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The Kolchin Seminar in Differential Algebra

Hunter College - Room HW706

Saturday February 21st 2004

Henri Gillet
University of Illinois at Chicago
Derivations, connections, exponential maps, and the Riemann-Roch Theorem (I)
2:00-3:00PM

Abstract: I will discuss joint work with my student Fatih Unlu, in which we apply ideas of Toledo, Tong and Kapranov to give an explicit proof of the Riemann­Roch theorem using ``formal'' methods. The key point is that a connection on the tangent bundle of an algebraic variety gives rise to a derivation on the coordinate ring of the variety. The original approach of Toledo and Tong used Cech cohomology, we replace the Cech methods with real and formal analytic techniques. However, I shall start with a "linear algebra" proof of the Gauss-Bonnet theorem.

Jerry Ianni
LaGuardia Community College
Synthetic musical scales and Busoni's problem
3:30-4:30PM

Abstract: This expository talk will begin with the definitions of synthetic musical scales and Busoni scales, relative to a literal alphabet. The number of Busoni scales obtained using three inflections will be presented, along with an enumeration of all possible Busoni scale structures. Algorithmic procedures for constructing Busoni scales will be described. A couple of theorems concerning the realizability of synthetic musical scales as Busoni scales will be discussed, and a few relevant computations will be presented using MAPLE. The mathematical techniques featured in this talk come primarily from enumerative combinatorics and matrix algebra. In particular, generating functions, precedence matrices, and computations with block matrices will be on display.
References:
J. Murray Barbour, Synthetic Musical Scales, The American Mathematical Monthly 36 (1929) 155 - 160.
Ferrucio Busoni, Entwurf einer neuen Aesthetik der Tonkunst (1907), English translation by Th. Baker (G. Schirmer, Inc., New York, 1911), pp. 29 - 30.
Robert M. Mason, Enumeration of Synthetic Musical Scales by Matrix Algebra and a Catalogue of Busoni Scales, Journal of Music Theory 14 (1970) 92 - 126.

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Henri Gillet
University of Illinois at Chicago
Derivations, connections, exponential maps, and the Riemann-Roch Theorem (II)
5:00-6:00PM

Abstract: TBA

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Coffee and tea will be served beginning at 1:30 PM
Room HW706

Click here for directions to Hunter College and location of room

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