My book Several Complex Variables and the Geometry of Real Hypersurfaces describes areas closely related to some of my research interests. For many years I have been interested in CR mappings between spheres of different dimensions. I am also interested in positivity conditions for real-analytic real-valued functions of several complex variables.For example, David Catlin and I gave the following necessary and sufficient condition for a bihomogeneous polynomial $p$ of $n$ complex variables to be positive away from the origin. There is an integer $d$ so that $p$ is the quotient of squared norms of homogeneous holomorphic polynomial mappings, the numerator vanishes only at the origin, and the denominator is the d-th power of the Euclidean norm. We have extended this result to an isometric embedding theorem for holomorphic bundles. See Math Research Letters 6 (1999).
My current research interests concern Hermitian analogues of Hilbert's 17th problem.
My book Hermitian Analysis: From Fourier Series to Cauchy-Riemann Geometry gives some indications of additional reserach interests. See especially Chapter 4.
In studying group invariant proper holomorphic mappings between balls in different dimensions I discovered a triangle of integers bearing a neat relationship to Pascal's triangle. Here are the first few rows! Can you figure out the rest?
1 1
1 3 1
1 5 5 1
1 7 14 7 1
1 9 27 30 9 1
1 11 44 77 55 11 1
1 13 65 156 182 91 13 1
1 15 90 275 450 378 140 15 1
Mathematical Thinking: Problem Solving and Proofs
Here are some typos and corrections in this book:
Page 7. Exercise 1.7. The final equation should be $|p(e^i\theta)|^2 = f(\theta)$.
Page 32. Proof of Corollary 1.7. In both displayed equations, exp(inx) should be exp(-inx) and exp(-inx) should be exp(inx).
Page 33, just below formula (50), "convolution of the Fourier series" is not quite correct. It is the convolution with the function h whose Fourier series is...
Page 35, proof of Theorem 1.11, replace "in" with "on compact subsets of" just before "the unit disk".
Page 40, Exercise 1.63 should be Exercise 1.64.
Page 68. In Exercise 2.35, the reference to Exercise 2.23 should be to Exercise 2.33.
Pages 79, 80. There is some confusion between the eigenvalues of L and the eigenvalues of (L-kI).
Page 113. Exercise 3.39 of Chapter 3. M should be multiplication by exp(-x^2/2) rather than by exp(-x^2).
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