound of Naber and Valtorta on the size of the superlevel sets for the frequency function to a sharp quadratic…

We prove a sharp stability estimate for Schur iterates of contractive analytic functions in the open unit disk. We then apply this result in the setting of the inverse scattering approach and obtain a fast algorithm for solving the discrete…

The integral Burau representation provides a map from the braid group into a group of integral matrices. This allows for a definition of congruence subgroups of the braid group as the preimage of the usual principal congruence subgroups of…

In this paper the authors show how to use Riemann-Hilbert techniques to prove various results, some old, some new, in the theory of Toeplitz operators and orthogonal polynomials on the unit circle (OPUC's). There are four main results: the…

We examine the internal geometry of a Kleinian surface group and its relations to the asymptotic geometry of its ends, using the combinatorial structure of the complex of curves on the surface. Our main results give necessary conditions for…

In this paper, we prove new identities for Bernoulli polynomials that extend Alzer and Kwong's results. The key idea is to use the Volkenborn integral over $\mathbb Z_p$ of the Bernoulli polynomials to establish recurrence relations on the…

Uniform bounds are developed for derivatives of solutions of the $2$-dimensional constant negative curvature equation and the Weil-Petersson metric for the Teichm\"{u}ller and moduli spaces. The dependence of the bounds on the geometry of…