Hearing Delzant Polytopes From the Equivariant Spectrum

Let M^{2n} be a symplectic toric manifold with a fixed T^n-action and with a toric K\"ahler metric g. Abreu asked whether the spectrum of the Laplace operator $\Delta_g$ on $\mathcal{C}^\infty(M)$ determines the moment polytope of M, and hence by Delzant's theorem determines M up to symplectomorphism. We report on some progress made on an equivariant version of this conjecture. If the moment polygon of M^4 is generic and does not have too many pairs of parallel sides, the so-called equivariant spectrum of M and the spectrum of its associated real manifold M_R determine its polygon, up to translation and a small number of choices. For M of arbitrary even dimension and with integer cohomology class, the equivariant spectrum of the Laplacian acting on sections of a naturally associated line bundle determines the moment polytope of M.

Perception of emotion in sounded and imagined music

WE STUDIED THE EMOTIONAL RESPONSES BY MUSICIANS to familiar classical music excerpts both when the music was sounded, and when it was imagined.We used continuous response methodology to record response profiles for the dimensions of valence and arousal simultaneously and then on the single dimension of emotionality. The response profiles were compared using cross-correlation analysis, and an analysis of responses to musical feature turning points, which isolate instances of change in musical features thought to influence valence and arousal responses. We found strong similarity between the use of an emotionality arousal scale across the stimuli, regardless of condition (imagined or sounded). A majority of participants were able to create emotional response profiles while imagining the music, which were similar in timing to the response profiles created while listening to the sounded music.We conclude that similar mechanisms may be involved in the processing of emotion in music when the music is sounded and when imagined.