Many of Hilbert’s 23 famous problems are not of a prove or disprove nature; rather, they are open-ended, “of a purely investigative nature,” [12] and may never be answered to satisfaction. One of the best examples of this is the sixth problem, that of the axiomatization of physics. Hilbert said, “The investigations on the foundations of geometry suggest the problem: To treat in the same manner,...

In this paper, we study the topological expansion in the cubic random matrix model, and we evaluate explicitly the expansion coefficients for genus 0 and 1. For genus 0 our formula coincides with the one in [6]. For higher genus, we obtain the asymptotic behavior of the coefficients in the expansion as the number of vertices of the associated graphs tends to infinity. Our study is based on the ...

Abstract. We show that the fourth-order nonlinear ODE which controls the pole dynamics in the general solution of equation P 2 I compatible with the KdV equation exhibits two remarkable properties: (1) it governs the isomonodromy deformations of a 2× 2 matrix linear ODE with polynomial coefficients, and (2) it does not possess the Painlevé property. We also study the properties of the Riemann–H...

We discuss non-compact SL(2, R) sectors in N=4 SYM and in AdS string theory and compare their integrable structures. We formulate and solve the Riemann-Hilbert problem for the finite gap solutions of the classical sigma model and show that at one loop it is identical to the classical limit of Bethe equations of the spin (-1/2) chain for the dilatation operator of SYM.

We study the problem of placing effective upper bounds for the number of zeros of solutions of Fuchsian systems on the Riemann sphere. The principal result is an explicit (non-uniform) upper bound, polynomially growing on the frontier of the class of Fuchsian systems of a given dimension n having m singular points. As a function of n, m, this bound turns out to be double exponential in the prec...

Bleher and Kuijlaars recently showed that the eigenvalue correlations from matrix ensembles with external source can be expressed by means of a kernel built out of special multiple orthogonal polynomials. We derive a Christoffel-Darboux formula for this kernel for general multiple orthogonal polynomials. In addition, we show that the formula can be written in terms of the solution of the Rieman...

We study zero temperature correlation functions of the spin-1 2 Heisenberg XXZ model in the critical regime −1 < ∆ ≤ 1 in a magnetic field by means of the Dual Field Approach. We show for one particular example how to derive determinant representations for correlation functions and how to use these to embed the correlation functions in integrable systems of integro-difference equations (IDE). T...

The perturbation theory based on the Riemann-Hilbert problem is developed for the modified nonlinear Schrödinger equation which describes the propagation of femtosecond optical pulses in nonlinear single-mode optical fibers. A detailed analysis of the adiabatic approximation to perturbation-induced evolution of the soliton parameters is given. The linear perturbation and the Raman gain are cons...

We construct large families of two-dimensional travelling water waves propagating under the influence of gravity in a flow of constant vorticity over a flat bed. A Riemann-Hilbert problem approach is used to recast the governing equations as a one-dimensional elliptic pseudo-differential equation with a scalar constraint. The structural properties of this formulation, which arises as the Euler-...

For an operator of a certain class in Hilbert space, we introduce axioms of an abstract intersection theory, which we prove to be equivalent to the Riemann Hypothesis concerning the spectrum of that operator. In particular if the nontrivial zeros of the Riemann zeta-function arise from an operator of this class, the original Riemann Hypothesis is equivalent to the existence of an abstract inter...