Regular homomorphisms of minimal sets

The classification of minimal sets is a central theme in abstract topological dynamics. Recently this work has been strengthened and extended by consideration of homomorphisms. Background material is presented in Chapter I. Given a flow on a compact Hausdorff space, the action extends naturally to the space of closed subsets, taken with the Hausdorff topology. These hyperspaces are discussed and used to give a new characterization of almost periodic homomorphisms. Regular minimal sets may be described as minimal subsets of enveloping semigroups. Regular homomorphisms are defined in Chapter II by extending this notion to homomorphisms with minimal range. Several characterizations are obtained. In Chapter III, some additional results on homomorphisms are obtained by relativizing enveloping semigroup notions. In Veech's paper on point distal flows, hyperspaces are used to associate an almost one-to-one homomorphism with a given homomorphism of metric minimal sets. In Chapter IV, a non-metric generalization of this construction is studied in detail using the new notion of a highly proximal homomorphism. An abstract characterization is obtained, involving only the abstract properties of homomorphisms. A strengthened version of the Veech Structure Theorem for point distal flows is proved. In Chapter V, the work in the earlier chapters is applied to the study of homomorphisms for which the almost periodic elements of the associated hyperspace are all finite. In the metric case, this is equivalent to having at least one fiber finite. Strong results are obtained by first assuming regularity, and then assuming that the relative proximal relation is closed as well.

Sonata Development at the Turn of the Century: From Liszt to Stravinsky

The sonata began to lose its position of predominance among compositions in the middle of the 19th century. Having been the platform for harmonic and thematic development of music since the late baroque period the sonata entered a process of reevaluation and experimentation with form. As a result fewer sonatas were being composed with some composers dropping the genre completely. This dissertation looks at the different approaches taken by the German, French and Russian schools of composition and compares the solo and chamber music applications of the sonata form. In the German tradition Franz Liszt's Sonata in b minor sets the standard for the revolutionary approach to form while the Berg Sonata is a very conservative application of form to an innovative use of extended chromaticism. Both composers chose to write one movement through composed pieces with Liszt working with a very expansive use of form and Berg being extremely compact and efficient. Among the Russian composers, Prokofieff's third sonata is also a one movement sonata, but he falls between Liszt and Berg in terms of the length of the piece and the use of innovative musical language. Scriabin uses a two movement approach, but keeps the element of a through composed piece with the same important material spanning both movements. Stravinsky is the most conservative of these with a three movement sonata that uses a mix of chromaticism and baroque and classical style influences. The French almost stopped composing true sonatas except for chamber music where Franck and Fauré write late romantic sonatas, while Debussy is very innovative within a three movement sonata. Estampes, by Debussy, are taken in almost as an afterthought to illustrate the direction Debussy takes in his piano solo music. While Estampes is by definition a set of character pieces they function like a sonata with three movements.

SELECTED VARIATIONS BY COMPOSERS IN GERMANY AND FRANCE

The Haward of Dictionary of Music (1983), defines variation as "a technique modifying a given musical idea." From the Baroque period on, the form and the techniques of variation were developed and enriched in Germany and France. Therefore, I presented the works of composers from these two nations. Even though there was a vast number of possibilities, I wanted to be scholastically fair and interesting in making my selections by choosing well-known pieces along with lesser-known ones. Haydn's well-known Variations in F minor consist of two sets of double variations which break into an improvisation fantasy. The first movement of Beethoven Sonata in A flat major, Op. 26, is a set of five variations on the composer's original theme. The variations are positioned in the first movement instead of Sonata-Allegro form. In 1861 Brahrns composed the Variations and Fugue, Op. 24, on the theme of Handel. Brahms displays a wealth of rhythmic, harmonic and textural contrasts in the variations. Chopin's E Major Variations without opus number are written on a Swiss influenced German folksong. Faure's Theme and Variations in C sharp minor, Op. 73, includes eleven variations. The work displays the composer's subtlety, grace and reticence. 12 Variationen iiber ein eigenes Thema were written by Alban Berg as a composition study with Schonberg. The Finale of Dutilleux's Piano Sonata, titled "Chorale with Variations", is written in an impressionistic style. A rich expressiveness is well blended in a classical form. In 1742, the remarkable Aria and thirty variations known as the Goldberg Variations were composed by J. S. Bach. The thirty Variations are unified by the bass line, which forms the foundation of the Aria. The pieces discussed above were presented in three recitals. Compact disc recordings of these recitals are available in the Michelle Smith Performing Arts Library of the Clarice Smith Performing Arts Center at the University of Maryland.

THE PIANO MUSIC OF ROBERT SCHUMANN (1810-1856) AND JOHANNES BRAHMS (1833- 1897): A SURVEY OF CONCERTI, VARIATIONS, AND CHARACTER PIECES

Robert Schumann (1810-1856) and Johannes Brahms (1833-1897), in some ways Robert Schumann's artistic descendant, are the most important and representative German piano composers during the Romantic period. Schumann was already a mature and established musician in 1853 when he first met the young Brahms and recognized his talents, an encounter that had a long-lasting affect on the lives and careers of both men. After Schumann’s mental breakdown and death, Brahms maintained his admiration of Schumann’s music and preserved an intimate relationship with Clara Schumann. In spite of the personal and musical closeness of the two men, Schumann’s music is stylistically distinct from that of Brahms. Brahms followed traditions from Baroque and Classical music, and avoided using images and expressive titles in his music. Brahms extraordinarily intermingled earlier musical forms with multicolored tones of German Romanticism. In contrast, Schumann saw himself as a radical composer devoted to personal emotionalism and spontaneity. He favored programmatic titles for his character pieces and extra-musical references in his music. While developing their own musical styles as German Romantic composers, Schumann and Brahms both utilized the piano as a resourceful tool for self-realization and compositional development. To investigate and compare the main characteristics of Schumann and Brahms’s piano music, I looked at three genres. First, in the category of the piano concerto, I chose two major Romantic works, Schumann’s A minor concerto and Brahms’s B-flat major concerto. Second, for the category of piano variations I included two sets by Brahms because the variation framework was such an important vehicle for him to express his musical thoughts. Schumann’s unique motivic approach to variation is displayed vividly in his character-piece cycle Carnaval. Third, the category of the character piece, perhaps the favorite medium of Romantic expression at the piano, is shown by Schumann’s Papillons and Brahms’s sets of pieces Op.118 and Op.119. This performance dissertation consists of three recitals performed in the Gildenhorn Recital Hall at the University of Maryland, College Park. These recitals are documented on compact disc recordings that are housed within the University of Maryland Library System.

Selected Piano Works by Schumann, Liszt, and Brahms Incorporating or Based on Pre-existing Music

This dissertation shows how Schumann, Liszt and Brahms composed piano works based in a variety of ways on other music that already existed. My idea to do this project came through my fascination with Brahms’s Variations on a Theme by Paganini, which was the first piece selected. Brahms composed six sets of variations for solo piano, and I also chose Variations and Fugue on a Theme by Handel. Besides the variations, I included Brahms’s Ballade in D minor, Op. 10 No. 1, based on the Scottish ballad “Edward”. This piece demonstrates that Brahms applied pre-existing music not only in the form of variations, but also in other genres. Among Romantic composers, Schumann and Liszt are two others besides Brahms whose music frequently quotes pre-existing materials. In Schumann’s output, the inspiration from Clara Wieck is significant. The best examples may be the Impromptus Op. 5 and the third movement of the Grand Sonata No. 3, Op. 14, in which Schumann quotes the music by Clara Wieck as the theme of the variations and in the other movements as well to unify the entire piece. In addition, Schumann quotes the old German folk song “Grossvater Tanz” (Grandfather Dance) in the finale of Papillons. The same tune also appears in Carnaval for a programmatic purpose. These two pieces are a clear illustration that Schumann applies pre-existing music, and in addition they represent the spirit of literary reference. Liszt is well known for his superb transformations of other composers’ works into glorious piano compositions. Liszt drew his inspirations from different genres, including both vocal and instrumental music. His ability to turn earlier musical materials into virtuosic solo piano pieces that demonstrate his brilliance in creating imaginative keyboard sounds is remarkable. Among those pieces composed by Liszt, terms such paraphrase, reminiscence, or fantasy frequently appear as titles. I selected two such pieces: Rigoletto: Paraphrase de concert, S. 434 and Rondeau fantastique sur un thème espagnol, S. 252. In addition, Liszt also uses variation form to explore the possibilities of pre-existing themes. The piece I chose to represent this is Variations on the Theme “Weinen, Klagen, Sorgen, Zagen” and Crucifixus from the Mass in B minor by J.S Bach, S.180. This dissertation comprises three piano recitals that were performed in 2010 and 2011 in Ulrich Recital Hall and Gildenhorn Recital Hall of the Clarice Smith Performing Arts Center of the University of Maryland. The recordings are documented on compact discs that are housed within the University of Maryland Library System.