## ResearchJohn P. D'Angelo

Please see Math Sci Net for my publications.

Several complex variables and geometry

I authored the book Several Complex Variables and the Geometry of Real Hypersurfaces; this book describes areas closely related to my research interests. In recent years I have been 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).

Here are some corrections and comments for the book Several Complex Variables and the Geometry of Real Hypersurfaces:

Proposition 3 on Page 148 is not true without an additional hypothesis. Jeff McNeal has an example of a polynomial of degree five in three complex variables whose zero set contains a singular variety, but for which the regular type is 4. One can add pseudoconvexity to the statement: "Let M be a pseudoconvex..." Also in Remark 3 on Page 147, add pseudoconvex to the first sentence.

The author has published the paper "A remark on finite type conditions" in the Journal of Geometric Analysis (2017) which provides a simple proof of a stronger assertion:

Suppose that the defining equation of M is written r = 2Re(z_n) + f(z', z'-bar, Im z_n) and that f has no pure terms in its Taylor expansion at 0. The following property suffices to prove that regular type four and singular type four coincide at 0. The function f(z',z'-bar, 0) vanishes to even order 2k at 0, and the Laplacian to order k on it is positive at 0. This property holds when M is pseudoconvex but much more generally as well.

Page 137. Equation (55). The denominator should be 2^(n-q-2). The proof is correct, but at the last line note that 2 * (1 / 2^{n-1-q}) = 1/ 2^{n-2-q} rather than 1/ 2^{n-q}.

Page 144. At the end of Remark 1, there is a reference to [D6]. It should be to [D5].

Page 153. Line 4 from the bottom: It should say that Aut(B_n) is the quotient of SU(n,1) by its center.

Page 154. First line of proof of Lemma 1. The second domain should be in $N$ space: $\Omega_2 \subset {\mathbb C}^N$

Page 195. "Although this result is not known". It is now known, for certain worm domains, that the particular solution orthogonal to the holomorphic functions does NOT have optimal regularity properties. See work of Barrett, Christ, and the survey article by Boas-Straube: Global regularity of the d-bar Neumann problem: a survey of the L2-Sobolev theory. in Several complex variables Math. Sci. Res. Inst. Publ., 37, Cambridge Univ. Press, Cambridge, 1999.

Page 195. 8 lines from the bottom: "there would a factor" should be "there would be a factor".

Page 197. Equation (14). In the first integral, there is a missing subscript on a \phi. It should be the partial derivative of \phi_j with respect to z_j. Note that equation (13) is correct as is the paragraph after (14).

Page 197. equation (18). The star should be a superscript, not a subscript.

Page 210. Equation (68). A ||\phi|| at the end should be a ||\phi||^2.

Page 209. One must be a bit careful about tangential vs. nontangential pseudodifferential operators.

Page 169. A few lines after (75), contractible should be replaced by "path-connected".

Page 171. "there are more than two hundred..." should be "there are more than one hundred". It should be noted that the sharp degree estimate $d \le 2N-3$ for monomial maps (when n=2) has been proved. See the paper by D'Angelo, Kos, and Riehl in Journal Geometric Analysis, 2003.

Hence the statement ''the maximum degree is 7'' is now part of a general theory.

The analogous statement for monomial maps in higher dimensions (n >2) is that $d \le {N-1 \over n-1}$. This result has been proved by Lebl and Peters. See Illinois J. Math 2012.

Page 175. The formula in (92) is correct, but it is worth noting that it can be also written as (2r+1)/s times {(2r-s) choose (s-1)}.

Page 115. Equation (127) the j should be a subscript. Thus the right-hand side is \sum_{i,j} |F_i P_j|^2

Page 114. The reference to [D8] should be to [D7].

Page 26. Lemma 3, "roots" should be "factors". The proof should fix this point as well.

Page 27. first line of proof of lemma 4, "roots" should be "factors".

Page 222. in Definition 6, after "For each choice of" add the word "generators".

Now for FUN STUFF:

In studying group invariant proper holomorphic mappings between balls in different dimensions I discovered a triangle of integers (page 175 in above book) 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

Hint: Solve the recurrence defined as follows:

p_0 = x

p_1 = x^3 + 3 xy

p_{n+2} = (x^2 + 2y)p_{n+1} - y^2 p_n

I have also discovered generalizations of this triangle where the coefficients are sometimes negative but the coefficients have remarkable combinatorial and number theoretic properties.

My current interests include complex variables analogues of Hilbert's seventeenth problem; when is a nonnegative real-analytic function a squared norm of a holomorphic mapping? More generally when is it the quotient of squared norms? I am also interested in the interpretation of such questions in terms of metrics on bundles.

See "Hermitian Analogues of Hilbert's 17th Problem", Advances in Math 226(2011) 4607-4637.

More on positivity conditions. My talk in Serra Negra, BRAZIL, August 2011. Go to https://math.uiuc.edu/~jpda/jpd-brazil-2011-beamer-copy.pdf Hermitian forms in CR Geometry

To see my talk at the AMS meeting in Kansas, 2012, go to https://math.uiuc.edu/~jpda/jpd-kansas-beamer.pdf Kansas 2012 talk

To see my talk at Johns Hopkins in April 2012, go to https://math.uiuc.edu/~jpda/jpd-hopkins-beamer-pub.pdf Hopkins 2012 talk

To see my talk at an REU in Fresno in June 2012, go to https://math.uiuc.edu/~jpda/jpd-fresno-beamer-12-pub.pdf Fresno 2012 REU talk

Here is my talk at the AMS Meeting at Akron, Ohio on Oct. 20, 2012. Akron2012

Here is my talk at Penn on Dec. 4, 2012. Penn2012

Here is my talk at Serra Negra on Aug. 5, 2013. Brazil2013

Here is my colloquium talk at Oklahoma State on March 28, 2014 Stillwater2014

Here is my colloquium talk at Penn on September 24, 2014: Penn2014

Here is my colloquium talk at Rutgers on September 26, 2014: Rutgers2014

Here is my talk at the Illinois sectional meeting of the MAA, March 27, 2015: MAA2015

Here is my talk at the Southern California Analysis and PDE meeting, May 3, 2015. SCAPDE 2015

Here is my talk at the Serra Negra CR Geometry and PDE meeting, August 5, 2015. Brazil 2015

Here is my talk at the Salt Lake City AMS meeting, April 9, 2016. Salt Lake City 2016

Here is my talk at the Toledo SCV meeting, May 13, 2016 Toledo 2016

Here is my talk at Oslo, Norway 8-22-16 Oslo 2016

Here is my talk at ECCAD, April 29, 2017 Take it to the limit

Here is my talk at SCAPDE, March 3, 2018. Iterated commutators

Here is my talk at AMS Meeting in Columbus, March 17, 2018. Hermitian invariant groups for holomorphic and CR maps

Here is my talk at CR Geometry Meeting in Levico, Italy. June 2018. Rational CR maps between spheres: old and new.

Here is my talk at the conference on holomorphic mappings in Ljubljana, September 2018. Groups associated with holomorphic mappings

Here is my talk at Brno, September 25, 2018. Rational CR maps between spheres: a compressed sensing problem.

I have published a book called Inequalities from Complex Analysis. It is in the Carus monograph series. This book considers questions such as this: Given a polynomial on complex Euclidean space that is positive on some set, does it agree with the quotient of squared norms of holomorphic polynomials on that set? The book begins with the definition of the complex numbers, and ends with recent research.

Here is a short list of typos and small corrections from the Carus book.

Page 65, line 9. "converges at $z$ should be "converges abosolutely at $z$"

Page 86, Exercise 9, z_1|^2 should be |z_1|^2.

Page 115. Lemma IV.5.8 should be restated as follows: Let (A_{jk}) be an n-by-n Hermitian matrix. Suppose that the principal (n-1) by (n-1) minor is diagonal. We then have formula (12). In the proof sketch drop the sentence that "Then show that the leading block is diagonal".

Comment on the previous. The lemma is used to prove Hadamard's inequality, and we may make this diagonal assumption inductively.

Page 127, statement of Theorem IV.6.4. It is assumed that "g is real-valued". That assumption is of course unnecessary. DROP "real-valued" and change "sup(g)" to "sup(|g|)".

Page 136, Theorem IV.6.3. It should be remarked that the constant $\pi$ is smallest possible, and that the reference [HLP] has considerable discussion of this inequality and it generalizations.

Page 141, comment to exercises 25 and 26. The reference to Exercise 24 should be to Exercise 25.

Page 211, (see also the last exercise on page 215). It is now known that P5) holds if and only if $a$ is less than $8$. See D'Angelo, J. and Varolin, D. Positivity conditions for Hermitian symmetric functions. Asian J. Math. 8 (2004), no. 2, 215–231.

Here is a typical result in the book: If a polynomial function is positive on the unit sphere in complex Euclidean space, then it agrees with a squared norm of a holomorphic polynomial mapping there.

By work of Putinar-Scheiderer, the previous statement does not hold for general algebraic strongly pseudoconvex hypersurfaces.

In July 2008 I gave a course at the Park City Mathematics Institute (PCMI). The lecture notes, called "Real and Complex Geometry Meet the Cauchy-Riemann Equations" appear in the PCMI lecture note series.

In August 2009 I gave a course at Serra Negra, Brazil on "The Unit Sphere and CR Geometry". The lecture notes are available: The Unit Sphere and CR Geometry

Recently I wrote two unusual articles in the Notices AMS. One of these, with Jeremy Tyson, is an Invitation to CR and sub-Riemannian Geometries (Feb. 2010). I also wrote my first recreational article, on Baseball and Markov Chains (April 2010).

Here is a short note on a lemma Martin Gardner would have loved. (2011) Gardner

Here is my talk at the Martin Gardner celebration 2012. Gardner2012

Here is my talk at Santa Barbara 2014 on Teaching complex variables to engineers. ECE-2014

Here is my talk at Santa Barbara 2014 on inverses and complex structures. SB-2014

I recently (2013) wrote "Hermitian Analysis: From Fourier Series to Cauchy-Riemann Geometry".

This book appears in the Birkhauser series "Cornerstones in Mathematics". Chapter 4 glimpses some of my research interests.

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".

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).

Page 125. The sketch of the second proof of Wirtinger inequality is not correct. The functions sin(n pi x) satisfy the ODE but not the integral equation because of boundary conditions. I will write a correction.

In 2017 I wrote the book "Linear and Complex Analysis for Applications".

Here are some typos and corrections from that book.

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 previous book, noted a few lines above.

In 2010 I wrote the book "An Introduction to Complex Analysis and Geometry", published by the AMS. It is an expanded version of a course I twice gave to honors freshmen. It includes a few tidbits from my research interests. Complex Analysis and Geometry Book (earlier version)

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