Asymptotics of determinants

Bernhard Riemann, in his inaugural dissertation, "Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen komplexen Grösse" (1851), raised the question of the boundary behavior of holomorphic functions, which today is known as the Riemann-Hilbert problem. This important question, clarified and generalized by Hilbert, is considered to be the cornerstone for the future development of Toeplitz operator theory, according to Nikolsky. The Riemann-Hilbert problem was first studied by Julian Vasilyevich Sokhotsky in Petersburg, and was later detailed by Nikolai Nikolaevich Luzin and Ivan Ivanovich Privalov in Moscow. Otto Toeplitz, one of the greats in the field of operator theory, did not focus on the operators that bear his name. The systematic study of these operators was apparently initiated by Gabor Szegő, and his first theorem, along with its generalizations by Andrei Kolmogorov and Mark Crane, will serve as the starting point for our discussion. We then turn to Szegő's second theorem, which determines the asymptotic behavior of the determinants of Toeplitz matrices. Additionally, we explore the Borodin-Okounkov-Geronimus-Case formula, which provides the residual term in this second theorem. Toeplitz determinants are found in a wide range of problems, and Szegő's theorems like the Borodin-Okunikov-Geronimos-Case formula have very different approaches: analytical, algebraic, or probabilistic. In the course, we will emphasize the applications of Toeplitz operators in the context of determinant point processes that arise in the study of random matrices and asymptotic combinatorics.