University of Illinois at Urbana-Champaign

John P. D'Angelo

Department of Mathematics
University of Illinois at Urbana-Champaign
1409 W. Green Street 355 Altgeld Hall
tel: (217) 333-6406
fax: (217) 333-9576

Short Curriculum Vita


Several 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.

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.


Office Hours

Since I am no longer teaching, I hold office hours only by making an e-mail appointment.