Contact Information
355 Altgeld Hall, MC-382
1409 W. Green Street
Urbana, IL 61801
Research Areas
Research Interests
Complex Analysis and CR Geometry
See https://publish.illinois.edu/jpda-math/ for additional information. That site includes information about all my books and much more.
Here is an (elementary sounding) open problem which 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.
Research Description
My research centers around complex analysis and CR Geometry.
I have worked on finite type conditions for real hypersurfaces, on Hermitian analogues of Hilbert's 17th problem, on proper holomorphic maps between balls in different dimensions, and on groups associated with holomorphic mappings.
Education
BA, University of Pennsylvania, 1972
PhD, Mathematics, Princeton University, 1976 (PhD thesis advisor J. J. Kohn)
Awards and Honors
Stefan Bergman Prize. 1999
Kenneth D. Schmidt Professorial Scholar. 2005
LAS Dean's Award for Excellence in Undergraduate Teaching, Univ. of Illinois, 2005.
Fellow, Amer. Math Society. 2014
External Links
Highlighted Publications
Books
Several Complex Variables and the Geometry of Real Hypersurfaces. CRC Press, 1992.
Mathematical Thinking: Problem Solving and Proofs. Prentice-Hall, 2000. (joint with Douglas B. West)
Inequalities from Complex Analysis. Mathematics Assoc. of America, 2002.
An Introduction to Complex Analysis and Geometry. American Math. Society, 2010.
Hermitian Analysis: From Fourier series to Cauchy-Riemann Geometry. Springer, 2013.
Linear and Complex Analysis for Applications. CRC Press, 2017.
Rational Sphere Maps. Springer-Birkhauser, 2021.