Wolfgang M Schmidt

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3 October 1933
Vienna, Austria

Wolfgang Schmidt is an Austrian mathematician who did his most important work in number theory.


Wolfgang Schmidt's parents were both secondary school teachers, his father teaching mathematics and physics, and his mother teaching German and French. Wolfgang has a sister Dietlinde who married the mathematician Wilfried Nöbauer (1928-1988) on 9 December 1958. Wolfgang attended schools in Vienna and by the time he was in the 3rd grade of "Middle School" his favourite subjects were mathematics and physics. He graduated from school in 1951 and later in that year began his studies at the University of Vienna.

At the University of Vienna, Schmidt specialised in mathematics and physics. He was taught analysis and number theory by Edmund Hlawka, functional analysis and measure theory by Johann Radon, and algebra by Nikolaus Hofreiter (1904-1990). He also attended lectures on probability theory by Leopold Schmetterer and on the distribution of prime numbers by Karl Pracher (1924-1994). Given this impressive collection of lecturers, it is not surprising that Schmidt should become interested in number theory.

Let us say a little about his lecturers. Edmund Hlawka was a number theorist who had been taught at the University of Vienna by Nikolaus Hofreiter and, in fact, Hofreiter had been his thesis advisor. Hofreiter had himself studied at the University of Vienna where his thesis advisor had been Philipp Furtwängler. While Schmidt was a student, Johann Radon, in addition to lecturing, served as dean during 1951-52 and rector in 1954. Karl Prachar was also a number theorist who had studied for his doctorate at the University of Vienna advised by Hlawka, and he is famed for his comprehensive treatise covering most aspects of the theory of the distribution of primes, namely Primzahlverteilung (1957).

Schmidt undertook research at the University of Vienna and was awarded a doctorate in 1955 for his thesis on the geometry of numbers Über höhere kritische Determinanten von Sternkörpern . In his thesis he made considerable progress towards verifying a conjecture made by Ambrose Rogers in his paper The number of lattice points in a star body (1951). His thesis was exceptional and J W S Cassels devoted a whole chapter in his classic monograph "An Introduction to the Geometry of Numbers" (1959) to explain the results Schmidt had obtained. Schmidt published his thesis in a paper with the same title as the thesis, in which he wrote:-
The present work owes its development and support to my revered teacher, Prof Dr E Hlawka.
Schmidt's next paper was Eine neue Abschätzung der kritischen Determinante von Sternkörpern (1956). In this paper Schmidt improved a bound given by the Minkowski-Hlawka theorem (see E Hlawka, Zur Geometrie der Zahlen (1943)). These two papers by Schmidt are the first of 202 publications listed in MathSciNet (when consulted in December 2019).

After graduating, Schmidt continued to undertake research at the University of Vienna first as a research assistant and then as a librarian at the Mathematical Institute. He began making research visits to the University of Montana at Missoula in the United States and, for example, his paper Mittelwerte über Gitter submitted on 13 February 1957 has a Missoula, Montana, address while his paper On the convergence of mean values over lattices submitted on 29 January 1957 has a Montana State University address. By 23 December 1957, when he submitted Mittelwerte über Gitter II , he gave a Vienna address. By June 1959, when he submitted On normal numbers, he was back at Montana State University. Schmidt habilitated at the University of Vienna in 1960, submitting his habilitation thesis Maßtheorie in der Geometrie der Zahlen .

When in Missoula, Montana, Schmidt met Patricia C Meyer, the daughter of Urban Joseph Meyer (1899-1945) and Caroline Weinzierl (1903-1932). Patricia (known as Pat) had been born in St Marys, Elk, Pennsylvania on 2 December 1929 but, following the death of her mother from polio when she was two years old, she and her sisters Helen and Martha had been brought up by relatives. After graduating from St Marys Catholic High School, Pat had moved to Alaska where she had worked as a medical secretary at Mt Edgecumbe Hospital in Sitka. She then moved to Missoula, Montana, where she met Wolfgang Schmidt. Wolfgang and Pat were married on 2 July 1960 in Vienna, Austria.

From 1960 to 1965 Schmidt spent time at the University of Vienna, at the University of Colorado in Boulder, USA and Columbia University, New York, USA. For example his paper Bounds for certain sums; a remark on a conjecture of Mahler, submitted in January 1961 had the University of Colorado address while the address on his paper Über die Normalität von Zahlen zu verschiedenen Basen , submitted in November 1961, is Columbia University in New York, USA. After moving back to Vienna, Wolfgang and Pat's two sons Michael and Johannes were born. In 1965 Schmidt became a professor at the University of Colorado in Boulder, a position he held until he retired in 2001.

Schmidt was a visiting member of the Institute for Advanced Study at Princeton on two occasions. The first was from September 1970 to June 1971 while the second was from September 1985 to April 1986.

In 1972 Schmidt was awarded the Cole Prize in Number Theory by the American Mathematical Society:-
... for the following papers: "On simultaneous approximation of two algebraic numbers by rationals (1967)", "T-numbers do exist (1970)", "Simultaneous approximation to algebraic numbers by rationals (1970)" and "On Mahler's T-numbers (1971)".
Hans Peter Schlickewei writes [6]:-
In view of his origins from Edmund Hlawka's Vienna school of mathematics it seems to be quite natural that a topic of recurrent interest in Schmidt's scientific activities have been problems from the theory of uniform distribution.
This topic forms the main part of Schmidt's book Lectures on irregularities of distribution (1977) based on lectures he gave at the Tata Institute of Fundamental Research, Bombay, in the autumn of 1972.

For extracts from reviews and prefaces of eight of Schmidt's books, see THIS LINK.

To obtain an overview of the topics that Schmidt worked on, a look at the information about his books is informative. In addition we list here the 14 sections in [6]: Geometry of numbers; Uniform distribution; Approximation of real numbers; Heights; Approximation of algebraic numbers by rationals; Norm form equations; Transcendental numbers; Riemann hypothesis for curves; Nonlinear approximation of real numbers; Zeros and small values of forms; Quadratic geometry of numbers; Approximation of algebraic numbers - quantitative results; Norm form equations - quantitative results; and Linear recurrence sequences.

Schmidt is among only four number theorists who have been invited to address the International Congress of Mathematicians three times. At the 1974 Congress in Vancouver he gave a plenary address entitled Applications of Time's Method in Various Branches of Number Theory; at the 1970 Congress in Nice he gave the invited lecture Some recent progress in Diophantine approximations; and at the 1983 Congress in Warsaw he gave the invited lecture Analytic methods for congruences, Diophantine equations and approximations.

In the 1970 lecture he ended his introduction with the words:-
Last winter I was able to extend Roth's famous theorem on rational approximation to an algebraic irrational to simultaneous approximations. I hope you will forgive me if I devote the rest of this lecture to this subject; I had not anticipated this result when I gave the title of my talk in the fall of 1969.
He began his 1983 lecture with the words:-
Our present methods for estimating the number of solutions of congruences, equations and inequalities have a number of common features. In particular, these methods involve exponential sums.
On Wednesday 15 July 2015, Schmidt was awarded an honorary degree by the University of York. The Presentation address by Professor Sanju Velani is given in [5]:-

It is a great honour, as well as a great pleasure, to present to you Wolfgang Schmidt, Distinguished Professor Emeritus of Mathematics at the University of Colorado in Boulder. One of the most gifted, creative and influential mathematicians today, he has made profound and far-reaching contributions to number theory.

Professor Schmidt was born in Vienna where he also received his early education. He studied Mathematics and Physics as an undergraduate and in 1955 obtained his PhD from the University of Vienna. His doctoral work was so revolutionary that a chapter was dedicated to its exposition in J W S Cassels', now classic, monograph "An Introduction to the Geometry of Numbers" published in 1959. He took up his Chair at the University of Colorado, Boulder in 1960, just five years after completing his PhD. He has remained there for his entire career not least because of an abiding passion for hiking and skiing. The mountains and snow conditions in Colorado are unrivalled! He has climbed all of the peaks in Colorado of 14,000 feet or more.

Since his PhD, exactly six decades ago, he has published over 200 research articles, many of them containing spectacular results and breakthroughs in various branches of number theory: Diophantine approximation, geometry of numbers, uniform distribution and normality. Most mathematicians would be happy with fifty in a lifetime! Even more outstanding is the fact that in this century alone, when he was past the age at which most of us would consider retiring, he has published 33 articles, some solving a longstanding open problem on recurrence sequences and some developing the new theory of "parametric' geometry of numbers. In short, Professor Schmidt continues to introduce novel ideas and techniques and produce work at the very highest level. He had told me that he is no longer quite the skier he once was, however.

Three examples of his fantastic creativity and vision that are still influencing current research, in particular at York, are the celebrated Schmidt Subspace Theorem; the axiomatic formulation of Schmidt Games, and the unifying aspect of Regular Systems. Apart from solving important, long-standing problems, Professor Schmidt has been widely influential in formulating new and exciting programmes of research. His conjectures have attracted the attention of many world experts, including some located here. This makes it especially appropriate that the University of York recognises his enormous, enriching and beautiful contributions to mathematics by the award of an honorary doctorate. Some of you graduating here today will have come directly into contact with Professor Schmidt's work - two of his books are recommended texts for our fourth year number theory course. He is a strikingly clear thinker and writer.

Professor Schmidt has received many honours including the American Mathematical Society Cole Prize, the Humboldt Prize and a Guggenheim Fellowship. He is the only mathematician to date to have been awarded the Austrian Decoration for Science and Art and is one of only four mathematicians invited to address the International Congress of Mathematicians three times. In addition, he is a fellow of the National Academy of Sciences, the Polish Academy of Sciences and the American Mathematical Society.
In addition to the honours mentioned above, he received honorary degrees from the University of Ulm (1981), the Sorbonne (1994), the University of Waterloo and the University of Marburg (1999). He was elected to the American Academy of Arts and Sciences in 1994.

Schmidt's wife Pat died on 26 November 2016. The following tribute is paid to her in [4]:-
She could laugh about almost any situation. She understood people and how to care for them. She made people happy, whether visiting scholars from all over the world or her beloved neighbours on Ithaca Drive. She volunteered for the homeless and causes for underprivileged populations. She was universally loved.
Let us end this biography with the following quote from Martin Neuwirther who for a time was Schmidt's assistant [3]:-
Although he restricted his studies primarily to the subject of number theory, his works have stimulated the evolution of several other branches of mathematics. For example, Schmidt developed the so-called theory of heights of polynomials. In the late 1980s algebraic geometer Bombieri applied Schmidt's results in order to prove effective assertions concerning the number of rational points of elliptic curves. This happened only a short time after Gerd Faltings had been awarded the Fields Medal, which is the highest award for mathematicians, for his very complicated proof of the finiteness of the number of such points. With the help of Schmidt's theory. Bombieri got more - and in an even easier way. As his assistant, I got to know Wolfgang toward the end of my studies in Vienna, when I was working in a very different - nearly opposite - branch of mathematics. Therefore I was quite amazed when I realized that the Wolfgang Schmidt, whom I had encountered in the bibliography of many publications, turned out to be the famous 'geometer of numbers.' His way of giving lectures is very factual, which corresponds well with the subject he is talking about, yet the most elegant wit of a very great thinker often gratifies the audience. Even in his dissertation, after reading several pages of tricky, hard-to-imagine assessment of the derivation of a new result, one comes across his dry comment, "After more precise consideration ... is capable of considerable improvement."

References (show)

  1. R F Tichy, H P Schlickewei and K Schmidt (eds.), Diophantine Approximation. Festschrift for Wolfgang Schmidt (Springer Verlag, Vienna, 2008).
  2. T Jackson, Review: Diophantine Approximation, by W M Schmidt, Mathematical Gazette 66 (438) (1982), 335-336.
  3. M Neuwirther, Wolfgang M Schmidt - The Geometer of Numbers, in Peter Weibel (ed.), Beyond Art: A Third Culture: A Comparative Study in Cultures, Art and Science in 20th Century Austria and Hungary (Springer Science & Business Media, 2005), 269.
  4. Obituary: Patricia C Schmidt, The Daily Camera (11 December 2016).
  5. Presentation address by Professor Sanju Velani on the occasion of the conferment of the honorary degree of Doctor of the University upon Professor Wolfgang Schmidt on Wednesday 15 July 2015, University of York (2015).
  6. H P Schlickewei, The mathematical work of Wolfgang Schmidt, Diophantine Approximation, in Robert F Tichy, Hans Peter Schlickewei and Klaus Schmidt (eds.), Diophantine Approximation. Festschrift for Wolfgang Schmidt (Springer Verlag, Vienna, 2008), 1-20.
  7. W Schmidt, Geometric intuition in number theory, Claremont Center for Mathematical Sciences, University of Colorado ( 2012).
  8. Wolfgang Schmidt, Institute for Advanced Study. https://www.ias.edu/scholars/wolfgang-schmidt
  9. Wolfgang Schmidt, University of York honours 11 for their contribution to society, University of York (2015). https://www.york.ac.uk/news-and-events/news/2015/events/honorary-graduates-july-2015/

Additional Resources (show)

Other pages about Wolfgang Schmidt:

  1. Wolfgang M Schmidt's Books

Honours (show)

Honours awarded to Wolfgang Schmidt

  1. AMS Cole Prize in Number Theory 1972

Written by J J O'Connor and E F Robertson
Last Update January 2020