## John P. D'Angelo

Professor
Department of Mathematics
University of Illinois at Urbana-Champaign
1409 W. Green Street 355 Altgeld Hall
tel: (217) 333-6406
fax: (217) 333-9576
email:jpda@math.uiuc.edu

### Short Curriculum Vita ResearchSeveral complex variables and geometry

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.

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

### Books

• Several Complex Variables and the Geometry of Real Hypersurfaces

• This book discusses points of finite type on real hypersurfaces in complex Euclidean space. (CRC Press 1993) Click on the research link above for a short list of corrections.
• Mathematical Thinking: Problem Solving and Proofs

• This book, coauthored with Doug West, helps students make the transition from elementary mathematics to upper division courses. See Doug's web page for more information including a list of typos. Go to home page for the book .

• Hermitian Analysis: From Fourier Series to Cauchy-Riemann Geometry.

• I have published this book (2013) in the Birkh\"auser Series Cornerstones in Math.

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).

• Inequalities from Complex Analysis

• I have published a Carus monograph on this topic. (MAA 2002) See my research page for a short list of typos.

• An Introduction to Complex Analysis and Geometry

• This book developed from Honors Classes on Complex Variables I gave to freshmen. It contains many exercises and many more advanced tidbits. (AMS 2011)

• Hermitian Analysis

• My teaching page has a link to a draft of this book, published by Springer-Birkhauser in 2013.

• Linear and complex analysis for applications

• My teaching page has a link to a draft of this book. The book was published by CRC Press in 2017.