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

**Official NSF Award Description for Mathematicians**

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The Faculty Early Career Development Award (CAREER) will support Alvaro Pelayo's research at the intersection of dynamical systems, spectral theory and symplectic geometry.

More specifically, the CAREER Award will fund Pelayo's investigations on classical and quantum semitoric
systems. A
classical semitoric system is an integrable Hamiltonian system with two degrees of freedom for which one component generates a periodic flow.

For mathematicians, semitoric systems form the next natural class of systems after toric systems. Semitoric systems retain some properties of toric systems, while at the same time they exhibit a much greater flexibility. This flexibility is reflected in the existence of singularities which are dynamically and symplectically rich.

Semitoric systems are commonly found in simple physical models and arise naturally as examples in analysis, partial differential equations, algebraic geometry and symplectic geometry.

As a matter of fact, a semitoric
system defines a singular toric fibration whose base comes endowed with a singular integral affine structure. These singular affine structures
are a central concept in symplectic topology and mirror symmetry.

A semitoric quantum system is given by two commuting self-adjoint semiclassical operators acting on a Hilbert space whose principal
symbols form a classical semitoric system. One of Pelayo's main goals is to
study the spectral theory of semitoric quantum systems and how it relates to classical systems.

Concretely, Pelayo's plan is to work towards
verifying that the semiclassical joint spectrum of a quantum semitoric system determines completely the system; this is the Spectral Conjecture,
widely considered the most spectacular problem in the area. Proving this conjecture requires establishing a number of results in semiclassical analysis, giving a tool for future research in the field, independent of the conjecture.

Another component of Pelayo's research plans is to continue studying semitoric systems in a more general context, in particular as it regards to the study of the convexity and connectivity properties of the
singular Lagrangian fibrations which semitoric systems induce, and which are of special interest in mirror symmetry.