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**1 - 5**of**5**### DESSINS D’ENFANTS AND DIFFERENTIAL EQUATIONS

, 2006

"... Abstract. We state and solve a discrete version of the classical Riemann-Hilbert problem. In particular, we associate a Riemann-Hilbert problem to every dessin d’enfants. We show how to compute the solution for a dessin that is a tree. This amounts to finding a Fuchsian differential equation satisfi ..."

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Abstract. We state and solve a discrete version of the classical Riemann-Hilbert problem. In particular, we associate a Riemann-Hilbert problem to every dessin d’enfants. We show how to compute the solution for a dessin that is a tree. This amounts to finding a Fuchsian differential equation satisfied by the local inverses of a Shabat polynomial. We produce a universal annihilating operator for the inverses of a generic polynomial. We classify those plane trees that have a representation by Möbius transformations and those that have a linear representation of dimension at most two. This yields an analogue for trees of Schwarz’s classical list, that is, a list of those plane trees whose Riemann-Hilbert problem has a hypergeometric solution of order at most two. 1. Review of the classical Riemann-Hilbert problem Consider a homogeneous linear differential equation of order n in one complex variable y (n) + p1y (n−1) + · · · + pn−1y ′ + pny = 0, where p1,...,pn are meromorphic at a point a in the complex plane C. If p1,...,pn are holomorphic at a, then a is called a regular point of the equation; otherwise a is called a singular point. Cauchy proved that at a regular point, there is an n-dimensional space of

### COUNTING COLORED PLANAR MAPS FREE-PROBABILISTICALLY

"... Abstract. Our main result is an explicit operator-theoretic formula for the number of colored planar maps with a fixed set of stars each of which has a fixed set of half-edges with fixed coloration. The formula transparently bounds the number of such colored planar maps and does so well enough to pr ..."

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Abstract. Our main result is an explicit operator-theoretic formula for the number of colored planar maps with a fixed set of stars each of which has a fixed set of half-edges with fixed coloration. The formula transparently bounds the number of such colored planar maps and does so well enough to prove convergence near the origin of generating functions arising naturally in the matrix model context. Such convergence is known but the proof of convergence proceeding by way of our main result is relatively simple. Our main technical tool is an integration identity representing the joint cumulant of several functions of a Gaussian random vector. In the case of cumulants of order 2 the identity reduces to one well-known as a means to prove the Poincaré inequality. Contents

### ON NODAL SETS AND NODAL DOMAINS ON S 2 AND R 2

"... Abstract. We discuss possible topological configurations of nodal sets, in particular the number of their components, for spherical harmonics on S 2.We also construct a solution of the equation ∆u = u in R 2 that has only two nodal domains. This equation arises in the study of high energy eigenfunct ..."

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Abstract. We discuss possible topological configurations of nodal sets, in particular the number of their components, for spherical harmonics on S 2.We also construct a solution of the equation ∆u = u in R 2 that has only two nodal domains. This equation arises in the study of high energy eigenfunctions. 1. Topological structure of nodal domains Homogeneous harmonic polynomials of degree n in three variables, when restricted to the unit sphere, become eigenfunctions of the Laplace–Beltrami operator on the sphere, with eigenvalue λ = n(n + 1) of multiplicity 2n +1, n =0, 1,.... We permit ourselves the liberty of calling them eigenfunctions of degree n. The zero set of an eigenfunction is called the nodal set. The nodal set is nonsingular if it is a union of disjoint closed analytic curves, and the nodal set of a generic eigenfunction is non-singular [12]. The known results about topology of nodal sets on the sphere are the following [5]. The nodal set is non-empty for n ≥ 1. A general theorem of Courant [2] implies that the nodal set of an eigenfunction of degree n consists of at most n2 components. H. Lewy [10] proved that for even n ≥ 2 the number of components

### NUMBER THEORY IN PHYSICS

"... always been the case for the theory of differential equations. In the early twentieth century, with the advent of general relativity and quantum mechanics, topics such as differential and Riemannian geometry, operator algebras and functional analysis, or group theory also developed a close relation ..."

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always been the case for the theory of differential equations. In the early twentieth century, with the advent of general relativity and quantum mechanics, topics such as differential and Riemannian geometry, operator algebras and functional analysis, or group theory also developed a close relation to physics. In the past decade, mostly through the influence of string theory, algebraic geometry also began to play a major role in this interaction. Recent years have seen an increasing number of results suggesting that number theory also is beginning to play an essential part on the scene of contemporary theoretical and mathematical physics. Conversely, ideas from physics, mostly from quantum field theory and string theory, have started to influence work in number theory. In describing significant occurrences of number theory in physics, we will, on the one hand, restrict our attention to quantum physics, while, on the other hand, we will assume a somewhat extensive definition of number theory, that will allow us to include arithmetic algebraic geometry. The territory is vast and an extensive treatment would go beyond the size limits imposed by the encyclopaedia. The