Children's Drawings and the Riemann-Hilbert Problem

A series of ceramic artworks are presented, inspired by the author's research connecting theoretical physics to the beautiful theory of Riemann surfaces. More specifically the research is related to the classification of curves on the surfaces based on a description of them as built from basic building blocks known as "pairs of pants". The relevant background on this mathematics of these two dimensional spaces is outlined, some of the artistic process is explained: Both the c...

The connections amongst (1) quivers whose representation varieties are Calabi-Yau, (2) the combinatorics of bipartite graphs on Riemann surfaces, and (3) the geometry of mirror symmetry have engendered a rich subject at whose heart is the physics of gauge/string theories. We review the various parts of this intricate story in some depth, for a mathematical audience without assumption of any knowledge of physics, emphasizing a plethora of results residing at the intersection...

We give a complete topological classification of germs of holomorphic foliations in the plane under rather generic conditions. The key point is the introduction of a new topological invariant called monodromy representation. This monodromy contains all the relevant dynamical information, in particular the projective holonomy representation whose topological invariance was conjectured in the eighties by Cerveau and Sad and proved here under mild hypotheses.

We introduce a systematic approach to express generating functions for the enumeration of maps on surfaces of high genus in terms of a single generating function relevant to planar surfaces. Central to this work is the comparison of two asymptotic expansions obtained from two different fields of mathematics: the Riemann-Hilbert analysis of orthogonal polynomials and the theory of discrete dynamical systems. By equating the coefficients of these expansions in a common region o...

We present drawings on the complex plane of the lines Im(zeta(s))=0 and Re(zeta(s))=0. This allow to illustrate many properties of the zeta function of Riemann. This is an expository paper. It does not pretend to prove any new result about the zeta function. But it gives a new perspective to many known results. Also it may prove useful to show to students of Complex Variables or Analytic Number Theory. (This was my initial motivation to make the drawings).

Let E be a finite set. Given permutations x and y of E that together generate a transitive subgroup, for which s is it true that x and the conjugate of y by s also generate a transitive subgroup? Such transitive permutation pairs encode dessins d'enfants, important graph-theoretic objects which are also known to have great arithmetic significance. The absolute Galois group acts on dessins d'enfants and permutes them in a very mysterious way. Two dessins d'enfants that share c...

Bipartite graphs, especially drawn on Riemann surfaces, have of late assumed an active role in theoretical physics, ranging from MHV scattering amplitudes to brane tilings, from dimer models and topological strings to toric AdS/CFT, from matrix models to dessins d'enfants in gauge theory. Here, we take a brief and casual promenade in the realm of brane tilings, quiver SUSY gauge theories and dessins, serving as a rapid introduction to the reader.

Riemann's Existence Theorem gives the following bijections: (1) Isomorphism classes of Belyi maps of degree $d$. (2) Equivalence classes of generating systems of degree $d$. (3) Isomorphism classes of dessins d'enfants with $d$ edges. In previous work, the first author and collaborators exploited the correspondence between Belyi maps and their generating systems to provide explicit equations for two infinite families of dynamical Belyi maps. We complete this picture by descri...

In this manuscript, by using Belyi maps and dessin d'enfants, we construct some concrete examples of Strebel differentials with four double poles on the Riemann sphere. As an application, we could give some explicit cone spherical metrics on the Riemann sphere.

This paper has been withdrawn by the author, due to a crucial error in page 5.