Expedition in Data and Harmonic Analysis on Graphs

The graph Laplacian operator is widely studied in spectral graph theory largely due to its importance in modern data analysis. Recently, the Fourier transform and other time-frequency operators have been defined on graphs using Laplacian eigenvalues and eigenvectors. We extend these results and prove that the translation operator to the i’th node is invertible if and only if all eigenvectors are nonzero on the i’th node. Because of this dependency on the support of eigenvectors we study the characteristic set of Laplacian eigenvectors. We prove that the Fiedler vector of a planar graph cannot vanish on large neighborhoods and then explicitly construct a family of non-planar graphs that do exhibit this property. We then prove original results in modern analysis on graphs. We extend results on spectral graph wavelets to create vertex-dyanamic spectral graph wavelets whose support depends on both scale and translation parameters. We prove that Spielman’s Twice-Ramanujan graph sparsifying algorithm cannot outperform his conjectured optimal sparsification constant. Finally, we present numerical results on graph conditioning, in which edges of a graph are rescaled to best approximate the complete graph and reduce average commute time.

RECORDING PROJECT: THE COMPLETE (PUBLISHED) PIANO WORKS OF ROBERT HELPS

This recording represents the complete solo piano works of Robert Helps (1928-2001). As of this writing (March, 2008), approx.120 minutes of Helps' solo piano music has been published, all of which is included on the Digital Media (CD). This project includes the following works: Trois Hommages, Quartet, Nocturne, Valse Mirage, In Retrospect, Three Etudes, Portrait, Three Etudes for the Left Hand, Starscape, Recollections, Shall We Dance and Image. (His few remaining pieces are officially "pending publication" and are therefore not included in this project.) Robert Helps, American pianist and composer, enjoyed a successful career on both fronts, teaching at such institutions as San Francisco Conservatory, Stanford University, the University of California, Berkeley, the New England Conservatory, the Manhattan School of Music and Princeton University. Helps, never the recipient of a university or conservatory degree, received private instruction from pianist Abby Whiteside and composer Roger Sessions. His recording of the Sessions' Sonatas is considered to be their benchmark performance. As a composer, he received commission and awards from the American Academy of Arts and Letters, the Ford Foundation, the Guggenheim Foundation and the National Endowment for the Arts. Helps' compositions were anachronistic in style: his compositional style ranges from Post-Impressionism, Neo­ Romanticsim and early 20th century Atonalism, although he never engaged in serial practices. Since his death in 2001, the Robert Helps Trust has been established at the University of South Florida. Funds are being used to support the continued publishing of his scores. The Robert Helps International Composition Competition and Festival was established in 2005.

THE COMPLETE PRELUDES OF ALEXANDER SCRIABIN: THE EVOLUTION OF HIS REVOLUTIONARY COMPOSITIONAL STYLE

During the second half of the nineteenth century, a series of remarkable advances in musical composition emerged in the works of such innovative spirits as Franz Liszt, Hector Berlioz and Richard Wagner. Their pioneering works exerted an extraordinary impact on the music of the subsequent generation of composers--of disparate nationalities-who were active at the dawn of the 20th century: Including most notably Claude-Achille Debussy, Gustav Mahler, Richard Strauss, and Alexander Nikolayevich Scriabin. These important musical figures, each one leaving an indelible and formative imprint on late-nineteenth century Romantic style, together launched the modern era in music. Scriabin stands alone as a transcendental visionary: His music, initiated in the fashion of Chopin and Liszt, wanders through the realms of Debussy and Wagner, and, ultimately abandoning late Romantic tradition, unlocks the heretofore unforeseen power of atonality, bitonality, polyrhythms and key-signature free compositions. Arguably, Scriabin's compositions count among the most innovative, idiosyncratic and bewitching of all time. The development of Scriabin's groundbreaking compositional style is best understood by means of his piano works, which comprise the majority of his oeuvre. Beyond the larger works-his twelve sonatas, a concerto and a fantasy-Scriabin's piano explorations are also represented by miniature gems: The mazurkas, impromptus, waltzes, poems, a polonaise, etudes, nocturnes, morceaux and, in particular, the preludes. Scriabin's 90 preludes for piano, arranged in several opus numbers, richly exemplify the striking evolution of his ingenious music, his idiosyncratic philosophy and his provocative personality.