We show how the Riemann–Hilbert problem can be used to compute correlation kernels for determinantal point processes arising in different models of asymptotic combinatorics and representation theory. The Whittaker kernel and the discrete Bessel kernel are computed as examples.

A vector functional-difference equation of the first order with a special matrix coefficient is analysed. It is shown how it can be converted into a Riemann-Hilbert boundary-value problem on a union of two segments on a hyperelliptic surface. The genus of the surface is defined by the number of zeros and poles of odd order of a characteristic function in a strip. An even solution of a symmetric...

Hilbert-Polya conjecture and Riemann hypothesis are proven. The construction of Hilbert-Polya operator is inspired by the conviction that Riemann Zeta function is associated with a physical system allowing superconformal transformations as its symmetries. The proof as such is elementary involving only basic facts about the theory of Hilbert space operators and complex analysis.

The construction of analogues of the Cauchy kernel is crucial for the solution of Riemann–Hilbert problems on compact Riemann surfaces. A formula for the Cauchy kernel can be given as an infinite sum over the elements of a Schottky group, and this sum is often used for the explicit evaluation of the kernel. In this paper a new formula for a quasi-automorphic analogue of the Cauchy kernel in ter...

Abstract We define and study Toeplitz operators on the Fréchet space of all holomorphic functions finitely connected domains in Riemann sphere. completely characterize Fredholm, semi-Fredholm invertible belonging to this class. As a result, we obtain characterization these classes unit disk case. motivation formulate analyze Riemann–Hilbert problem which consider.

A Riemann-Hilbert problem for a q-difference Painlevé equation, known as $$q{\text {P}}_{{\text {IV}}}$$ , is shown to be solvable. This yields bijective correspondence between the transcendental solutions of and corresponding data on an associated q-monodromy surface. We also construct moduli space explicitly.