A theory is more impressive the greater the simplicity of its premises, the more kinds of things it relates, and the more extended its area of applicability.
A. Einstein
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Summary of G2 : Spectral Theory and Semiclassical Analysis 
In the 1970s and 1980s Colin de Verdiere, Duistermaat, Guillemin, and Sjostrand, proved a series of major results in spectral theory of (differential, pseudodifferential) operators. Among the most striking ones are Colin de Verdiere's inverse results for collections of commuting pseudodifferential operators on cotangent bundles. The problems in G2 belong to the realm of classical questions in inverse spectral theory going back to these pioneer works.
I am primarily concerned with studying semiclassical spectral theory of quantum integrable systems using microlocal analysis for pseudodifferential operators and BerezinToeplitz operators. Microlocal analysis is a branch of analysis which establishes a deep connection between geometry and PDEs (it is connected at its core with harmonic analysis and representation theory). Even though quantum integrable systems date back to the early days of quantum mechanics, such as the work of Bohr, Sommerfeld and Einstein, the theory did not blossom at the time. The development of semiclassical analysis with microlocal techniques in the last forty years now permits a constant interplay between spectral theory and symplectic geometry. The goal of my research in G2 is to use this interplay to prove inverse results about quantum systems.
I have been working on isospectral problems for quantum integrable systems, that is, studying when the semiclassical joint spectrum of the system, given by a sequence of commuting BerezinToeplitz or pseudodifferential operators on a sequence of Hilbert spaces, determines the classical integrable system given by the symplectic manifold and commuting Hamiltonians. Recent results at the intersection of symplectic and spectral geometry suggest that symplectic invariants are better encoded in spectral information than Riemannian invariants (Charles, myself, and Vu Ngoc proved this recently for quantum toric systems).
The question of isospectrality in Riemannian geometry may be traced back to Weyl (1910) and is most well known thanks to an article by Kac (1966), who himself attributes the question to Bochner. Kac popularized the sentence: "can one hear the shape of a drum?", to refer to this type of isospectral problem. Bochner and Kac's question has a negative answer in this case, even for planar domains with Dirichlet boundary conditions; major results in this direction were proven OsgoodPhillipsSarnak, and Zelditch, among others. An important observation of Sarnak is that in this context a much better question to ask is whether the set of isospectral domains is finite or compact.
Specific topics I have worked on:

Semiclassical Inverse Spectral theory for Singularities:
studying the semiclassical spectral theory of singularities of quantum integrable systems, combining analytic methods developed in the past fifteen years in semiclassical analysis with results from symplectic geometry going back to the foundational works of Weinstein in the 1970s;

Global Semiclassical Spectral Theory:
studying and describing the semiclassical spectral theory in detail for large classes of quantum integrable systems. Then using this theory to study isospectrality questions for these systems, in collaboration with Charles, Polterovich, and Vu Ngoc. In some cases one can prove that the semiclassical spectrum of a quantum system determines the classical system given by the principal symbols. One of the main results in this group is an analogue of the Colin de Verdiere inverse spectral results on pseudodifferential operators, for the case of any prequantizable compact manifold and BerezinToeplitz operators (it uses microlocal methods developed since around 2003);

Quantization and Quantum Integrable Systems:
studying the existence of different types of quantizations and for different types of classical systems. Currently studying the relations to important works on the more algebraic side by Reshetikhin, Yakimov, and others. In the past five years I worked with Bertram Kostant in writing a treatment on geometric quantization from the angle of Lie theory and representation theory;

Spectral Theory for Integrable Systems from Physics:
I have worked on understanding quantum versions of physical systems typical in mechanics (see Marsden and Ratiu book for several examples). This includes the JaynesCummings model from quantum optics.
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