Hermann Klaus Hugo Weir

Born November 9, 1885 in Elmshorn, Schleswig-Holstein, Germany (near Hamburg)

Died December 8, 1955, in Zurich, Switzerland

From 1923 to 38, Weyl developed the concept of continuous groups using matrix representations. By applying group theory to quantum mechanics, he established the modern discipline.

Hermann Weyl was known to him by his close friend Peter. His parents were Anna Dieck and Ludwig Weyl, the latter of whom was a director of a bank. Herman had already shown his talent for mathematics and broader science as a child. After taking his abiturarbeit (high school graduation exam) (see [17]), he was prepared for university studies. In 1904 he entered the university of Munich, where he undertook courses in mathematics and physics, and then continued to study the same subject at the University of Göttingen. He was completely captivated by Hilbert. He later wrote:-

I was determined to study anything this man wrote. At the end of the first year, I took home the "zahlbericht", and during the summer I worked hard to complete it— without any knowledge of elementary number theory or Galois theory before. It was the happiest months of my life, and its light still comforted my soul in the years of our shared doubts and failures.

His PhD was from Göttingen and his supervisor was Hilbert. After submitting his doctoral thesis singuläre integralgleichungen mit besonder berücksichtigung des fourierschen integraltheorems (T) he received his degree in 1908. This paper examines the singular integral equations and takes an in-depth look at Fourier's integral theorem. It was at Göttingen that he held his first teaching post as privatdozent, which he held until after 1913. His qualification paper Über gewöhnliche differentialgleicklungen mit singularitäten und die zugehörigen entwicklungen willkürlicher funktionen (T) studied the spectral theory of the singular sturm-liouville problem. During his time in Göttingen, Weil was known as a brilliant mathematician, and his work had a major impact on the progress of mathematics. His adaptability thesis is one such job, but there are many more. He delivered the lecture course Riemann Face in 1911 - 12 in and out of this course came his first book Mold Edremannschenfläche (T) published in 1913. It combined analysis, geometry, and topology to make the theory of geometric functions developed by Riemann rigorous. The book introduces for the first time the concept of [59]:-

... Two-dimensional micromanifolds, a covering surface, and duality between differential and 1 cycles. ... Weyl's concept of space also includes the famous separation features of felix hausdorff (1914), which were later introduced and generally attributed.

L Sario wrote in 1956 weyl 1913 text:-

... Undoubtedly had a greater impact on the development of geometric function theory than any other publication since Riemann's doctoral dissertation.

Notably, the 1913 text was reprinted in 1997. Weil himself later produced two editions, the third (and final of which) appeared in 1955, covering the same subject matter as the original text, but with a more modern approach. This is the original 1913 edition, however, this is reprinted in 1997 out of the mathematical development of the 1913 text may be more significant and fuller than the later version originally was.

As a private scholar at the University of Göttingen, Weil was influenced by Edmund Husserl, where he served as president of philosophy from 1901 to 1916. In 1913, Weil married Helen Joseph, a student of Husserl; they had two sons. Helen, from a Jewish background, is a philosopher who is working as a translator for Spanish. Weil and his wife not only shared a common interest in philosophy, but also had a real talent for language. Weyl's language is particularly important. Not only did he write beautifully in German, but later he wrote amazing English prose, albeit in his own words from the 1939 English text:

...... The gods imposed the shackles on me of writing a foreign language that was not sung in my cradle.

From 1913 to 1930, Weil was head of the Department of Mathematics at the Zurich Institute of Technology. During his first academic year in this new position, he was a colleague of Einstein's, when he was working on the details of general relativity. This was an event that had a great impact on Weyl, who soon became fascinated by the mathematical principles behind the theory.

Shortly after Weil assumed the presidency in Zurich, World War I broke out. As a German citizen, he was conscripted into the army in 1915, but the Swiss government made a special request to allow him to return to Zurich, which was approved in 1916. In 1917 Weyl opened another course that introduced innovative approaches to the study of relativity through differential geometry. These lectures formed the basis of Weyl's second book, Raum-Zeit-Materie (T), which first appeared in 1918 and was published in more editions in 1919, 1920, and 1923, each showing how his ideas developed. These later ideas included gauge measures (Weyer metrics) that led to gauge field theory. However, Einstein, Pauli, Eddington, and others did not fully accept Weil's method. During this period, Weyer also contributed to the uniform distribution of modulus 1 numbers, which is the basis in analytic number theory.

In 1921 Schrödinger was appointed to Zurich, where he became Weyl's colleague and soon became his closest friend. They share common interests in mathematics, physics, and philosophy. As Moore says in [5], their personal lives also become entangled:-

Those familiar with princeton's serious, plump stature could hardly recognize this slender, handsome young man in his twenties with a romantic black moustache. His wife, Helene Joseph, had a Jewish background and was a philosopher and writer. Her friends called her Hela, and a certain bold and careless character made her the undisputed leader of a social group of scientists and their wives. Anne [Schrödinger's wife] is almost the exact opposite of the fashionable and intellectual Hella, but perhaps it is for this reason that [Weil] finds her amusing, and soon she is madly in love with him. ...... The particular circle in which they lived in Zurich had already enjoyed the sexual revolution before [the United States]. Extramarital affairs are not only forgiven, but expected, and seem to cause anxiety infrequently. Anne will find a lover in Hermann Weil, whose wife, Hella, has a crush on Paul Scheler.

From 1923 to 38, Weyl developed the concept of continuous groups using matrix representations. In particular his theory represents the semi-single group, developed in 1924-26, very deeply, considered by Weil himself to be his greatest achievement. Hurwitz and Shure have introduced the thinking behind this theory, but it was Weil who came up with his general character formula. However, he was not the only mathematician to develop this theory, as Cartan also studied this important subject.

From 1930 to 1933, Weil served as head of the mathematics department in Göttingen, where he was appointed to fill vacancies that arose after Hilbert's retirement. Given the different political circumstances, he is likely to spend the rest of his life in Göttingen. However[8] :-

...... The rise of the Nazis persuaded him to accept the position at the newly established Institute for Advanced Study in Princeton in 1933, where Einstein also went. Here, Weil found a very harmonious working environment, where he was able to mentor and influence the younger generation of mathematicians, a task he was well suited to.

One must also understand that Weyl's wife was Jewish, which must have played an important role in their decision to leave Germany in 1933. Weil remained at the Institute for Advanced Study in Princeton until his retirement in 1952. His wife, Helene, died in 1948, and two years later he married ellen Lohnstein Bär, a sculptor from Zurich.

Weil certainly took on important work at Princeton, but his most productive period was undoubtedly the years he spent in Zurich. He tried to incorporate electromagnetism into the geometric form of general relativity. He proposed the first unified field theory, the emergence of Maxwell electromagnetic fields and gravitational fields as geometric properties of space-time. By applying group theory to quantum mechanics, he established the modern discipline. In 1927–28 he taught group theory and quantum mechanics in Zurich, which led to the publication of his third major book, gruppentheorie und quantenmechanik (T), in 1928. John Wheeler writes [56]:-

Every time I read that book, there's some great new information.

More recently, Wheeler has tried to incorporate electromagnetism into general relativity. Wheeler's theory, like Weyl's theory, lacks an association with quantum phenomena, which are important for interactions beyond gravity. Wheeler writes about the first time he saw weyl in [56]:-

When I took me to Princeton in 1937, I saw Herman Weil's flesh upright, bright eyes, and smiling for the first time. There, I attended his lecture on Élie cartan calculus in differential form and its application in electromagnetism – eloquent, simple, and insightful.

We've already seen above how Weyl's great work was first offered as a lecture course. This is a deliberate design of weyl [ 56 ]:-

At one point, Weil arranged for a course on the history of mathematics at Princeton University. One day he explained to me that it was absolutely necessary for him to take a comprehensive look at the topics he was concerned with through his speeches. Only then, he said, can he see the huge gaps, the places where deeper understanding is needed, and where the work should be concentrated.

Weil also published many other great books during his time at Princeton University. These include the basic theory of invariants (1935), classical groups (1939), algebra theory (1940), philosophy of mathematics and natural science (1949), symmetry (1952), and the concept of Riemann surfaces (1955). There's too much to say about all of these works, but we're limited to looking at symmetry, because that's probably what best tells us about weyl's full interests. Coaster commented on the book, and his comments beautifully capture the spirit of the book:-

This is a slightly modified version of the Louis Clark vanuxem lectures held at Princeton University in 1951... The first lesson first showed how the idea of bilateral symmetry influenced painting and sculpture, especially in antiquity. This naturally leads to a discussion of "left and right philosophy", including the following questions. Is there one of two enantiomeric forms of optically active substances unique to living matter in nature? At what stage of embryonic development is the symmetrical plane determined? The second lecture contains a concise exposition of the theory of transformation groups, with special emphasis on similar groups and their subgroups: full equal transformation groups, kinematic groups, translation groups, rotation groups, and finally any given symmetry group numbers. ... Cyclic and dihedral groups are illustrated by snowflakes and flowers, animals called Medusas, and floor plans of symmetrical buildings. Similarly, the arrangement of florets in the nautilus shell and sunflower illustrates the infinite cyclic group produced by spiral similarity. The third lecture gives the basic steps of enumerating 17 spatial groups of two-dimensional crystallography... [In Lecture IV] he shows how special relativity essentially studies the intrinsic symmetry of four-dimensional space-time continuums, where the symmetry operation is the Lorentz transform; and how the symmetry operation of atoms, according to quantum mechanics, includes the arrangement of their peripheral electrons. Moving from physics to mathematics, he gives a very succinct epitome of Galois's theory, leading to the statement of his guiding principle: "Whenever you have to deal with a structure-giving entity, try to determine its automorphism group".

In 1951 Welle retired from the Institute for Advanced Study at Princeton University. In fact, he described the symmetrical book as his "masterpiece." After retiring, Weyl and his wife Ellen spent part of their time in Princeton and part of it in Zurich. He died unexpectedly while in Zurich. He sent letters of thanks to those who wished him good luck on his seventieth birthday and walked home.

We have to talk a little bit about another aspect of Weyl's work that we haven't really mentioned yet, namely his work on mathematical philosophy and mathematical foundations. Interestingly, the large number of references we cite deal with this aspect of his work, the importance of which lies not only in the work itself, but also in the extent to which Weyl's ideas on these topics are the basis of his mathematics and other parts. Physical contributions. Weil's views were heavily influenced by Husserl and shared many ideas with Brower. Both argue that Cantor's set-theoretic continuum is not an accurate representation of the intuitive continuum. Wheeler [ 56 ] wrote:-

Continuity..., Weyl tells us, is an illusion. This is an idealization. It's a dream.

Weyl summed up his approach to mathematics, writing:-

My own mathematical work is always very unsystematic, without patterns, without connections. Expression and shape are almost more important to me than knowledge itself. But I believe, leaving aside my own peculiarities, mathematics itself has characteristics that are closer to freely creative art than experimental disciplines.

Comments he quotes frequently:-

My work always tries to combine truth and beauty, but when I have to choose one of them, I usually choose beauty...

Although half joking, summed up his personality.