Jani Virtanen

Asymptotics of Toeplitz determinants with singularities and their applications in mathematical physics

The asymptotic study of the determinants of Toeplitz matrices (easily defined as matrices constant along the parallels to the main diagonal) goes back to the work of Szegö in the early 1900s, when he considered Toeplitz matrices whose entries are given by the Fourier coefficients of sufficiently smooth functions. Later in the 1960s, in connection with problems in statistical physics, Fisher and Hartwig singled out symbols that possess singularities and conjectured the asymptotics of the corresponding Toeplitz determinants. The asymptotic study of Toeplitz determinants continues to play an important role in applications in random matrix theory and mathematical physics. In this talk, I will discuss the case of matrix-valued symbols with singularities and some of their applications using operator theoretic methods (instead of the Riemann-Hilbert analysis).