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Simon Grünzweig at Lincoln University
The Lion Yearbook, 1949
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This blogpost continues the blogposts "Intro to Simon Grünzweig," "Lincoln in the 1940s," and "Grünzweig at Lincoln."
Grünzweig planned to start earning his PhD in the summer of 1949, shortly before moving to Lincoln. However, he found himself busier than expected and was only able to start work later, probably the next summer. In the summer of 1952, one year after he left Lincoln, he received his PhD from the University of Pittsburgh.
By contemporary standards, completing a PhD in 2 years, especially while also teaching 5 classes per semester, is remarkably short. (Today, a typical teaching load for an assistant professor would be between 1 and 4 classes.) While an impressive accomplishment, this was less unusual in the 1950s. Grünzweig's education in Europe (7 years at the University of Vienna) compared well with a master's degree from a U.S. university. Receiving a PhD a few years after completing a master's degree was not unusual. For example, at The Ohio State University, many such students graduated within 2 years and most spent less than 6 years. By contrast, today (in 2020), the average PhD student at Ohio State spends 6 years in graduate school.
Grünzweig planned to start earning his PhD in the summer of 1949, shortly before moving to Lincoln. However, he found himself busier than expected and was only able to start work later, probably the next summer. In the summer of 1952, one year after he left Lincoln, he received his PhD from the University of Pittsburgh.
By contemporary standards, completing a PhD in 2 years, especially while also teaching 5 classes per semester, is remarkably short. (Today, a typical teaching load for an assistant professor would be between 1 and 4 classes.) While an impressive accomplishment, this was less unusual in the 1950s. Grünzweig's education in Europe (7 years at the University of Vienna) compared well with a master's degree from a U.S. university. Receiving a PhD a few years after completing a master's degree was not unusual. For example, at The Ohio State University, many such students graduated within 2 years and most spent less than 6 years. By contrast, today (in 2020), the average PhD student at Ohio State spends 6 years in graduate school.
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| James S. Taylor From The Owl Yearbook, 1951 |
At the University of Pittsburgh, Grünzweig's PhD advisor was James S. Taylor. Professor Taylor had graduated from Berkeley in 1918. There he had been advised by Mellen Woodman Haskell and wrote a dissertation on "A set of five postulates for Boolean algebras in terms of the operations 'exception'." It seems that he never published his dissertation, but he further developed these ideas in two publications: "Complete Existential Theory of Bernstein's Set of Four Postulates for Boolean Algebras" in Annals of Mathematics and "Sheffer's set of five postulates for Boolean algebras in terms of the operation 'rejection' made completely independent" in the Bulletin of the American Mathematical Society.
Both papers build on earlier work of Sheffer on axiomatizing Boolean algebras. Sheffer had observed that a Boolean algebra can be defined as a set with a NOR operation that satisfies 5 axioms. Taylor's second paper shows how to modify these axioms so that they are completely independent (meaning there are no non-trivial logical implications among the axioms and their negations). His first paper studies a different set of axioms that had been proposed by Bernstein.
The two papers on Boolean algebras are the only ones that appear on the database MathSciNet, but MathSciNet is an incomplete record for U.S. mathematicians in the early 20th century. In particular, it misses some of Taylor's publications.
Grünzweig's PhD work is based on one of Taylor's publications that does not show up on MathSciNet: the paper "A Four-Space Representation of Complex Plane Analytics." The paper was published in the Journal of Mathematics and Physics in 1923, while Taylor was working MIT. The paper studies the standard 2-dimensional complex vector space. This vector space can also be considered as a 4-dimensional real vector space, and Taylor studies the interplay between the real and complex geometry.
Every complex vector subspace of complex dimension 1 is a real vector subspace of real dimension 2, and Taylor defines real vector subspaces that arise in this manner as "regular" subspaces. His paper focuses on describing which real rigid motions send complex vector subspaces to complex vector subspaces. Many rigid motions fail to preserve complex vector subspaces, and Taylor shows that the rigid motions preserving complex subspaces are those that are complex linear or anti-linear.
Grünzweig's dissertation, titled "Affine and non-affine projective transformations in four-space which leave invariant the family of surfaces representing regular functions of a complex variable" extends Taylor's work. The dissertation was unpublished, and I haven't gotten a copy of it. However, thanks to Pittsburgh's University Archivist Zach Brodt, I was able to get a copy of a 6-page abstract which was published in a university bulletin.
Grünzweig's dissertation generalizes Taylor's result. He proves that the analogue of Taylor's results on real rigid motions holds for real affine transformation, and he proves a similar result for real projective transformations. The results are proven by detailed explicit computations, and as a by-product of his computations, he obtains information about how an affine transformation can act on complex subspaces.
The "regular functions" referenced in the dissertation title are holomorphic functions, and they are connected to the work as follows. A holomorphic function of 1 variable can be studied geometrically by studying its graph in the complex plane. The graph is naturally a Riemann surface (in fact, is biholomoprhic to the complex line). A real affine transformation maps the graph into another real surface, but that real surface could fail to have complex structure, i.e. could fail to be a Riemann surface. However, a basic result is that a real transformation which sends complex vector subspaces to complex vector subspaces must also send Riemann surfaces to Riemann surfaces. Grünzweig's results thus provide a description of the real affine transformation which act on Riemann surfaces.
As far as I can tell, Grünzweig did not publish his dissertation, and he does not have seemed to have continued this line of research.
As Grünzweig was finishing up his dissertation, he began to look for further academic employment. Although the Second World War had ended seven years ago, the International Rescue Committee was still helping war refugees, and it sent out inquiries on Grünzweig's behalf. The next blogpost will discuss what response they got, and where Grünzweig moved to next.
Both papers build on earlier work of Sheffer on axiomatizing Boolean algebras. Sheffer had observed that a Boolean algebra can be defined as a set with a NOR operation that satisfies 5 axioms. Taylor's second paper shows how to modify these axioms so that they are completely independent (meaning there are no non-trivial logical implications among the axioms and their negations). His first paper studies a different set of axioms that had been proposed by Bernstein.
The two papers on Boolean algebras are the only ones that appear on the database MathSciNet, but MathSciNet is an incomplete record for U.S. mathematicians in the early 20th century. In particular, it misses some of Taylor's publications.
Grünzweig's PhD work is based on one of Taylor's publications that does not show up on MathSciNet: the paper "A Four-Space Representation of Complex Plane Analytics." The paper was published in the Journal of Mathematics and Physics in 1923, while Taylor was working MIT. The paper studies the standard 2-dimensional complex vector space. This vector space can also be considered as a 4-dimensional real vector space, and Taylor studies the interplay between the real and complex geometry.
Every complex vector subspace of complex dimension 1 is a real vector subspace of real dimension 2, and Taylor defines real vector subspaces that arise in this manner as "regular" subspaces. His paper focuses on describing which real rigid motions send complex vector subspaces to complex vector subspaces. Many rigid motions fail to preserve complex vector subspaces, and Taylor shows that the rigid motions preserving complex subspaces are those that are complex linear or anti-linear.
Grünzweig's dissertation, titled "Affine and non-affine projective transformations in four-space which leave invariant the family of surfaces representing regular functions of a complex variable" extends Taylor's work. The dissertation was unpublished, and I haven't gotten a copy of it. However, thanks to Pittsburgh's University Archivist Zach Brodt, I was able to get a copy of a 6-page abstract which was published in a university bulletin.
Grünzweig's dissertation generalizes Taylor's result. He proves that the analogue of Taylor's results on real rigid motions holds for real affine transformation, and he proves a similar result for real projective transformations. The results are proven by detailed explicit computations, and as a by-product of his computations, he obtains information about how an affine transformation can act on complex subspaces.
The "regular functions" referenced in the dissertation title are holomorphic functions, and they are connected to the work as follows. A holomorphic function of 1 variable can be studied geometrically by studying its graph in the complex plane. The graph is naturally a Riemann surface (in fact, is biholomoprhic to the complex line). A real affine transformation maps the graph into another real surface, but that real surface could fail to have complex structure, i.e. could fail to be a Riemann surface. However, a basic result is that a real transformation which sends complex vector subspaces to complex vector subspaces must also send Riemann surfaces to Riemann surfaces. Grünzweig's results thus provide a description of the real affine transformation which act on Riemann surfaces.
As far as I can tell, Grünzweig did not publish his dissertation, and he does not have seemed to have continued this line of research.
As Grünzweig was finishing up his dissertation, he began to look for further academic employment. Although the Second World War had ended seven years ago, the International Rescue Committee was still helping war refugees, and it sent out inquiries on Grünzweig's behalf. The next blogpost will discuss what response they got, and where Grünzweig moved to next.
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| Simon Grünzweig The Lion Yearbook, 1951 |
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