**The Kolchin Seminar in Differential
Algebra **

**Saturday February 21 ^{st}
2004**

**Henri Gillet****
**University of Illinois at Chicago

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

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 **