Metaplectic-c quantization was developed by Robinson and Rawnsley as an alternative to the classical Kostant-Souriau quantization procedure with half-form correction. Given a metaplectic-c quantized symplectic manifold and a real-valued function on that manifold, we propose a condition under which a regular value of the function is a quantized energy level for the system. We discuss the properties of this condition, and we give the quantized energy levels of the harmonic oscillator and the hydrogen atom.

This week we continue with:

---The index problem for elliptic PDE's, characteristic classes via Chern-Weil theory,

---Miraculous cancellations, Getzler's supersymmetric proof of the Atiyah-Singer index theorem, special cases: Gauss-Bonnet-Chern, Hirzebruch signature theorem, and Riemann-Roch.

This talk will consider the problem of characterizing the sum of CR functions from two competing (oppositely-oriented) CR structures sharing the same maximal complex subspace, in two specific scenarios.

In the first scenario the two structures are simply conjugate to each other and the functions in question are pluriharmonic boundary values. (This problem has an extensive history, but some new results will be presented.) In the second scenario the two structures are related by projective duality considerations.

In both cases special attention will be paid to two-dimensional circular domains. This is joint work with Dusty Grundmeier.

We will not assume any prior knowledge of stacks for this talk. We introduce a notion of "stacky resolution" which is gives a way to study mildly singular spaces. We then discuss applications of these resolutions to group theory, Hodge theory, and toric geometry.

Following the unpublished work of C. Rezk, Toposes and Homotopy Toposes, we present a formulation of the notion of model topos, intended as a model-categorical version of the classical concept of Grothendieck topos. Such a definition will be sensible enough to establish a Giraud-type theorem for model topoi. We will start by reviewing the notion of Grothendieck topos, albeit from a slightly unusual perspective which avoids the use of Grothendieck topologies. We will then state one of the possible formulation of the classical Giraud's theorem for Grothendieck topoi which characterises them axiomatically as categories satisfying suitable internal properties. An important role in this result is played by the concept of weak descent. Using our definition of Grothendieck topoi and its equivalent interpretation which involve categories admitting a left exact small presentation, it will be relatively easy to explain how to homotopify the ordinary categorical setting (substituting presheaves categories with simplicial presheaves categories and localizations with Bousfield localizations) and get the desired notion of model topoi. We will finally state and sketch the proof of a meaningful version of Giraud's theorem for such model topoi and, if time permits, we will see how it applies to provide a nice class of examples of model topoi which present the homotopy theory of homotopy sheaves on a Grothendieck site.