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Details of Grant 

EPSRC Reference: EP/M024784/1
Title: Riemann-Hilbert problems, infinite matrices and their applications
Principal Investigator: Virtanen, Professor J
Other Investigators:
Researcher Co-Investigators:
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Department: Mathematics and Statistics
Organisation: University of Reading
Scheme: First Grant - Revised 2009
Starts: 01 August 2015 Ends: 01 November 2017 Value (£): 98,560
EPSRC Research Topic Classifications:
Mathematical Analysis
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
03 Mar 2015 EPSRC Mathematics Prioritisation Panel March 2015 Announced
Summary on Grant Application Form
The Riemann-Hilbert problem (RHP) has a long and impressive history going back to Riemann's dissertation (1851) and Hilbert's related results at the beginning of the 20th century. The Riemann-Hilbert problem, which can be described as a problem of finding an analytic function in the complex plane with a prescribed jump across a given curve, is closely connected to one-dimensional singular integral operators, convolution operators, Toeplitz operators, and Wiener-Hopf operators. In these several different forms, the problem has attracted many famous mathematicians. The research area continues to expand rapidly and find new applications in many (applied) fields of mathematics, in (mathematical) physics and even in chemistry.

A great deal of the current importance of the RHP is due to its use in random matrix theory, orthogonal polynomials and integrable systems. The Riemann-Hilbert method for integrable PDEs originated in the works of Manakov, Shabat, and Zakharov in 1975-1979, and since then it has been widely used in soliton theory. The Riemann-Hilbert approach to quantum exactly solvable models was most recently developed in the 1990s in the series of works by Its, Izergin, Korepin, Slavnov, Deift, and Zhou. The Riemann-Hilbert approach to orthogonal polynomials and matrix models was initiated in 1991 by Its, Fokas, and Kitaev, which has led to solving some of the long-standing problems in the asymptotics of orthogonal polynomials related to universalities in random matrices.

The relations between random matrix models and classical integrable systems appear in deformation theory, when parameters characterizing the measures or the domain of localization of the eigenvalues are varied. The resulting differential equations determining the partition function and correlation functions are, remarkably, of the same type as certain equations appearing in the theory of integrable systems. They may be analyzed effectively through methods based on the RHP and by related approaches to the study of nonlinear asymptotics in the large N limit.

Many of the aforementioned applications arise via certain classes of operators and infinite matrices, namely Toeplitz and Hankel matrices, and require the study of the asymptotics of their determinants when the size of the matrix goes to infinity. The study of Toeplitz matrices and determinants was initiated by Otto Toeplitz in 1907 to find concrete examples of Hilbert's general theory of functional analysis. A great variety of problems in mathematics, physics and engineering can be expressed in terms of these matrices; in particular in areas such as function theory, probability theory, statistics, and statistical mechanics, including Kaufman and Onsager's work on the Ising model.

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Organisation Website: http://www.rdg.ac.uk