Professor Gilbert is the leading analyst in the field of complex analytic methods for partial differential equations throughout the world. In a natural way these methods were applied long time ago in mathematical physics as e.g. for potential theory and fluid flows. In the 30th and 40th of the last century mainly in the Soviet Union and in the USA this area was intensively developed through e.g. I. Muskhelishvili, I.N. Vekua, L. Bers, W. Haack and others. On one hand the applications to elasticity theory and shell theory on the other the treatment of certain elliptic systems and equations in the plane via complex analysis had shown how powerful and elegant complex methods are.
As a young mathematician with a physical background R.P. Gilbert has started in the late 50s with his investigation of singularities of solutions to certain differential equations in higher dimensions in particular for GASP (generalized axially symmetric potentials). In connection with GASP-Theory he has studied Bergman integral operators and Riemann functions. This has led him to the so-called "method of ascent" and the Bergman-Gilbert operator. This latter has served R. Carroll as a motivation for his general "transmutation theory". After one decade of research he had become known to the Georgian school around I. Muskhelishvili and I.N. Vekua already. This contact was the beginning of his important world wide international cooperation.
In connection with the Ph.D. project of G.N. Hile he initiated the theory of generalized hyperanalytic functions, describing the theory of solutions to first order elliptic systems of 2n equations in the plane. Moreover, basic integral representation formulas in Clifford analysis have been given in this thesis. This theory was later further developed by the Gent school around R. Delanghe. And it was R.P. Gilbert who in the eighties made this Clifford analysis popular in mainland China. His 1983 jointly written monograph appeared at the beginning of the newly risen interest in Clifford analysis. It also has anticipated later interest in applying function theory of several complex variables to partial differential equations. Besides having graduated in mathematics R.P. Gilbert has also graduated in physics. Hence, he always was and is interested in applied problems especially from mathematical physics. He has studied problems in fluid dynamics, underwater acoustics, nonlinear waves, Hele-Shaw flows, planar filtration, porous media, biological mechanics. The methods applied are from complex analysis, potential theory, inverse problems as e.g. inverse scattering, homogenization, approximation theory, numerical analysis.
Many of his publications are joint ones with a variety of co-authors from all over the world. He has built these international co-operations with visiting prestigious institutions often on the basis of highly ranked awards like the Alexander von Humboldt Senior Scientist Award and the British Science Council Research Award. Through his grants many of his coworkers have visited the University of Delaware. A second group originated from his former Ph.D.-students. At present he is leading a strong USA-French research group on underwater acoustics. This group is very productive and its publications culminate in various book projects.
More than many other scientists Dr. Gilbert takes care of his students in particular his Ph.D. students. Instead of just advising them he immediately starts to collaborate with them and this collaboration often continues beyond the day they receive their Ph.D. degrees. Many of his co-authors are thus former Ph.D. students of his. In the just mentioned USA-French research group two of his former Ph.D. students are involved.
R.P. Gilbert has a deep interest in teaching. Once in the eighties visiting the University of Delaware myself I was witness of his teaching on symbolic computation - at that time using Macsyma - applied to problems from differential equations. His enthusiasm in this subject led to several book publications and one of the co-authors, W. Koepf, before a pure function theorist, became a specialist in symbolic computation and only because of this has found a tenure position at a German university.
Besides his eminent, important and numerous scientific work including several monographs, course books and edited proceedings initiated and co-organized by him beginning as early as 1970. R.P. Gilbert has two more great contributions for the mathematical community. They both originated from his desire to promote the area of analysis in a time where discrete mathematics and numerical methods attract the young generation of mathematicians. One of these contributions is the foundation of two journals "Applicable Analysis" and "Complex Variables, Theory and Applications". Years before the grants for mathematicians in the US were shifted from pure to applied mathematics he had the vision of the importance of applied mathematics in conjunction to pure mathematics. Both journals have shown to be perfect junction between pure and applied mathematics and proved to be very successful scientific journals (despite their extremely high prices). The same philosophy led R.P. Gilbert to found ISAAC, the International Society for Analysis, its Applications and Computation. Analysis with its applications and together with computational methods has to be considered as a unity. To promote this area and to attract young mathematicians to this field regularly international congresses are organized, thus opening a certain window on this huge field in mathematics. Moreover, at each congress, young talented researchers are awarded for their achievements in analysis, its applications and computation. Between these congresses workshops and special conferences are organized. How successful ISAAC is can be seen from their publications. In the ISAAC series with Kluwer 10 proceedings volumes have appeared since 1998, 5 more with other publishers.
R.P. Gilbert has become a central figure in mathematical analysis. He is the world leading expert in complex analysis devoted to partial differential equations. Since decades he is leading an international, very productive research team. He has attained some results in analysis which will forever be linked to his name. His contributions to the mathematical community in the form of his journals and the ISAAC distinguish him from all other mathematicians of our time. R.P. Gilbert is a remarkable analyst with great merits for the mathematical science.