My book Several Complex Variables and the Geometry of Real Hypersurfaces describes areas closely related to some of my research interests. Click on the research link for a short list of corrections. 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 and proper holomorphic mappings between balls in different dimensions.
For the last thirty years much of my research has concerned Rational Sphere Maps. In 2021, I published a book with this title in the Springer-Birkhauser series PROGRESS IN MATHEMATICS. See my research page for more information, including a short list of typos and clarifications. The results and examples in this book include connections to many other branches of mathematics, including CR Geometry and combinatorial number theory.
My book Hermitian Analysis: From Fourier Series to Cauchy-Riemann Geometry gives some indications of additional research interests. See especially Chapter 4 of the first edition and Chapters 4 and 5 of the second edition.
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
Here is an elementary talk I gave on "Algebraic Combinatorics Arising in CR Geometry": Algebraic Combinatorics Arising in CR Geometry
OPEN PROBLEM: This fascinating open problem is discussed in the book on Rational Sphere Maps and in my paper An optimization problem arising in CR geometry, Illinois J. Math. 65 (2021), no. 2, 475-498.
Suppose that p(x,y) is a polynomial of degree d, that p(x,0)= x^d, that p(0,y)=y^d, that p(x,1-x)=1 for all x, and that the coefficients of p are all non-negative. What is the minimum value m(2,d) of p(1,1)? The values of 2d-m(2,d) are known for all d up to 202 and appear in the book.
Mathematical Thinking: Problem Solving and Proofs
I published a second edition in 2019. Here is the link:
The second edition includes a new Chapter on groups associated with holomorphic mappings.
Here are some typos and corrections in the first edition:
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 97. In Proposition 3.4, item (2), the second term should not have a $\sigma$ and the third term should be mulitplied by ${1 \over \sigma}$.
Page 113. Exercise 3.39 of Chapter 3. M should be multiplication by exp(-x^2/2) rather than by exp(-x^2).
Page 125. The sketch of the second proof of the Wirtinger inequality is not quite correct.
The functions sin(n pi x) satisfy the ODE but not the integral equation because of boundary conditions.
There are several similar correct approaches to this problem. We can consider either T*T or TT*
If we consider T*T, then we get the boundary conditions f(1)=0 and f'(0) = 0. The eigenfunctions in this case are cos(x (2n+1) pi/2 ). The maximum eigenvalue becomes 4/pi^2, hence the norm of T^*T is 4/pi^2, and hence the norm of T is 2/pi. We obtain the inequality (in L^2 norms on [0,1]) ||f||_2 \le (2/pi) ||f'||_2 when we assume f(0)=0. If we further assume f vanishes at 1, by symmetry, we get the inequality ||f||_2 \le (1\pi) ||f'||_2. The Wirtinger inequality follows, as noted below.
Remark: If we consider TT^*, then we get boundary conditions f(0)=0 and f'(1) = 0. Now the eigenfunctions are sin(x (2n+1) pi/2) and again we obtain the norm of T is 2/pi.
Once we know that the norm is 2/pi for the operator T on L^2[0,1], the Wirtinger inequality follows. Setting x=2t-1 then rescales things to the interval [-1,1]. The even function f'(x) = cos(x pi/2) vanishes at both endpoints of [-1,1] and rescales to cos((2t-1) pi/2) = sin (pi t), which vanishes at both 0 and 1. The norm of the integral operator on functions vanishing at both endpoints is then 1/pi, and the Wirtinger inequality follows.
One cannot assume at the start that f vanishes at both endpoints; there are no such eigenfunctions for T*T.
Page 126. Formula (P.1) should have a C^2 on the right-hand side.
click here for additions, errata, etc.
I have placed my story "Listing My Life Away" on this site.
This humorous story includes anecdotes about pop music, lists of my favorite songs, and numerous other amusements.
Here are some comments, typos, and corrections.
Page 6. Item 1) of Theorem 1. "For each B \in V" should be "For each B \in W".
Page 37. In the first sentence of section 10, "there is basis" should be "there is a basis".
Page 75. Exercise 1.2. "minimum" should be "local minimum".
Page 82. In the third paragraph of the section called "Line integrals and work", after the second sentence, one should add three sentences: Continuous simple closed curves in the plane ${\mathbb R}^2$ are called {\bf Jordan curves}. The Jordan curve theorem asserts that each Jordan curve divides the complement of the image of the curve into two regions, the interior and the exterior. We assume this theorem without proof, although we only apply it in situations where it is easy to prove.
Page 94. Three lines above the picture; in both line integrals $d{\bf S}$ should be $d{\bf l}$.
Page 121. Definition 6.1 should say "(1) p is called removable if lim_{z \to p} f(z) exists".
Page 67. Four lines from the bottom, the matrices on the left should be multiplied in the opposite order.
Page 143. Just above Corollary 1.9, there is a missing F(s) in the limit.
Page 153. The operator T should be a bold letter.
Page 154. Example 2.6. In the first displayed equation, the far left and far right terms are correct, but the middle term has a typo. The plus sign between the two log terms should be a minus sign.
Page 160. 11 lines from the bottom, a $c_n$ should be an $a_n$.
Page 165. Lemma 4.7. The value of the integral should be $\sqrt{\pi}$ instead of $\sqrt{2 \ pi}$.
Page 166. In the subsequent Example 4.2, the value of the integral is correct. BUT, the sentence saying what the Fourier transform of G(x,\sigma) is has included a factor of \sigma which shouldn't be there. The sentence should read "Its Fourier transform is ${1 \over \sqrt{2 \pi}} e^{- w^2 \sigma^2 \over 2}, which is (1 over \sigma) times the density function of a Gaussian..."
Page 174. In equation (1), the exponent in the denominator of the third term in the middle expression should be 4 (not 2).
Page 196. Proposition 8.1. The word "if" should be added after "if and only". One could also add "Assume q and r are real-valued continuous functions." as the first sentence of the statement.
Page 141. A right parenthesis is missing in the infinite product formula for sinc(x).
Page 203. Example 10.2 has the same error as on page 125 of the Hermitian Analysis book, noted above. The reasoning can then be used to give a second proof of Theorem 9.1.