Analysis and Probability seminar

Abstract: The framework of Berezin-Toeplitz quantization is a popular approach in the quantization of Kähler manifolds, which brings together geometric and representation-theoretic considerations. It is the non-flat analogue of the anti-Wick quantization scheme. In this talk, I will introduce some fundamental concepts in the theory of BT quantization, and explain how they can be obtained from a representation-theoretic approach. I will then present recent work with B. Ruba and J. P. Solovej on the particular case of complex projective spaces. These are the homogeneous spaces associated with the symmetric representations of SU(d), and their rich symmetry allows for a particularly explicit analysis. I will explain how to get sharp asymptotic expansions of the product of Toeplitz operators and introduce an interesting notion of positivity of the remainder in these expansions.