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Thursday, January 20, 2005

**Abstract:** I will consider a class of functions, called Hermitian algebraic, whose domains are the total space of a holomorphic line bundle. I will define these functions, and show that they are generalizations of very natural mathematical objects, namely bihomogeneous Hermitian polynomials. I will talk about a few elementary properties of Hermitian algebraic functions, and then proceed to state a Hermitian algebraic analogue of Hilbert's 17th problem for such functions. (Recall Hilbert's 17th problem = Artin's Theorem: Any non-negative polynomial is a sum of squares of rational functions.) The analog can be vaguely stated as follows: Is any non-negative Hermitian algebraic function a quotient of squares of holomorphic mappings? This Hermitian algebraic analogue of Hilbert's 17th problem was considered by D. Quillen in a certain important case, and later by D'Angelo and Catlin in the general setting. Both Quillen and Catlin-D'Angelo settled the problem in the affirmative under the hypothesis that the Hermitian function is strictly positive away from the zero section. D'Angelo also showed that in the Hermitian algebraic setting, there are non-negative Hermitian algebraic functions that are not quotients of squares. I will state necessary and sufficient conditions under which a non-negative Hermitian algebraic function is a quotient of squares. I will then indicate how to prove this result in the case considered by D'Angelo-Catlin. If time permits, I will indicate how one goes on to the general case.

Thursday, February 10, 2005

Thursday, February 24, 2005

Thursday, March 3, 2005

Thursday, March 10, 2005

Thursday, March 17, 2005

Thursday, April 21, 2005

Thursday, April 28, 2005

Thursday, August 25, 2005

Thursday, September 8, 2005

Thursday, September 15, 2005

Thursday, September 29, 2005

Thursday, October 6, 2005

Thursday, October 13, 2005

Thursday, October 20, 2005

Friday, October 28, 2005

Friday, November 4, 2005

Thursday, November 10, 2005

Thursday, November 17, 2005

Thursday, December 1, 2005

Friday, December 2, 2005

Thursday, December 8, 2005