Title: Quantization and Kähler geometry
Abstract: Quantization
is a process which associates a classical mechanics problem (symplectic
geometry) with a quantum mechanics problem (spectral theory or time
evolution of a linear operator, notably of linear PDEs). Many symplectic
objects have a quantum counterpart, and there is an ongoing dictionary
between the properties on both sides.
During
the last two decades, there has been a growing interest for a global
quantization on (compact, boundaryless, integrable) Kähler or almost
Kähler manifolds, the Berezin-Toeplitz quantization, which encompasses
Donaldson's peak sections, spin thermodynamics, automorphic forms, and
pseudo-differential operators.
I will review recent progress in the development of Berezin-Toeplitz quantization, with a special emphasis on regularity, and will present some personal perspectives which I hope to be interesting from the point of view of symplectic geometry.
Title: Homological invariants of codimension 2 contact embeddings
Abstract: There is a rich theory of transverse knots in 3-dimensional contact manifolds. It was a major open question in contact topology whether non-trivial transverse knots (i.e. codimension 2 contact embeddings) also exist in higher dimensions. This question was recently settled in the affirmative by Casals and Etnyre. Motivated by their result, I will talk about recent work with Francois-Simon Fauteux-Chapleau in which we develop invariants of codimension 2 contact embeddings using the machinery of symplectic field theory.
