2011 • 90 Pages • 60.93 MB • English

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A Personal Mathematical History I wasn’t always a mathematician Institute for Fluid Dynamics and Applied Mathematics 1961-65 Assistant Professor and Associate Professor Indiana University Bloomington, Professor 1966-75 Analytic Methods in Mathematical Physics My First Monograph Generalized Hyperanalytic Function Theory The Georgians Free University Berlin and the Hahn-Meitner Institute 1974-75 University of Delaware, Unidel Professor 1975 Some Work in Elasticity, etc. I visit China for the First Time, 1984 Research in Underwater Acoustics The 65th Birthday Conference in Graz The Cold War Ends: No More Underwater Acoustics The Birth of ISAAC I am 63 and I begin to work in Homogenization My 70th Birthday Conference in Frejus, France The 75thBirthdayConferenceinOrlando Medical Mathematics A Personal Mathematical History Robert Pertsch-Ru¨sterholz Gilbert Department of Mathematics

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A Personal Mathematical History I wasn’t always a mathematician Institute for Fluid Dynamics and Applied Mathematics 1961-65 Assistant Professor and Associate Professor Indiana University Bloomington, Professor 1966-75 Analytic Methods in Mathematical Physics My First Monograph Generalized Hyperanalytic Function Theory The Georgians Free University Berlin and the Hahn-Meitner Institute 1974-75 University of Delaware, Unidel Professor 1975 Some Work in Elasticity, etc. I visit China for the First Time, 1984 Research in Underwater Acoustics The 65th Birthday Conference in Graz The Cold War Ends: No More Underwater Acoustics The Birth of ISAAC I am 63 and I begin to work in Homogenization My 70th Birthday Conference in Frejus, France The 75thBirthdayConferenceinOrlando Medical Mathematics I wasn’t always a mathematician Originally I was a student of physics and obtained a BSc and MSc in Physics before switching to mathematics. The cause was the Heisenberg Uncertainty Principle, which led me to write to Albert Einstein about my uncertainty about Uncertainty. After switching to the Mathematics Department I worked on a project with Hirsh Cohen and it resulted in a paper. Hirsh left Carnegie for Rennselaer. I began working with Zeev Nehari, who had recently joined the Department from Washington University. This was professionally a good move, as Zeev Nehari was one of the preeminent complex analysts during the 1950s.

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th th th A Personal Mathematical History I wasn’t always a mathematician Institute for Fluid Dynamics and Applied Mathematics 1961-65 Assistant Professor and Associate Professor Indiana University Bloomington, Professor 1966-75 Analytic Methods in Mathematical Physics My First Monograph Generalized Hyperanalytic Function Theory The Georgians Free University Berlin and the Hahn-Meitner Institute 1974-75 University of Delaware, Unidel Professor 1975 Some Work in Elasticity, etc. I visit China for the First Time, 1984 Research in Underwater Acoustics The 65 Birthday Conference in Graz The Cold War Ends: No More Underwater Acoustics The Birth of ISAAC I am 63 and I begin to work in Homogenization My 70 Birthday Conference in Frejus, France The 75 BirthdayConferenceinOrlando Medical Mathematics

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th th th A Personal Mathematical History I wasn’t always a mathematician Institute for Fluid Dynamics and Applied Mathematics 1961-65 Assistant Professor and Associate Professor Indiana University Bloomington, Professor 1966-75 Analytic Methods in Mathematical Physics My First Monograph Generalized Hyperanalytic Function Theory The Georgians Free University Berlin and the Hahn-Meitner Institute 1974-75 University of Delaware, Unidel Professor 1975 Some Work in Elasticity, etc. I visit China for the First Time, 1984 Research in Underwater Acoustics The 65 Birthday Conference in Graz The Cold War Ends: No More Underwater Acoustics The Birth of ISAAC I am 63 and I begin to work in Homogenization My 70 Birthday Conference in Frejus, France The 75 BirthdayConferenceinOrlando Medical Mathematics This is the paper with Hirsh Cohen on free-boundary problems, an area I would return to after many years.

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th th th A Personal Mathematical History I wasn’t always a mathematician Institute for Fluid Dynamics and Applied Mathematics 1961-65 Assistant Professor and Associate Professor Indiana University Bloomington, Professor 1966-75 Analytic Methods in Mathematical Physics My First Monograph Generalized Hyperanalytic Function Theory The Georgians Free University Berlin and the Hahn-Meitner Institute 1974-75 University of Delaware, Unidel Professor 1975 Some Work in Elasticity, etc. I visit China for the First Time, 1984 Research in Underwater Acoustics The 65 Birthday Conference in Graz The Cold War Ends: No More Underwater Acoustics The Birth of ISAAC I am 63 and I begin to work in Homogenization My 70 Birthday Conference in Frejus, France The 75 BirthdayConferenceinOrlando Medical Mathematics For my PhD thesis I used Integral Operator Methods, ´a le Stefan Bergman, which were used to determine the analytic and singular behavior of various classes of elliptic partial diﬀerential equations. The Bergman-Whittaker Operator generates harmonic functions in 3 R , ∫ d ζ H(X) = B3(f ) := f (u, ζ) , ζ C from functions of two complex variable f (u, ζ), where X = (x1, x2, x3), u = x1 (ζ − 1/ζ) + ix2 (ζ + 1/ζ) + x3 is an 1 auxillary variable and C is a contour in C . I proved necessary and suﬃcient conditions for H(X) to be singular in terms of its analytic associate.

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th th th A Personal Mathematical History I wasn’t always a mathematician Institute for Fluid Dynamics and Applied Mathematics 1961-65 Assistant Professor and Associate Professor Indiana University Bloomington, Professor 1966-75 Analytic Methods in Mathematical Physics My First Monograph Generalized Hyperanalytic Function Theory The Georgians Free University Berlin and the Hahn-Meitner Institute 1974-75 University of Delaware, Unidel Professor 1975 Some Work in Elasticity, etc. I visit China for the First Time, 1984 Research in Underwater Acoustics The 65 Birthday Conference in Graz The Cold War Ends: No More Underwater Acoustics The Birth of ISAAC I am 63 and I begin to work in Homogenization My 70 Birthday Conference in Frejus, France The 75 BirthdayConferenceinOrlando Medical Mathematics Theorem −1 Let the deﬁning function for the set of singularities of f (u, ζ)ζ 2 be a global function in C . Then if h(u, ζ) := S(X; ζ) = 0 is such a function we have that H(X) = B3f is regular for all points X which may be reached by continuation along a curve Γ starting at an initial point of deﬁnition X0, provided X and hence, the curve Γ, do not lie on the intersection S := {X : S(X; ζ) = 0} ∩ {X : Sζ(X; ζ) = 0} .

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th th th A Personal Mathematical History I wasn’t always a mathematician Institute for Fluid Dynamics and Applied Mathematics 1961-65 Assistant Professor and Associate Professor Indiana University Bloomington, Professor 1966-75 Analytic Methods in Mathematical Physics My First Monograph Generalized Hyperanalytic Function Theory The Georgians Free University Berlin and the Hahn-Meitner Institute 1974-75 University of Delaware, Unidel Professor 1975 Some Work in Elasticity, etc. I visit China for the First Time, 1984 Research in Underwater Acoustics The 65 Birthday Conference in Graz The Cold War Ends: No More Underwater Acoustics The Birth of ISAAC I am 63 and I begin to work in Homogenization My 70 Birthday Conference in Frejus, France The 75 BirthdayConferenceinOrlando Medical Mathematics Figure: Zeev Nehari, ne´e Wili Weissbach

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th th th A Personal Mathematical History I wasn’t always a mathematician Institute for Fluid Dynamics and Applied Mathematics 1961-65 Assistant Professor and Associate Professor Indiana University Bloomington, Professor 1966-75 Analytic Methods in Mathematical Physics My First Monograph Generalized Hyperanalytic Function Theory The Georgians Free University Berlin and the Hahn-Meitner Institute 1974-75 University of Delaware, Unidel Professor 1975 Some Work in Elasticity, etc. I visit China for the First Time, 1984 Research in Underwater Acoustics The 65 Birthday Conference in Graz The Cold War Ends: No More Underwater Acoustics The Birth of ISAAC I am 63 and I begin to work in Homogenization My 70 Birthday Conference in Frejus, France The 75 BirthdayConferenceinOrlando Medical Mathematics Figure: Graduation, Carnegie-Mellon 1958

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th th th A Personal Mathematical History I wasn’t always a mathematician Institute for Fluid Dynamics and Applied Mathematics 1961-65 Assistant Professor and Associate Professor Indiana University Bloomington, Professor 1966-75 Analytic Methods in Mathematical Physics My First Monograph Generalized Hyperanalytic Function Theory The Georgians Free University Berlin and the Hahn-Meitner Institute 1974-75 University of Delaware, Unidel Professor 1975 Some Work in Elasticity, etc. I visit China for the First Time, 1984 Research in Underwater Acoustics The 65 Birthday Conference in Graz The Cold War Ends: No More Underwater Acoustics The Birth of ISAAC I am 63 and I begin to work in Homogenization My 70 Birthday Conference in Frejus, France The 75 BirthdayConferenceinOrlando Medical Mathematics Using the methods of projective diﬀerential geometry we were able to prove, among other results, Theorem Let H(X) be a harmonic function given by H(X) = B3f where the singulaities of f (u, ζ) have a global representation, h(u, ζ) := ζu − φ(ζ) = 0. Then the singularities of H(X) must lie on a develople in P. Further, if φ(ζ) is a polynomial of degree th n, n > 2, then the singularities are twisted n degree (complex) space curves.

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th th th A Personal Mathematical History I wasn’t always a mathematician Institute for Fluid Dynamics and Applied Mathematics 1961-65 Assistant Professor and Associate Professor Indiana University Bloomington, Professor 1966-75 Analytic Methods in Mathematical Physics My First Monograph Generalized Hyperanalytic Function Theory The Georgians Free University Berlin and the Hahn-Meitner Institute 1974-75 University of Delaware, Unidel Professor 1975 Some Work in Elasticity, etc. I visit China for the First Time, 1984 Research in Underwater Acoustics The 65 Birthday Conference in Graz The Cold War Ends: No More Underwater Acoustics The Birth of ISAAC I am 63 and I begin to work in Homogenization My 70 Birthday Conference in Frejus, France The 75 BirthdayConferenceinOrlando Medical Mathematics 4 The above results were generalized to harmonic functions in R using the operator ∫ ∫ 1 dξ dη H(X) = G4f := − f (τ, ξ, η) X;= (x1, x2, x3, x4) 2 4π ξ η C1 C2 where f is a holomorphic function of three complex variables and 1 1 1 1 1 1 τ := x1(1 + ) + ix2(1 − ) + x3( − ) + ix4( + ). ηξ ηξ ξ η ξ η Here the curves Ck are homologous to zero. Kreyzig inverted this operator and referred to G4 as Gilbert’s operator. Using Kreyzig’s inversion the entire three-dimensional theory was carried over to 4 4 either R or C .

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