Jani Virtanen (Univ. of Reading) : Spectral theory of Toeplitz operators Kasia Koslowska (Univ. of Reading) : Transition asymptotics of Toeplitz determinants and their applications

Wednesday 11 Jan 2017, 15:00 → 17:00 Europe/Brussels

CYCL02 (M. de Hemptinne)

Jani Virtanen : Toeplitz matrices are easily defined as matrices constant along parallels to the main diagonal. We can think of the entries of a Toeplitz matrix as the Fourier coefficients of a measurable function, which we refer to as the symbol of the Toeplitz matrix it generates. Despite their simple definition, Toeplitz matrices appear in a variety of (physical) problems and possess an extremely rich spectral theory. They also form one of the most important classes of non-selfadjoint operators. In this talk spectral properties of Toeplitz operators are discussed with emphasis on geometric descriptions of their essential spectra and the existence of eigenvalues embedded in these sets. The situation is well understood up to piecewise continuous symbols but it becomes much more difficult for general (bounded) symbols. ≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠≠ Kasia Kozlowska: The study of the asymptotics of Toeplitz determinants is important because of a vast number of applications in random matrix theory and mathematical physics. These asymptotics are well understood for many symbol classes, such as smooth Szegö symbols, and symbols with Fisher-Hartwig (F-H) singularities of jump and/or root type. By introducing an additional parameter in the symbol, we can consider what is called transition asymptotics. In the paper titled ‘Emergence of a singularity for Toeplitz determinants and Painleve V’, the authors Claeys, Its and Krasovsky considered the transition case between a Szego and a F-H symbol with one singularity. In this talk I will present my work on the transition case in which we see emergence of additional singularities. In that case however, we need to consider so-called F-H representations and the Tracy-Basor conjecture (proven by Deift, Its and Krasovsky). These types of results model phase transitions in numerous problems arising in statistical mechanics, one of which will be mentioned in the talk.