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Shiing-Shen Chern

Shiing-Shen Chern (, ; October 28, 1911 – December 3, 2004) was a Chinese American mathematician and poet. He made fundamental contributions to differential geometry and topology. He has been called the "father of modern differential geometry" and is widely regarded as a leader in geometry and one of the greatest mathematicians of the twentieth century, winning numerous awards and recognition including the Wolf Prize and the inaugural Shaw Prize. In memory of Shiing-Shen Chern, the International Mathematical Union established the Chern Medal in 2010 to recognize "an individual whose accomplishments warrant the highest level of recognition for outstanding achievements in the field of mathematics."

Chern worked at the Institute for Advanced Study (1943–45), spent about a decade at the University of Chicago (1949-1960), and then moved to University of California, Berkeley, where he cofounded the Mathematical Sciences Research Institute in 1982 and was the institute's founding director. Renowned coauthors with Chern include Jim Simons, an American mathematician and billionaire hedge fund manager. Chern's work, most notably the Chern-Gauss-Bonnet Theorem, Chern–Simons theory, and Chern classes, are still highly influential in current research in mathematics, including geometry, topology, and knot theory, as well as many branches of physics, including string theory, condensed matter physics, general relativity, and quantum field theory.

Name spelling
Chern's surname (陳/陈, pinyin: Chén) is a common Chinese surname which is now usually romanized as Chen. The spelling "Chern" is from the Gwoyeu Romatzyh (GR) romanization system. In English, Chern pronounced his own name as "Churn".

Early years in China
Chern was born in Xiushui, Jiaxing, China in 1911. He graduated from Xiushui Middle School and subsequently moved to Tianjin in 1922 to accompany his father. In 1926, after spending four years in Tianjin, Chern graduated from Fulun High School.

At age 15, Chern entered the Faculty of Sciences of the Nankai University in Tianjin and was interested in physics, but not so much the laboratory, so he studied mathematics instead. Chern graduated with a Bachelor of Science degree in 1930. At Nankai, Chern's mentor was mathematician Jiang Lifu, and Chern was also heavily influenced by Chinese physicist Rao Yutai, considered to be one of the founding fathers of modern Chinese informatics.

Chern went to Beijing to work at the Tsinghua University Department of Mathematics as a teaching assistant. At the same time he also registered at Tsinghua Graduate School as a student. He studied projective differential geometry under Sun Guangyuan, a University of Chicago-trained geometer and logician who was also from Zhejiang. Sun is another mentor of Chern who is considered a founder of modern Chinese mathematics. In 1932, Chern published his first research article in the Tsinghua University Journal. In the summer of 1934, Chern graduated from Tsinghua with a master's degree, the first ever master's degree in mathematics issued in China.

Yang Chen-Ning's father, Yang Ko-Chuen, another Chicago-trained professor at Tsinghua, but specializing in algebra, also taught Chern. At the same time, Chern was Chen-Ning Yang's teacher of undergraduate maths at Tsinghua. At Tsinghua, Hua Luogeng, also a mathematician, was Chern's colleague and roommate.

In 1932, Wilhelm Blaschke from the University of Hamburg visited Tsinghua and was impressed by Chern and his research.

1934–1937 in Europe
In 1934, Chern received a scholarship to study in the United States at Princeton and Harvard, but at the time he wanted to study geometry and Europe was the center for the maths and sciences.

He studied with the well-known Austrian geometer Wilhelm Blaschke. Co-funded by Tsinghua and the Chinese Foundation of Culture and Education, Chern went to continue his study in mathematics in Germany with a scholarship.

Chern studied at the University of Hamburg and worked under Blaschke's guidance first on the geometry of webs then on the Cartan-Kähler theory and invariant theory. He would often eat lunch and chat in German with fellow colleague Erich Kähler.

He had a three-year scholarship but finished his degree very quickly in two years. He obtained his Dr. rer.nat. (Doctor of Science, which is equivalent to PhD) degree in February, 1936. He wrote his thesis in German, and it was titled Eine Invariantentheorie der Dreigewebe aus $$r$$-dimensionalen Mannigfaltigkeiten im $$R_{2r}$$ (English: An invariant theory of 3-webs of $$r$$-dimensional manifolds in $$R_{2r}$$).

For his third year, Blaschke recommended Chern to study at the University of Paris.

It was at this time that he had to choose between the career of algebra in Germany under Emil Artin and the career of geometry in France under Élie-Joseph Cartan. Chern was tempted by what he called the "organizational beauty" of Artin's algebra, but in the end, he decided to go to France in September 1936.

He spent one year at the Sorbonne in Paris. There he met Cartan once a fortnight. Chern said: "Usually the day after [meeting with Cartan] I would get a letter from him. He would say, “After you left, I thought more about your questions...”—he had some results, and some more questions, and so on. He knew all these papers on simple Lie groups, Lie algebras, all by heart. When you saw him on the street, when a certain issue would come up, he would pull out some old envelope and write something and give you the answer. And sometimes it took me hours or even days to get the same answer... I had to work very hard."In August 1936, Chern watched the Summer Olympics in Berlin together with Chinese mathematician Hua Luogeng who paid Chern a brief visit. During that time, Hua was studying at the University of Cambridge in Britain.

1937-1943 WW2
In the summer of 1937, Chern accepted the invitation of Tsinghua University and returned to China. He was promoted to professor of mathematics at Tsinghua.

In late 1937, however, the start of World War 2 forced Tsinghua and other academic institutions to move away from Beijing to west China. Three universities including Peking University, Tsinghua, and Nankai formed the temporary National Southwestern Associated University (NSAU), and relocated to Kunming, Yunnan province. Chern never reached Beijing.

In 1939, Chern married Shih-Ning Cheng, and the couple had two children, Paul and May.

The war prevented Chern from having regular contacts with the outside mathematical community. He wrote to Cartan about his situation, to which Cartan sent him a box of his reprints. Chern spent a considerable amount of time pondering over Cartan's papers and published despite relative isolation. In 1943, his papers gained international recognition, and Oswald Veblen invited him to the IAS. Because of the war, it took him a week to reach Princeton via US military aircraft.

1943-1945 visit to the IAS, the Chern theorem
In July 1943, Chern went to the United States, and worked at the Institute for Advanced Study (IAS) in Princeton on characteristic classes in differential geometry. There he worked with André Weil on the Chern–Weil homomorphism and theory of characteristic classes, later to be foundational to the Atiyah–Singer index theorem. Shortly afterwards, he was invited by Solomon Lefschetz to be an editor of Annals of Mathematics.

Between 1943-1964 he was invited back to the IAS on several occasions. On Chern, Weil wrote: "... we seemed to share a common attitude towards such subjects, or towards mathematics in general; we were both striving to strike at the root of each question while freeing our minds from preconceived notions about what others might have regarded as the right or the wrong way of dealing with it."It was at the IAS that his work culminated in his publication of the generalization of the famous Gauss–Bonnet theorem to higher dimensional manifolds, now known today as the Chern theorem. It is widely considered to be his magnum opus. This period at the IAS was a turning point in career, having a major impact on mathematics, while fundamentally altering the course of differential geometry and algebraic geometry. In a letter to the then director Frank Aydelotte, Chern wrote: "“The years 1943–45 will undoubtedly be decisive in my career, and I have profited not only in the mathematical side. I am inclined to think that among the people who have stayed at the Institute, I was one who has profited the most, but the other people may think the same way.”"

1945-48 first return to China
Chern returned to Shanghai in 1945 to help found the Institute of Mathematics of the Academia Sinica. Chern was the acting president of the institute. Wu Wenjun was Chern's graduate student at the institute.

In 1948, Chern was elected one of the first academicians of the Academia Sinica. He was the youngest academician elected (at age 37).

In 1948, he accepted an invitation by Weyl and Veblen to return to Princeton as a professor.

1948-60 Back in the USA, University of Chicago
By the end of 1948, Chern returned to the United States and IAS. He brought his family with him. In 1949, he was invited by Weil to become professor of mathematics at the University of Chicago and accepted the position as chair of geometry. Coincidentally, Ernest Preston Lane, former Chair at UChicago Department of Mathematics, was the doctoral advisor of Chern's undergraduate mentor at Tsinghua—Sun Guangyuan.

In 1950 he was invited by the International Congress of Mathematicians in Cambridge, Massachusetts. He delivered his address on the Differential Geometry of Fiber Bundles. According to Hans Samelson, in the lecture Chern introduced the notion of a connection on a principal fiber bundle, a generalization of the Levi-Civita connection. Shii

Berkeley and MSRI
In 1960 Chern moved to the University of California, Berkeley. He worked and stayed there until he became an emeritus professor in 1979. In 1961, Chern became a naturalized citizen of the United States. In the same year, he was elected member of the United States National Academy of Sciences. "My election to the US National Academy of Sciences was a prime factor for my US citizenship. In 1960 I was tipped about the possibility of an academy membership. Realizing that a citizenship was necessary, I applied for it. The process was slowed because of my association to Oppenheimer. As a consequence I became a US citizen about a month before my election to academy membership."In 1964, Chern was a vice president of American Mathematical Society (AMS).

Chern retired from UC Berkeley in 1979. In 1981, together with colleagues Calvin C. Moore and Isadore Singer, he founded the Mathematical Sciences Research Institute (MSRI) at Berkeley, serving as the director until 1984. Afterward he became the honorary director of the institute. MSRI now is one of the largest and most prominent mathematical institutes in the world. Shing-Tung Yau was one of his PhD students during this period, and he later won the Fields Medal in 1982.

During WW2, the US did not have much of a scene in geometry (which is why he chose to study in Germany). Chern was largely responsible in making the US a leading research hub in the field, but he remained modest about his achievements, preferring to say that he is a man of 'small problems' rather than 'big views.'

Visits to China and bridging East and West
The Shanghai Communiqué was issued by the United States and the People's Republic of China on February 27, 1972. The relationship between these two nations started to normalize, and American citizens were allowed to visit China. In September 1972, Chern visited Beijing with his wife. During this period of time, Chern visited China 25 times, of which 14 were to his home province Zhejiang.

He was admired and respected by Chinese leaders Mao Zedong, Deng Xiaoping, and Jiang Zemin. Because of foreign prestigious scientific support, Chern was able to revive mathematical research in China, producing a generation of influential Chinese mathematicians.

Chern founded the Nankai Institute for Mathematics (NKIM) at his alma mater Nankai in Tianjin. The institute was formally established in 1984 and fully opened on October 17, 1985. NKIM was renamed the Chern Institute of Mathematics in 2004 after Chern's death. He was treated as a rock star and cultural icon in China. Regarding his influence in China and help raising a generation of new mathematicians, ZALA films says: "Several world-renowned figures, such as Gang Tian and Shing-Tung Yau, consider Chern the mentor who helped them study in western countries following the bleak years of the Cultural Revolution, when Chinese universities were closed and academic pursuits suppressed. By the time Chern started returning to China regularly during the 1980s, he had become a celebrity; every school child knew his name, and TV cameras documented his every move whenever he ventured forth from the institute he established at Nankai University."He has said that back then the main obstruent to the growth of math in China is the low pay, which is important considering that after the cultural revolution many families were impoverished. But he has said that given China's size, it naturally has a large talent pool of budding mathematicians.

Nobel Prize winner and former student CN Yang has said "Chern and I and many others felt that we have the responsibility to try to create more understanding between the American people and the Chinese people, and... all of us shared the desire to promote more exchanges."

Final years and death
In 1999, Chern moved from Berkeley back to Tianjin, China permanently until his death.

Based on Chern's advice, a mathematical research center was established in Taipei, Taiwan, whose co-operational partners are National Taiwan University, National Tsing Hua University and the Academia Sinica Institute of Mathematics.

In 2002, he convinced the Chinese government (the PRC) for the first time to host the International Congress of Mathematicians in Beijing. In the speech at the opening ceremony he said: "The great Confucius guided China spiritually for over 2,000 years. The main doctrine is “仁” pronounced “ren”, meaning two people, i.e., human relationship. Modern science has been highly competitive. I think an injection of the human element will make our subject more healthy and enjoyable. Let us wish that this congress will open a new era in the future development of math."

Chern was also a director and advisor of the Center of Mathematical Sciences at Zhejiang University in Hangzhou, Zhejiang.

Chern died of heart failure at Tianjin Medical University General Hospital in 2004 at age 93.

In 2010 George Csicsery featured him in the documentary short Taking the Long View: The Life of Shiing-shen Chern.

His former residence, Ningyuan, is still in campus of Nankai University, kept in the way when he was living there. Every year on December 3, Ningyuan is open for visitors for memorial of him.

Research
Physics Nobel Prize winner (and former student) C. N. Yang has said that Chern is on par with Euclid, Gauss, Riemann, Cartan. Two of Chern's most important contributions that have reshaped the fields of geometry and topology include

 * Chern-Gauss-Bonnet Theorem, the generalization of the famous Gauss–Bonnet theorem (100 years earlier) to higher dimensional manifolds. Chern considers this his greatest work. Chern proved it by developing his geometric theory of fiber bundles.
 * Chern classes, the complexification of Pontryagin classes, which have found wide-reaching applications in modern physics, especially string theory, quantum field theory, condensed matter physics, in things like the magnetic monopole. His main idea was that one should do geometry and topology in the complex case.

In 2007, Chern's disciple and IAS director Phillip Griffiths edited Inspired by S. S. Chern: A Memorial Volume in Honor of A Great Mathematician (World Scientific Press). Griffiths wrote: "“More than any other mathematician, Shiing-Shen Chern defined the subject of global differential geometry, a central area in contemporary mathematics. In work that spanned almost seven decades, he helped to shape large areas of modern mathematics... I think that he, more than anyone, was the founder of one of the central areas of modern mathematics.”"His work extended over all the classic fields of differential geometry as well as more modern ones including general relativity, invariant theory, characteristic classes, cohomology theory, Morse theory, Fiber bundles, Sheaf theory, Cartan's theory of differential forms, etc. His work included areas currently-fashionable, perennial, foundational, and nascent:

 * Chern–Simons theory arising from a 1974 paper written jointly with Jim Simons; and also gauge theory, Chern–Simons form, Chern-Simons field theory. CS theory now has great importance in knot theory and modern string theory and condensed matter physics research, including Topological phases of matter and Topological quantum field theory.
 * Chern–Weil theory linking curvature invariants to characteristic classes from 1944
 * class theory for Hermitian manifolds
 * Chern-Bott theory, including the Chern-Bott theorem, a famous result on complex geometrizations of complex value distribution functions
 * value distribution theory of holomorphic functions
 * Chern-Lashof theory on tight immersions, compiled in a monograph over 30 years with Richard Lashof at Chicago
 * Chern-Lashof theorem: a proof was announced in 1989 by Sharpe
 * projective differential geometry
 * webs
 * integral geometry, including the 'moving theorem', in collaboration with Yan Zhida
 * minimal surfaces, minimal submanifolds and harmonic mappings
 * Exterior Differential Systems and Partial Differential Equations

He was a follower of Élie Cartan, working on the 'theory of equivalence' in his time in China from 1937 to 1943, in relative isolation. In 1954 he published his own treatment of the pseudogroup problem that is in effect the touchstone of Cartan's geometric theory. He used the moving frame method with success only matched by its inventor; he preferred in complex manifold theory to stay with the geometry, rather than follow the potential theory. Indeed, one of his books is entitled "Complex Manifolds without Potential Theory".

Differential forms
Along with Cartan, Chern is one of the mathematicians known for popularizing the use of differential forms in math and physics. In his biography, Richard Palais and Chuu-Lian Terng have written "... we would like to point out a unifying theme that runs through all of it: his absolute mastery of the techniques of differential forms and his artful application of these techniques in solving geometric problems. This was a magic mantle, handed down to him by his great teacher, Élie Cartan. It permitted him to explore in depth new mathematical territory where others could not enter. What makes differential forms such an ideal tool for studying local and global geometric properties (and for relating them to each other) is their two complementary aspects. They admit, on the one hand, the local operation of exterior differentiation, and on the other the global operation of integration over cochains, and these are related via Stokes's Theorem."While at the IAS, there were two competing methods of geometry: the tensor calculus and the newer differential forms. Chern wrote: "I usually like to say that vector fields is like a man, and differential forms is like a woman. Society must have two sexes. If you only have one, it’s not enough."In the last years of his life, he advocated the study of Finsler geometry, writing several books and articles on the subject. His research on Finsler geometry is continued through Tian Gang, Paul C. Yang, and Sun-Yung Alice Chang of Princeton University.

He was known for unifying geometric and topological methods to prove stunning new results.

Honors and awards
Chern received numerous honors and awards in his life, including:
 * 1970, Chauvenet Prize, of the Mathematical Association of America;
 * 1975, National Medal of Science;
 * 1982, Humboldt Prize, Germany;
 * 1983, Leroy P. Steele Prize, of the American Mathematical Society;
 * 1984, Wolf Prize in Mathematics, Israel;
 * 2002, Lobachevsky Medal;
 * 2004 May, Shaw Prize in mathematical sciences, Hong Kong;
 * 1948, Academician, Academia Sinica;
 * 1950, Honorary Member, Indian Mathematical Society;
 * 1950, Honorary Fellow, Tata Institute of Fundamental Research
 * 1961, Member, United States National Academy of Sciences;
 * 1963, Fellow, American Academy of Arts and Sciences;
 * 1971, Corresponding Member, Brazilian Academy of Sciences;
 * 1983, Associate Founding Fellow, TWAS;
 * 1985, Foreign Fellow, Royal Society of London, UK;
 * 1986, Honorary Fellow, London Mathematical Society, UK;
 * 1986, Corresponding Member, Accademia Peloritana, Messina, Sicily;
 * 1987, Honorary Life Member, New York Academy of Sciences;
 * 1989, Foreign Member, Accademia dei Lincei, Italy;
 * 1989, Foreign Member, Académie des sciences, France;
 * 1989, Member, American Philosophical Society;
 * 1994, Foreign Member, Chinese Academy of Sciences.

Chern was given a number of honorary degrees, including from The Chinese University of Hong Kong (LL.D. 1969), University of Chicago (D.Sc. 1969), ETH Zurich (Dr.Math. 1982), Stony Brook University (D.Sc. 1985), TU Berlin (Dr.Math. 1986), his alma mater Hamburg (D.Sc. 1971) and Nankai (honorary doctorate, 1985), etc.

Chern was also granted numerous honorary professorships, including at Peking University (Beijing, 1978), his alma mater Nankai (Tianjin, 1978), Chinese Academy of Sciences Institute of Systems Science (Beijing, 1980), Jinan University (Guangzhou, 1980), Chinese Academy of Sciences Graduate School (1984), Nanjing University (Nanjing, 1985), East China Normal University (Shanghai, 1985), USTC (Hefei, 1985), Beijing Normal University (1985), Zhejiang University (Hangzhou, 1985), Hangzhou University (1986, the university was merged into Zhejiang University in 1998), Fudan University (Shanghai, 1986), Shanghai University of Technology (1986, the university was merged to establish Shanghai University in 1994), Tianjin University (1987), Tohoku University (Sendai, Japan, 1987), etc.

Publications

 * Shiing Shen Chern, Topics in Differential Geometry, The Institute for Advanced Study, Princeton 1951
 * Shiing Shen Chern, Differential Manifolds, University of Chicago 1953
 * Shiing Shen Chern, Complex Manifolds, University of Chicago, 1956
 * Shiing Shen Chern: Complex manifolds Without Potential Theory, Springer-Verlag, New York 1979
 * Shiing Shen Chern, Minimal Submanifolds in a Riemannian Manifold, University of Kansas 1968
 * Bao, David Dai-Wai; Chern, Shiing-Shen; Shen, Zhongmin, Editors, Finsler Geometry American Mathematical Society 1996
 * Shiing-Shen Chern, Zhongmin Shen, Riemann Finsler Geometry, World Scientific 2005
 * Shiing Shen Chern, Selected Papers, Vol I-IV, Springer
 * Shiing-Shen Chern, A Simple Intrinsic Proof of the Gauss-Bonnet Formula for Closed Riemannian Manifolds, Annals of Mathematics, 1944
 * Shiing-Shen Chern, Characteristic Classes of Hermitian Manifolds, Annals of Mathematics, 1946
 * Shiing Shen Chern, Geometrical Interpretation of the sinh-Gordon Equation
 * Shiing Shen Chern, Geometry of a Quadratic Differential Form, Journal of the Society for Industrial and Applied Mathematics 1962
 * Shiing Shen Chern, On the Euclidean Connections in a Finsler Space, Proceedings of the National Academy of Sciences 1943
 * Shiing Shen Chern, General Relativity and differential geometry
 * Shiing Shen Chern, Geometry and physics
 * Shiing Shen Chern, Web geometry
 * Shiing Shen Chern, Deformation of surfaces preserving principle curvatures
 * Shiing Shen Chern, Differential Geometry and Integral Geometry
 * Shiing Shen Chern, Geometry of G-structures
 * Shiing-Shen Chern, Wei-Huan Chen, K. S. Lam, Lectures on Differential Geometry, World Scientific, 1999
 * David Dai-Wai Bao, Shiing-Shen Chern, Zhongmin Shen, An Introduction to Riemann-Finsler Geometry, GTM 200, Springer 2000
 * David Bao, Robert L. Bryant, Shiing-Shen Chern, Zhongmin Shen, Editors, A Sampler of Riemann-Finsler Geometry, MSRI Publications 50, Cambridge University Press 2004
 * David Dai-Wai Bao, Shiing-Shen Chern, Zhongmin Shen, An Introduction to Riemann-Finsler Geometry, GTM 200, Springer 2000
 * David Bao, Robert L. Bryant, Shiing-Shen Chern, Zhongmin Shen, Editors, A Sampler of Riemann-Finsler Geometry, MSRI Publications 50, Cambridge University Press 2004

Namesake and persona

 * The asteroid 29552 Chern is named after him;
 * The Chern Medal, of the International Mathematical Union (IMU);
 * The Shiing-Shen Chern Prize, of the Association of Chinese Mathematicians;
 * The Chern Institute of Mathematics at Nankai University, Tianjin, renamed in 2005 in honor of Chern;
 * The Chern Lectures, and the Shiing-Shen Chern Chair in Mathematics, both at the Department of Mathematics, UC Berkeley.

Chern liked to play contract bridge, Go (game), read Wuxia-literature of Jin Yong and had an interest in Chinese philosophy and history.

In 1975, Chen Ning Yang and Chern found out that their research in non-abelian gauge theory and Fiber bundle describe the same theoretical structure, which showed a surprising connection between physics and mathematics. Therefore, Chern asked Fan Zeng to finish a chinese painting named Shiing-Shen Chern and Chen Ning Yang for that. The Painting was later donated to the Nankai University.

A polyglot, he spoke German, French, English, Wu and Mandarin Chinese."“Whenever we had to go to the chancellor to make some special request, we always took Chern along, and it always worked,” says Berkeley mathematician Rob Kirby. “Somehow he had a presence, a gravitas. There was something about him that people just listened to him, and usually did things his way.”"

The Chern Song
In 1979 a Chern Symposium offered him a honorary song in tribute: ''Hail to Chern! Mathematics Greatest!''

He made Gauss-Bonnet a household word,

Intrinsic proofs he found,

Throughout the World his truths abound,

Chern classes he gave us,

and Secondary Invariants,

Fibre Bundles and Sheaves,

Distributions and Foliated Leaves!

All Hail All Hail to CHERN. It's called the Chern song.

Chern professorships
Allyn Jackson writes S. S. Chern is the recipient of many international honors, including six honorary doctorates, the U.S. National Medal of Science, Israel’s Wolf Prize, and membership in learned academies around the world. He has also received a more homegrown honor, the dream-turned-reality of an appreciative student of 30 years ago, who grew up in the Bay Area.

When Robert Uomini would buy his 10 tickets for the California State Lottery, he had an unusual “what if I win?” fantasy: He wanted to endow a professorship to honor S. S. Chern. While an undergraduate at U.C. Berkeley in the 1960s, Uomini was greatly inspired by a differential geometry course he took from Chern. With Chern’s support and encouragement, Uomini entered graduate school at Berkeley and received his Ph.D. in mathematics in 1976. Twenty years later, while working as a consultant to Sun Microsystems in Palo Alto, Uomini won $22 million in the state lottery. He could then realize his dream of expressing his gratitude in a concrete way.

Uomini and his wife set up the Robert G. Uomini and Louise B. Bidwell Foundation to support an extended visit of an outstanding mathematician to the U.C. Berkeley campus. There have been three Chern Visiting Professors so far: Sir Michael Atiyah of the University of Cambridge (1996), Richard Stanley of the Massachusetts Institute of Technology (1997), and Friedrich Hirzebruch of the Max Planck Institute for Mathematics in Bonn (1998). Jean-Pierre Serre of the Collège de France was the Chern Visiting Professor for 1999. [sic]

The foundation also helps to support the Chern Symposium, a yearly one-day event held in Berkeley during the period when the Chern Visiting Professor is in residence. The March 1998 Symposium was co-sponsored by the Mathematical Sciences Research Institute and was expanded to run for three days, featuring a dozen speakers. The MSRI also set up a Chern Professorship, funded by Chern's children May and Paul as well as James Simons.

Biographies on Chern and other memorabilia
Abraham Pais wrote about Chern in his book Subtle is the Lord. To paraphrase one passage: the outstanding mathematician Chern has two things to say, 1) I feel very mysterious that in the fields I'm working on (general relativity and differential geometry) there is so much more that can be explored; and 2) when talking with Albert Einstein (his colleague at the IAS) about his problem of a Grand Unified Theory, I realized the difference between mathematics and physics is at the heart of the journey towards a theory of everything.

Manfredo Do Carmo dedicated his book on Riemannian Geometry to Chern, his PhD advisor.

In Yau's autobiography, he talks a lot about his advisor Chern. In 1982, while on sabbatical at the New York University Courant Institute, he visited Stony Brook to see his friends and former students CN Yang and Simons.

In 2011 ZALA films published a documentary titled Taking the Long View: the Life of Shiing-shen Chern . In 2013 it was broadcast on US public television. It was compiled with the help of his friends including Alan Weinstein, Chuu-Lian Terng, Calvin C. Moore, Marty Shen, Robert Bryant, Robert Uomini, Robert Osserman, Hung-Hsi Wu, Rob Kirby, CN Yang, Paul Chu, Udo Simon, Phillip Griffiths, etc.

Dozens of other biographies have been written on Chern. See the citations for more info.

Poetry
Chern was an expressive poet as well. On his 60th birthday he wrote a love letter re-affirming his gratitude towards his wife and celebrating their 'beautiful, long, happy, marriage': Thirty-six years together

Through times of happiness

And times of worry too.

Time’s passage has no mercy.

We fly the Skies and cross the Oceans

To fulfill my destiny;

Raising the children fell

Entirely on your shoulders.

How fortunate I am

To have my works to look back upon,

I feel regrets you still have chores.

Growing old together in El Cerrito is a blessing.

Time passes by,

And we hardly notice.

Students
Chern has 43 students, including Fields medalist Shing-Tung Yau, Nobel Prize winner Chen-Ning Yang; and over 1000 descendants.

His student James Harris Simons at Stony Brook (co-author of the Chern–Simons theory) later founded the hedge fund Renaissance Technologies and became a billionaire. Simons talks about Chern in his TED talk.

Two of his students Manfredo do Carmo and Katsumi Nomizu have written influential textbooks in geometry.

Former director of the IAS Phillip Griffiths wrote "[Chern] took great pleasure in getting to know and working with and helping to guide young mathematicians. I was one of them."

Family
His wife, Shih-ning Cheng, whom he married in 1939, died in 2000. He also had a daughter, May Chu, wife of the physicist Chu Ching-wu, and a son named Paul. On his wife he writes (also see Selected Papers): "I would not conclude this account without mentioning my wife's role in my life and work. Through war and peace and through bad and good times we have shared a life for forty years, which is both simple and rich. If there is credit for my mathematical works, it will be hers as well as mine."May Chu described her father as an easygoing parent. Paul added that he often saw what was best for you before you realized it.

title: Shiing-Shen Chern


text:
Kähler

Kähler may refer to:

People

 * Birgit Kähler (born 1970), German high jumper
 * Erich Kähler (1906–2000), German mathematician
 * Heinz Kähler (1905–1974), German art historian and archaeologist
 * Luise Kähler (1869–1955), German trade union leader and politician
 * Martin Kähler (1835–1912), German theologian
 * Otto Kähler (1894–1967), German admiral
 * Wilhelmine Kähler (1864–1941), German politician

Other

 * Kähler Keramik, a Danish ceramics manufacturer
 * Kähler manifold, an important geometric complex manifold

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Kähler


text:
Chern–Simons theory

The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type developed by Edward Witten. It was discovered first by mathematical physicist Albert Schwarz. It is named after mathematicians Shiing-Shen Chern and James Harris Simons, who introduced the Chern–Simons 3-form. In the Chern–Simons theory, the action is proportional to the integral of the Chern–Simons 3-form.

In condensed-matter physics, Chern–Simons theory describes the topological order in fractional quantum Hall effect states. In mathematics, it has been used to calculate knot invariants and three-manifold invariants such as the Jones polynomial.

Particularly, Chern–Simons theory is specified by a choice of simple Lie group G known as the gauge group of the theory and also a number referred to as the level of the theory, which is a constant that multiplies the action. The action is gauge dependent, however the partition function of the quantum theory is well-defined when the level is an integer and the gauge field strength vanishes on all boundaries of the 3-dimensional spacetime.

It is also the central mathematical object in theoretical models for topological quantum computers (TQC). Specifically, an SU(2) Chern–Simons theory describes the simplest non-abelian anyonic model of a TQC, the Yang–Lee–Fibonacci model.

The dynamics of Chern–Simons theory on the 2-dimensional boundary of a 3-manifold is closely related to fusion rules and conformal blocks in conformal field theory, and in particular WZW theory.

Mathematical origin
In the 1940s S. S. Chern and A. Weil studied the global curvature properties of smooth manifolds M as de Rham cohomology (Chern–Weil theory), which is an important step in the theory of characteristic classes in differential geometry. Given a flat G-principal bundle P on M there exists a unique homomorphism, called the Chern–Weil homomorphism, from the algebra of G-adjoint invariant polynomials on g (Lie algebra of G) to the cohomology $$H^*(M,\mathbb{R})$$. If the invariant polynomial is homogeneous one can write down concretely any k-form of the closed connection ω as some 2k-form of the associated curvature form Ω of ω.

In 1974 S. S. Chern and J. H. Simons had concretely constructed a (2k − 1)-form df(ω) such that
 * $$dTf(\omega)=f(\Omega^k),$$

where T is the Chern–Weil homomorphism. This form is called Chern–Simons form. If df(ω) is closed one can integrate the above formula
 * $$Tf(\omega)=\int_C f(\Omega^k),$$

where C is a (2k − 1)-dimensional cycle on M. This invariant is called Chern–Simons invariant. As pointed out in the introduction of the Chern–Simons paper, the Chern–Simons invariant CS(M) is the boundary term that cannot be determined by any pure combinatorial formulation. It also can be defined as
 * $$\operatorname{CS}(M)=\int_{s(M)}\tfrac{1}{2}Tp_1\in\mathbb{R}/\mathbb{Z},$$

where $$p_1$$ is the first Pontryagin number and s(M) is the section of the normal orthogonal bundle P. Moreover, the Chern–Simons term is described as the eta invariant defined by Atiyah, Patodi and Singer.

The gauge invariance and the metric invariance can be viewed as the invariance under the adjoint Lie group action in the Chern–Weil theory. The action integral (path integral) of the field theory in physics is viewed as the Lagrangian integral of the Chern–Simons form and Wilson loop, holonomy of vector bundle on M. These explain why the Chern–Simons theory is closely related to topological field theory.

Configurations
Chern–Simons theories can be defined on any topological 3-manifold M, with or without boundary. As these theories are Schwarz-type topological theories, no metric needs to be introduced on M.

Chern–Simons theory is a gauge theory, which means that a classical configuration in the Chern–Simons theory on M with gauge group G is described by a principal G-bundle on M. The connection of this bundle is characterized by a connection one-form A which is valued in the Lie algebra g of the Lie group G. In general the connection A is only defined on individual coordinate patches, and the values of A on different patches are related by maps known as gauge transformations. These are characterized by the assertion that the covariant derivative, which is the sum of the exterior derivative operator d and the connection A, transforms in the adjoint representation of the gauge group G. The square of the covariant derivative with itself can be interpreted as a g-valued 2-form F called the curvature form or field strength. It also transforms in the adjoint representation.

Dynamics
The action S of Chern–Simons theory is proportional to the integral of the Chern–Simons 3-form

 * $$S=\frac{k}{4\pi}\int_M \text{tr}\,(A\wedge dA+\tfrac{2}{3}A\wedge A\wedge A).$$

The constant k is called the level of the theory. The classical physics of Chern–Simons theory is independent of the choice of level k.

Classically the system is characterized by its equations of motion which are the extrema of the action with respect to variations of the field A. In terms of the field curvature

 * $$F = dA + A \wedge A \, $$

the field equation is explicitly

 * $$0=\frac{\delta S}{\delta A}=\frac{k}{2\pi} F.$$

The classical equations of motion are therefore satisfied if and only if the curvature vanishes everywhere, in which case the connection is said to be flat. Thus the classical solutions to G Chern–Simons theory are the flat connections of principal G-bundles on M. Flat connections are determined entirely by holonomies around noncontractible cycles on the base M. More precisely, they are in one-to-one correspondence with equivalence classes of homomorphisms from the fundamental group of M to the gauge group G up to conjugation.

If M has a boundary N then there is additional data which describes a choice of trivialization of the principal G-bundle on N. Such a choice characterizes a map from N to G. The dynamics of this map is described by the Wess–Zumino–Witten (WZW) model on N at level k.

Quantization
To canonically quantize Chern–Simons theory one defines a state on each 2-dimensional surface Σ in M. As in any quantum field theory, the states correspond to rays in a Hilbert space. There is no preferred notion of time in a Schwarz-type topological field theory and so one can require that Σ be a Cauchy surface, in fact, a state can be defined on any surface.

Σ is of codimension one, and so one may cut M along Σ. After such a cutting M will be a manifold with boundary and in particular classically the dynamics of Σ will be described by a WZW model. Witten has shown that this correspondence holds even quantum mechanically. More precisely, he demonstrated that the Hilbert space of states is always finite-dimensional and can be canonically identified with the space of conformal blocks of the G WZW model at level k.

For example, when Σ is a 2-sphere, this Hilbert space is one-dimensional and so there is only one state. When Σ is a 2-torus the states correspond to the integrable representations of the affine Lie algebra corresponding to g at level k. Characterizations of the conformal blocks at higher genera are not necessary for Witten's solution of Chern–Simons theory.

Wilson loops
The observables of Chern–Simons theory are the n-point correlation functions of gauge-invariant operators. The most often studied class of gauge invariant operators are Wilson loops. A Wilson loop is the holonomy around a loop in M, traced in a given representation R of G. As we will be interested in products of Wilson loops, without loss of generality we may restrict our attention to irreducible representations R.

More concretely, given an irreducible representation R and a loop K in M, one may define the Wilson loop $$W_R(K)$$ by

 * $$ W_R(K) =\operatorname{Tr}_R \, \mathcal{P} \exp\left(i \oint_K A\right)$$

where A is the connection 1-form and we take the Cauchy principal value of the contour integral and $$\mathcal{P} \exp$$ is the path-ordered exponential.

HOMFLY and Jones polynomials
Consider a link L in M, which is a collection of ℓ disjoint loops. A particularly interesting observable is the ℓ-point correlation function formed from the product of the Wilson loops around each disjoint loop, each traced in the fundamental representation of G. One may form a normalized correlation function by dividing this observable by the partition function Z(M), which is just the 0-point correlation function.

In the special case in which M is the 3-sphere, Witten has shown that these normalized correlation functions are proportional to known knot polynomials. For example, in G = U(N) Chern–Simons theory at level k the normalized correlation function is, up to a phase, equal to

 * $$\frac{\sin(\pi/(k+N))}{\sin(\pi N/(k+N))}$$

times the HOMFLY polynomial. In particular when N = 2 the HOMFLY polynomial reduces to the Jones polynomial. In the SO(N) case, one finds a similar expression with the Kauffman polynomial.

The phase ambiguity reflects the fact that, as Witten has shown, the quantum correlation functions are not fully defined by the classical data. The linking number of a loop with itself enters into the calculation of the partition function, but this number is not invariant under small deformations and in particular, is not a topological invariant. This number can be rendered well defined if one chooses a framing for each loop, which is a choice of preferred nonzero normal vector at each point along which one deforms the loop to calculate its self-linking number. This procedure is an example of the point-splitting regularization procedure introduced by Paul Dirac and Rudolf Peierls to define apparently divergent quantities in quantum field theory in 1934.

Sir Michael Atiyah has shown that there exists a canonical choice of 2-framing, which is generally used in the literature today and leads to a well-defined linking number. With the canonical framing the above phase is the exponential of 2πi/(k + N) times the linking number of L with itself.

"The original Jones polynomial was defined for 1-links in the 3-sphere (the 3-ball, the 3-space R3). Can you define the Jones polynomial for 1-links in any 3-manifold?"
 * Problem (Extension of Jones polynomial to general 3-manifolds)

See section 1.1 of this paper for the background and the history of this problem. Kauffman submitted a solution in the case of the product manifold of closed oriented surface and the closed interval, by introducing virtual 1-knots. It is open in the other cases. Witten's path integral for Jones polynomial is written for links in any compact 3-manifold formally, but the calculus is not done even in physics level in any case other than the 3-sphere (the 3-ball, the 3-space R3). This problem is also open in physics level. In the case of Alexander polynomial, this problem is solved.

Topological string theories
In the context of string theory, a U(N) Chern–Simons theory on an oriented Lagrangian 3-submanifold M of a 6-manifold X arises as the string field theory of open strings ending on a D-brane wrapping X in the A-model topological string theory on X. The B-model topological open string field theory on the spacefilling worldvolume of a stack of D5-branes is a 6-dimensional variant of Chern–Simons theory known as holomorphic Chern–Simons theory.

WZW and matrix models
Chern–Simons theories are related to many other field theories. For example, if one considers a Chern–Simons theory with gauge group G on a manifold with boundary then all of the 3-dimensional propagating degrees of freedom may be gauged away, leaving a two-dimensional conformal field theory known as a G Wess–Zumino–Witten model on the boundary. In addition the U(N) and SO(N) Chern–Simons theories at large N are well approximated by matrix models.

Chern–Simons gravity theory
In 1982, S. Deser, R. Jackiw and S. Templeton proposed the Chern–Simons gravity theory in three dimensions, in which the Einstein–Hilbert action in gravity theory is modified by adding the Chern–Simons term.

In 2003, R. Jackiw and S. Y. Pi extended this theory to four dimensions and Chern–Simons gravity theory has some considerable effects not only to fundamental physics but also condensed matter theory and astronomy.

The four-dimensional case is very analogous to the three-dimensional case. In three dimensions, the gravitational Chern–Simons term is
 * $$\operatorname{CS}(\Gamma)=\frac{1}{2\pi^2}\int d^3x\varepsilon^{ijk}\biggl(\Gamma^p_{iq}\partial_j\Gamma^q_{kp}+\frac{2}{3}\Gamma^p_{iq}\Gamma^q_{jr}\Gamma^r_{kp}\biggr).$$

This variation gives the Cotton tensor
 * $$=-\frac{1}{2\sqrt{g}}\bigl(\varepsilon^{mij}D_i R^n_j+\varepsilon^{nij}D_i R^m_j).$$

Then, Chern–Simons modification of three-dimensional gravity is made by adding the above Cotton tensor to the field equation, which can be obtained as the vacuum solution by varying the Einstein–Hilbert action.

Chern–Simons matter theories
In 2013 Kenneth A. Intriligator and Nathan Seiberg solved these 3d Chern–Simons gauge theories and their phases using monopoles carrying extra degrees of freedom. The Witten index of the many vacua discovered was computed by compactifying the space by turning on mass parameters and then computing the index. In some vacua, supersymmetry was computed to be broken. These monopoles were related to condensed matter vortices.

The N = 6 Chern–Simons matter theory is the holographic dual of M-theory on $$AdS_4\times S_7$$.

Four-dimensional Chern–Simons theory
In 2013 Kevin Costello defined a closely related theory defined on a four-dimensional manifold consisting of the product of a two-dimensional 'topological plane' and a two-dimensional (or one complex dimensional) complex curve. He later studied the theory in more detail together with Witten and Masahito Yamazaki,  demonstrating how the gauge theory could be related to many notions in integrable systems theory, including exactly solvable lattice models (like the six-vertex model or the XXZ spin chain), integrable quantum field theories (such as the Gross–Neveu model, principal chiral model and symmetric space coset sigma models), the Yang–Baxter equation and quantum groups such as the Yangian which describe symmetries underpinning the integrability of the aforementioned systems.

The action on the 4-manifold $$M = \Sigma \times C$$ where $$\Sigma$$ is a two-dimensional manifold and $$C$$ is a complex curve is

where $$\omega$$ is a meromorphic one-form on $$C$$.

Chern–Simons terms in other theories
The Chern–Simons term can also be added to models which aren't topological quantum field theories. In 3D, this gives rise to a massive photon if this term is added to the action of Maxwell's theory of electrodynamics. This term can be induced by integrating over a massive charged Dirac field. It also appears for example in the quantum Hall effect. The addition of the Chern–Simons term to various theories gives rise to vortex- or soliton-type solutions Ten- and eleven-dimensional generalizations of Chern–Simons terms appear in the actions of all ten- and eleven-dimensional supergravity theories.

One-loop renormalization of the level
If one adds matter to a Chern–Simons gauge theory then, in general it is no longer topological. However, if one adds n Majorana fermions then, due to the parity anomaly, when integrated out they lead to a pure Chern–Simons theory with a one-loop renormalization of the Chern–Simons level by −n/2, in other words the level k theory with n fermions is equivalent to the level k − n/2 theory without fermions.

title: Chern–Simons theory


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71 (number)

71 (seventy-one) is the natural number following 70 and preceding 72.

In mathematics
71 is the 20th prime number. Because both rearrangements of its digits (17 and 71) are prime numbers, 71 is an emirp and more generally a permutable prime. It is the largest number which occurs as a prime factor of an order of a sporadic simple group, the largest (15th) supersingular prime.

It is a Pillai prime, since $$9!+1$$ is divisible by 71, but 71 is not one more than a multiple of 9. It is part of the last known pair (71, 7) of Brown numbers, since $$71^{2}=7!+1$$.

It is centered heptagonal number.

title: 71 (number)


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73 (number)

73 (seventy-three) is the natural number following 72 and preceding 74. In English, it is the smallest natural number with twelve letters in its spelled out name.

In mathematics
73 is the 21st prime number, and emirp with 37, the 12th prime number. It is also the eighth twin prime, with 71. It is the largest minimal primitive root in the first 100,000 primes; in other words, if $p$ is one of the first one hundred thousand primes, then at least one of the numbers 2, 3, 4, 5, 6, ..., 73 is a primitive root modulo $p$. 73 is also the smallest factor of the first composite generalized Fermat number in decimal: $$10^{4}+1=10,001=73\times 137$$, and the smallest prime congruent to 1 modulo 24, as well as the only prime repunit in octal (1118). It is the fourth star number.

Sheldon prime
Where 73 and 37 are part of the sequence of permutable primes and emirps in base-ten, the number 73 is more specifically the unique Sheldon prime, named as an homage to Sheldon Cooper and defined as satisfying "mirror" and "product" properties, where:

 * 73 has 37 as the mirroring of its decimal digits. 73 is the 21st prime number, and 37 the 12th. The "mirror property" is fulfilled when 73 has a mirrored permutation of its digits (37) that remains prime. Similarly, their respective prime indices (21 and 12) in the list of prime numbers are also permutations of the same digits (1, and 2).
 * 73 is the 21st prime number. It satisfies the "product property" since the product of its decimal digits is precisely in equivalence with its index in the sequence of prime numbers. i.e., 21 = 7 × 3. On the other hand, 37 does not fulfill the product property, since, naturally, its digits also multiply to 21; therefore, the only number to fulfill this property between these two numbers is 73, and as such it is the only "Sheldon prime".

Further properties ligating 73 and 37
Arithmetically, from sums of 73 and 37 with their prime indexes, one obtains:

Meanwhile, 73 and 37 have a range of 37 numbers, inclusive of both 37 and 73; their difference, on the other hand, is 36, or thrice 12. Also,

Where 73 is the ninth member of Hogben's central polygonal numbers, which enumerates the maximal number of interior regions formed by nine intersecting circles, members in this sequence also include 307, 343, and 703 as the 18th, 19th, and 27th indexed numbers, respectively (where $777 = 3 × 37 × 7 = 21 × 37$); while 3, 7 and 21 are also in this sequence, as the 2nd, 3rd, and 5th members.

73 and 37 are also consecutive star numbers, equivalently consecutive centered dodecagonal (12-gonal) numbers (respectively the 4th and the 3rd). They are successive lucky primes and sexy primes, both twice over,  and successive Pierpont primes, respectively the 9th and 8th. 73 and 37 are consecutive values of $$g(k)$$ such that every positive integer can be written as the sum of 73 or fewer sixth powers, or 37 or fewer fifth powers (and 19 or fewer fourth powers; see Waring's problem).

In binary, 73 is represented as $703$, while 21 in binary is $3.7$, with 7 and 3 represented as $373$ and $337 + 373 + 733 = 1443$ respectively, all which are palindromic. Of the seven binary digits representing 73, there are three 1s. In addition to having prime factors 7 and 3, the number $343 = 7 × 7 × 7 = 7^{3}$ represents the ternary (base-3) equivalent of the decimal numeral 7, that is to say: $3 + 4$.

Sierpiński numbers
73 and 37 are consecutive primes in the seven-integer covering set of the first known Sierpiński number 78,557 of the form $$k \times 2^{n} + 1$$ that is composite for all natural numbers $$n$$, where 73 is the largest member: $$\{3, 5, 7, 13, 19, 37, 73\}.$$ More specifically, $$78,557 \times 2^{n} + 1$$ modulo 36 will be divisible by at least one of the integers in this set.

Consider the following sequence $$A(n)$$:
 * Let $$k$$ be a Sierpiński number or Riesel number divisible by $$2n - 1$$, and let $$p$$ be the largest number in a set of primes which cover every number of the form $$k \times 2^{m} + 1$$ or of the form $$k \times 2^{m} - 1$$, with $$m \geq 1$$;
 * $$A(n)$$ equals $$p$$ if and only if there exists no number $$k$$ that has a covering set with largest prime greater than $$p$$.

Known such index values $$n$$ where $$p$$ is equal to 73 as the largest member of such covering sets are: $$\{1, 6, 9, 12, 15, 16, 21, 22, 24, 27\}$$, with 37 present alongside 73. In particular, $$A(n)$$ ≥ 73 for any $$n$$.

In addition, 73 is the largest member in the covering set $$\{5,7,13,73\}$$ of the smallest proven generalized Sierpiński number of the form $$k \times b^{n} + 1$$ in nonary $$(2,344 \times 9^{n} + 1)$$, while it is also the largest member of the covering set $$\{7, 11, 13, 73\}$$ that belongs to the smallest such provable number in decimal $$(9,175 \times 10^n + 1)$$ — both in congruencies $$\text {mod } 6$$.

Other properties

73 is one of the fifteen left-truncatable and right-truncatable primes in decimal, meaning it remains prime when the last "right" digit is successively removed and it remains prime when the last "left" digit is successively removed; and because it is a twin prime (with 71), it is the only two-digit twin prime that is both a left-truncatable and right-truncatable prime.

The row sum of Lah numbers of the form $$L(n,k) = \textstyle {\left\lfloor {n \atop k} \right\rfloor}$$ with $$n = 4$$ and $$k = {1, 2, 3, 4}$$ is equal to $$73$$. These numbers represent coefficients expressing rising factorials in terms of falling factorials, and vice-versa; equivalently in this case to the number of partitions of $$\{{1,2,3,4}\}$$ into any number of lists, where a list means an ordered subset.

73 requires 115 steps to return to 1 in the Collatz problem, and 37 requires 21: {37, 112, 56, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1}. Collectively, the sum between these steps is 136, the 16th triangular number, where {16, 8, 4, 2, 1} is the only possible step root pathway.

There are 73 three-dimensional arithmetic crystal classes that are part of 230 crystallographic space group types. These 73 groups are specifically symmorphic groups such that all operating lattice symmetries have one common fixed isomorphic point, with the remaining 157 groups nonsymmorphic (the 37th prime is 157).

In five-dimensional space, there are 73 Euclidean solutions of 5-polytopes with uniform symmetry, excluding prismatic forms: 19 from the $$\mathrm A_{5}$$ simplex group, 23 from the $$\mathrm D_{5}$$ demihypercube group, and 31 from the $$\mathrm B_{5}$$ hypercubic group, of which 15 equivalent solutions are shared between $$\mathrm D_{5}$$ and $$\mathrm B_{5}$$ from distinct polytope operations.

In moonshine theory of sporadic groups, 73 is the first non-supersingular prime greater than 71 that does not divide the order of the largest sporadic group $$\mathrm {F_{1}}$$. All primes greater than or equal to 73 are non-supersingular, while 37, on the other hand, is the smallest prime number that is not supersingular. $$\mathrm {F_{1}}$$ contains a total of 194 conjugacy classes that involve 73 distinct orders (without including multiplicities over which letters run).

73 is the largest member of a 17-integer matrix definite quadratic that represents all prime numbers: {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 67, 73}, with consecutive primes between 2 through 47.

In science

 * The atomic number of tantalum.

In astronomy

 * Messier object M73, a magnitude 9.0 apparent open cluster in the constellation Aquarius.
 * The New General Catalogue object NGC 73, a barred spiral galaxy in the constellation Cetus.
 * The number of seconds it took for the Space Shuttle Challenger OV-099 shuttle to explode after launch.
 * 73 is the number of rows in the 1,679-bit Arecibo message, sent to space in search for extraterrestrial intelligence.

In chronology

 * The year AD 73, 73 BC, or 1973.
 * The number of days in 1/5 of a non-leap year.
 * The 73rd day of a non-leap year is March 14, also known as Pi Day.

In other fields
73 is also:
 * The number of books in the Catholic Bible.
 * Amateur radio operators and other morse code users commonly use the number 73 as a "92 Code" abbreviation for "best regards", typically when ending a QSO (a conversation with another operator). These codes also facilitate communication between operators who may not be native English speakers. In Morse code, 73 is an easily recognized palindrome: ( - - · · ·    · · · - - ).
 * 73 (also known as 73 Amateur Radio Today) was an amateur radio magazine published from 1960 to 2003.
 * 73 was the number on the Torpedo Patrol (PT) boat in the TV show McHale's Navy.
 * The registry of the U.S. Navy's nuclear aircraft carrier USS George Washington (CVN-73), named after U.S. President George Washington.
 * No. 73 was the name of a 1980s children's television programme in the United Kingdom. It ran from 1982 to 1988 and starred Sandi Toksvig.
 * Pizza 73 is a Canadian pizza chain.
 * Game show Match Game '73 in 1973.
 * Fender Rhodes Stage 73 Piano.
 * Sonnet 73 by William Shakespeare.
 * The number of the French department Savoie.
 * On a CB radio, 10-73 means "speed trap at..."

In sports

 * In international curling competitions, each side is given 73 minutes to complete all of its throws.
 * In baseball, the single-season home run record set by Barry Bonds in 2001.
 * In basketball, the number of games the Golden State Warriors won in the 2015–16 season (73–9), the most wins in NBA history.
 * NFL: In the 1940 NFL championship game, the Bears beat the Redskins 73–0, the largest score ever in an NFL game. (The Redskins won their previous regular season game, 7–3.)

Doctor Who
In a 2024 episode of Doctor Who, "73 Yards", the character Ruby Sunday is haunted by a mysterious woman who is always standing exactly 73 yards away from her.

The Big Bang Theory
73 is Sheldon Cooper's favorite number in The Big Bang Theory. He first expresses his love for it in "The Alien Parasite Hypothesis, the 73rd episode of The Big Bang Theory.". Jim Parsons was born in the year 1973. He often wears a t-shirt with the number 73 on it.

title: 73 (number)


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74 (number)

74 (seventy-four) is the natural number following 73 and preceding 75.

In mathematics
74 is:

 * the twenty-first distinct semiprime and the eleventh of the form (2.q), where q is a higher prime.
 * with an aliquot sum of 40, within an aliquot sequence of three composite numbers (74,40,50,43,1,0) to the Prime in the 43-aliquot tree.
 * a palindromic number in bases 6 (2026) and 36 (2236).
 * a nontotient.
 * the number of collections of subsets of {1, 2, 3} that are closed under union and intersection.
 * φ(74) = φ(σ(74)).

There are 74 different non-Hamiltonian polyhedra with a minimum number of vertices.

In science

 * The atomic number of tungsten

In astronomy

 * Messier object M74, a magnitude 10.5 spiral galaxy in the constellation Pisces.
 * The New General Catalogue object NGC 74, a galaxy in the constellation Andromeda.

In music

 * Seventy-four, one of the Number Pieces by John Cage
 * "Seventy-Four", a song by the American band Bright from the album The Albatross Guest House

In other fields
Seventy-four is also:
 * The year AD 74, 74 BC, or 1974
 * Designates the 7400 series of Integrated Chips. 74xx xx=00-4538
 * A seventy-four was a third-rate warship with 74 guns.
 * The registry of the U.S. Navy's nuclear aircraft carrier USS John C. Stennis (CVN-74), named after U.S. Senator John C. Stennis
 * A hurricane or typhoon is a system with sustained winds of at least 74 mph (64 knots).
 * The number of the French department Haute-Savoie
 * In the Bible it is the number of people that were in the presence of God on Mount Sinai and saw God without dying "Exodus 24:9-11"

title: 74 (number)


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75 (number)

75 (seventy-five) is the natural number following 74 and preceding 76.

In mathematics
75 is a self number because there is no integer that added up to its own digits adds up to 75. It is the sum of the first five pentagonal numbers, and therefore a pentagonal pyramidal number, as well as a nonagonal number.

It is also the fourth ordered Bell number, and a Keith number, because it recurs in a Fibonacci-like sequence started from its base 10 digits: 7, 5, 12, 17, 29, 46, 75...

75 is the count of the number of weak orderings on a set of four items.

Excluding the infinite sets, there are 75 uniform polyhedra in the third dimension, which incorporate star polyhedra as well. Inclusive of 7 families of prisms and antiprisms, there are also 75 uniform compound polyhedra.

In other fields
Seventy-five is:

 * The atomic number of rhenium
 * The age limit for Canadian senators
 * A common name for the Canon de 75 modèle 1897, a French World War I gun
 * The department number of the city of Paris
 * The number of balls in a standard game of Bingo in the United States

title: 75 (number)


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Streamline Pictures

Streamline Pictures was an American media company. Founded by screenwriter Carl Macek and animation historian Jerry Beck, it was one of the earliest distributors of English-dubbed Japanese animation.

Founding
Founded in Los Angeles, California, in late 1988, Streamline Pictures was one of the first North American companies that was created primarily with the intention of distributing translated anime uncut and faithful to the original content. The founders of Streamline were television writer and producer Carl Macek, who had worked for Harmony Gold USA on the series Robotech, and film distributor Jerry Beck. At one point or another, Gregory Snegoff, Steve Kramer, Ardwight Chamberlain, Tom Wyner, and Mike Reynolds, all of whom served as series staff writers (and voice actors) on Robotech, worked as independent writers and voice actors for Streamline.

High-profile releases
The first high-profile release by Streamline was the film Laputa: Castle in the Sky from director Hayao Miyazaki, premiering at the Roxy Screening Room in Philadelphia on March 24, 1989. The company was also known for its theatrical release of the film Akira at The Biograph in Georgetown, Washington, D.C. on December 25, 1989, as well as its 1992 video dubbing of The Castle of Cagliostro (earlier screened with subtitles at the Film Forum in New York City in April 1991). Streamline also dubbed My Neighbor Totoro, and Kiki's Delivery Service, both adapted and directed by Gregory Snegoff, in 1988 for Tokuma Shoten, although these were only used as in-flight films by Japan Airlines at the time. As a fan of Miyazaki's films, Macek did not think Laputa (of which Streamline was the distributor) had received the quality of dubbing that a Miyazaki film deserved. He thought that it could have been done better, so Tokuma Shoten gave him the opportunity to prove his words. For his first project, he asked to dub My Neighbor Totoro, one of his favorite Miyazaki works. The Streamline dub of My Neighbor Totoro was released theatrically in the U.S. by Troma Films in 1993; but its dub of Kiki's Delivery Service appeared only on the 1990s Japanese laserdisc release of that title. (Since then, however, all three afore-mentioned films by Miyazaki at Studio Ghibli have been redubbed by Disney.)

Tokuma Shoten was pleased with the quality of the initial work, and it immediately hired Streamline to produce the English language version of Kiki's Delivery Service. Soon after its release, the film was bought by Japan Airlines, who showed it during their flights between Japan and the U.S. Streamline also licensed and dubbed other popular anime series and movies, such as Fist of the North Star, Wicked City, Lensman, Vampire Hunter D and The Professional: Golgo 13.

Home video market
Streamline, in being the first company dedicated to making anime accessible to the English-speaking world, was notable for releasing a wide variety of anime that fit in a wide variety of genres, many of which do not fit completely in any genre such as Twilight of the Cockroaches. During the 1990s VHS era, before the common availability of hybrid DVDs, anime distributors released anime via subtitled or dubbed tapes with the subtitled editions being noticeably more expensive than the dubs which were expected to sell better. Streamline is also notable in being the only such company to eschew this practice and release only dubs of its anime. The only exceptions were the later Akira subtitled release and their Robotech Collection, which included episodes of the original Macross, Southern Cross and Mospeada episodes on the same tape along with their Robotech counterparts. (Since then, these three titles were re-released by ADV Films through Harmony Gold.)

Criticism
Because of Macek's notoriety with a certain branch of fandom, Streamline became one of the early catalysts of the sub vs. dub debate. Macek's philosophy towards anime dubbing, as stated in several interviews, most notably published Protoculture Addicts and Animag, has become largely synonymous with the negative connotation concerning "Americanized dubs".

Later years
Streamline Pictures stopped producing new anime releases in 1997, but continued to distribute its complete library in North America. In 1996, the company began to release foreign films under its "Independent Filmworks" label until 2000 when the company closed. The company's Modelworks division was sold in 1998 and the "Streamline Pictures Modelworks" division became a separate company operating under the name, "Avatar Creations."

Today, rights to much of the Streamline film and television library (such as Akira and The Castle of Cagliostro) have either reverted to the Japanese rights holder and licensed to other companies; while others have never been re-released due to licensing issues and lack of demand.

title: Streamline Pictures


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Robert Lacroix

Robert Lacroix, (born April 15, 1940) is a professor of economics at the Université de Montréal in Montreal, Quebec, Canada.

After finishing his Ph.D. in economics at Leuven, in Belgium, in 1970, he became professor at the Department of Economics of the Université de Montréal. Throughout his career, he published many articles in peer-reviewed journals, mainly in the field of labor economics. From 1987 to 1993, he served as Dean of the Faculty of Arts and Sciences at the Université de Montréal. He served as Rector of the university from 1998 until 2005.

In 2005, he was among the twelve signatories of the political manifesto Pour un Québec lucide.

He is a Fellow of the Royal Society of Canada, an Officer of the Order of Canada, an Officer of the National Order of Quebec, and recipient of the Quebec Award for Science (Prix Armand-Frappier). He is also an active Fellow at the CIRANO, a Montréal-based think tank in economics and public policy.

title: Robert Lacroix


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Lusatian Neisse

The Lusatian Neisse  (Lausitzer Neiße; Nysa Łużycka; Lužická Nisa; Upper Sorbian: Łužiska Nysa; Lower Sorbian: Łužyska Nysa), or Western Neisse, is a 252 km river in northern Central Europe. It rises in the Jizera Mountains, near Nová Ves nad Nisou, at the Czech border becoming the Polish–German border for its remaining 197 km, to flow into the similarly northward-flowing Oder from the left.

Its drainage basin covers 4403 km2, of which 2201 km2 is in Poland, the rest is mainly in Germany. The river reaches the tripoint of the three nations by Zittau, a German town/city, after 54 km, leaving the Czech Republic. It is a left-bank tributary of the Oder, into which it flows between Neißemünde-Ratzdorf and Kosarzyn – north of the towns of Guben and Gubin. The river was a motivations to found Gubin as a craftmanship and trading port in the 13th Century.

Since the 1945 Potsdam Agreement in the aftermath of World War II, the river has partially demarcated the German-Polish border (along the Oder–Neisse line). The German population east of the river was expelled from Poland to Germany.

It is the longest and most watered of the three rivers of its non-adjectival name in both the main languages (the two other rivers being the Eastern Neisse (Nysa Kłodzka; Glatzer Neisse) and Raging Neisse (Polish: Nysa Szalona; German: Wütende Neiße or Jauersche Neiße)). It is usually simply referred to as the Neisse.

Name
Since the river runs through the historic region of Lusatia, the adjective "Lusatian" or "Western" before the name of the river Neisse is used whenever differentiating this border river from the Eastern Neisse (Polish: Nysa Kłodzka, German: Glatzer Neisse) and the smaller Raging Neisse (Polish: Nysa Szalona; German: Wütende Neisse or Jauersche Neisse), both in Poland.

Towns and villages
At Bad Muskau the Neisse flows through Muskau Park, a UNESCO World Heritage Site. Cities and towns on the river from source to mouth include:
 * Jablonec nad Nisou, Czech Republic
 * Vratislavice, Czech Republic
 * Liberec, Czech Republic
 * Chrastava, Czech Republic
 * Hrádek nad Nisou, Czech Republic
 * Zittau, Germany
 * Bogatynia, Poland
 * Görlitz, Germany; Zgorzelec, Poland
 * Pieńsk, Poland
 * Bad Muskau, Germany; Łęknica, Poland
 * Forst (Lausitz), Germany
 * Guben, Germany; Gubin, Poland

Tributaries
Right bank:
 * Lubsza

Left bank:
 * Mandau

title: Lusatian Neisse


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32 (number)

32 (thirty-two) is the natural number following 31 and preceding 33.

Mathematics
32 is the fifth power of two ($$2^{5}$$), making it the first non-unitary fifth-power of the form $$p^{5}$$ where $$p$$ is prime. 32 is the totient summatory function $$\Phi(n)$$ over the first 10 integers, and the smallest number $$n$$ with exactly 7 solutions for $$\varphi (n)$$.

The aliquot sum of a power of two is always one less than the number itself, therefore the aliquot sum of 32 is 31.

$$ \begin{align} 32 & = 1^{1} + 2^{2} + 3^{3} \\ 32 & = (1\times4)+(2\times5)+(3\times6) \\ 32 & = (1\times2)+(1\times2\times3)+(1\times2\times3\times4) \\ \end{align}$$

The product between neighbor numbers of 23, the dual permutation of the digits of 32 in decimal, is equal to the sum of the first 32 integers: $$22 \times 24= 528$$.

32 is also a Leyland number expressible in the form $$x^y + y^x$$, where: $$32=2^{4} + 4^{2}.$$

The eleventh Mersenne number is the first to have a prime exponent (11) that does not yield a Mersenne prime, equal to: $$2047 = 32^{2} + (31 \times 33) = 1024 + 1023 = 2^{11} - 1.$$

The product of the five known Fermat primes is equal to the number of sides of the largest regular constructible polygon with a straightedge and compass that has an odd number of sides, with a total of sides numbering $$2^{32} - 1 = 3\cdot5\cdot17\cdot257\cdot65\;537 = 4\;294\;967\;295.$$

The first 32 rows of Pascal's triangle read as single binary numbers represent the 32 divisors that belong to this number, which is also the number of sides of all odd-sided constructible polygons with simple tools alone (if the monogon is also included).

There are also a total of 32 uniform colorings to the 11 regular and semiregular tilings.

There are 32 three-dimensional crystallographic point groups and 32 five-dimensional crystal families, and the maximum determinant in a 7 by 7 matrix of only zeroes and ones is 32. In sixteen dimensions, the sedenions generate a non-commutative loop $$\mathbb {S}_{L}$$ of order 32, and in thirty-two dimensions, there are at least 1,160,000,000 even unimodular lattices (of determinants 1 or −1); which is a marked increase from the twenty-four such Niemeier lattices that exists in twenty-four dimensions, or the single $$\mathrm E_{8}$$ lattice in eight dimensions (these lattices only exist for dimensions $$ d \propto 8$$). Furthermore, the 32nd dimension is the first dimension that holds non-critical even unimodular lattices that do not interact with a Gaussian potential function of the form $$ f_{\alpha} (r) = e^{-\alpha {r}}$$ of root $$r$$ and $$\alpha > 0$$.

32 is the furthest point in the set of natural numbers $$\mathbb {N}_{0}$$ where the ratio of primes (2, 3, 5, ..., 31) to non-primes (0, 1, 4, ..., 32) is $$\tfrac {1}{2}.$$

In science

 * The atomic number of germanium
 * The freezing point of water at standard atmospheric pressure in degrees Fahrenheit
 * In the Standard Model of particle physics, there are 32 degrees of freedom among the leptons and all bosons that interact with them (including the graviton, which is generally expected to exist, and assuming there are no right-handed neutrinos)

Astronomy

 * Messier 32, a magnitude 9.0 galaxy in the constellation Andromeda which is a companion to M31.
 * The New General Catalogue object NGC 32, a star in the constellation Pegasus

In music

 * A thirty-second note or demisemiquaver is a note played for 1/32 of the duration of a whole note
 * The number of completed, numbered piano sonatas by Ludwig van Beethoven
 * "32 Footsteps", a song by They Might Be Giants
 * "The Chamber of 32 Doors", a song by Genesis, from their 1974 concept album The Lamb Lies Down on Broadway
 * "32", a song on Mr. Mister's debut album I Wear the Face
 * "32", a song by electro-rock group Carpark North
 * "Thirty Two", a song by Van Morrison on the album New York Sessions '67
 * ThirtyTwo is the fourth album by English band Reverend and the Makers
 * "32 Pennies", a song on Warrant's 1989 debut album Dirty Rotten Filthy Stinking Rich
 * The number of rays in the Japanese Rising Sun on the cover of Incubus' 2006 album Light Grenades
 * "32 Ways To Die", a song on Sum41's album Half Hour of Power
 * The shortened pseudonym of UK rapper Wretch 32

In religion
In the Kabbalah, there are 32 Kabbalistic Paths of Wisdom. This is, in turn, derived from the 32 times of the Hebrew names for God, Elohim appears in the first chapter of Genesis.

One of the central texts of the Pāli Canon in the Theravada Buddhist tradition, the Digha Nikaya, describes the appearance of the historical Buddha with a list of 32 physical characteristics.

The Hindu scripture Mudgala Purana also describes Ganesha as taking 32 forms.

In sports

 * In chess, the total number of black squares on the board, the total number of white squares, and the total number of pieces (black and white) at the beginning of the game.
 * The number of teams in the National Football League.
 * The number of teams in the National Hockey League.
 * In association football:
 * The FIFA World Cup final tournament has featured 32 men's national teams from 1998 through 2022, after which the field will expand to 48.
 * The FIFA Women's World Cup final tournament will feature 32 national teams starting with the next edition in 2023.
 * The ball used in association football is most often made with 32 panels of leather or synthetic material.

In other fields
Thirty-two could also refer to:
 * The number of teeth of a full set of teeth in an adult human, including wisdom teeth
 * The size of a databus in bits: 32-bit
 * The size, in bits, of certain integer data types, used in computer representations of numbers
 * IPv4 uses 32-bit (4-byte) addresses
 * ASCII and Unicode code point for space
 * The code for international direct dial phone calls to Belgium
 * In the title Thirty-Two Short Films About Glenn Gould, starring Colm Feore
 * The Article 32 of the UCMJ concerns pre-trial investigations. Such a hearing is often called an "Article 32 hearing"
 * The caliber .32 ACP
 * The number of the French department Gers
 * The traditional 32 counties of Ireland

title: 32 (number)


text:
34 (number)

34 (thirty-four) is the natural number following 33 and preceding 35.

In mathematics
34 is the twelfth semiprime, with four divisors including 1 and itself. Specifically, 34 is the ninth distinct semiprime, it being the sixth of the form $$2 \times q$$. Its neighbors 33 and 35 are also distinct semiprimes with four divisors each, where 34 is the smallest number to be surrounded by numbers with the same number of divisors it has. This is the first distinct semiprime treble cluster, the next being (85, 86, 87).

The number 34 has an aliquot sum of 20, and is the seventh member in the aliquot sequence (34, 20, 22, 14, 10, 8, 7, 1, 0) that belongs to the prime 7-aliquot tree.

34 is the sum of the first two perfect numbers 6 + 28, whose difference is its composite index (22).

Its reduced totient and Euler totient values are both 16 (or 42 = 24). The sum of all its divisors aside from one equals 53, which is the sixteenth prime number.

There is no solution to the equation φ(x) = 34, making 34 a nontotient. Nor is there a solution to the equation x − φ(x) = 34, making 34 a noncototient.

It is the third Erdős–Woods number, following 22 and 16.

It is the ninth Fibonacci number and a companion Pell number. Since it is an odd-indexed Fibonacci number, 34 is a Markov number.

34 is also the fourth heptagonal number, and the first non-trivial centered hendecagonal (11-gonal) number.

This number is also the magic constant of $$n-$$Queens Problem for $$n=4$$.

There are 34 topologically distinct convex heptahedra, excluding mirror images.

34 is the magic constant of a $$4 \times 4$$ normal magic square, and magic octagram (see accompanying images); it is the only $$n$$ for which magic constants of these $$n \times n$$ magic figures coincide.

In science

 * The atomic number of selenium
 * Messier object M34, a magnitude 6.0 open cluster in the constellation Perseus
 * The New General Catalogue object NGC 34, a peculiar galaxy in the constellation Cetus
 * The human vertebral column is made up of up to 34 vertebrae.

Literature

 * In The Count of Monte Cristo, Number 34 is how Edmond Dantès is referred to during his imprisonment in the Château d'If.

Transportation

 * 34th Street (Manhattan), a major cross-town street in New York City
 * 34th Street station (disambiguation), multiple rail stations
 * U.S. Route 34, an east-west highway between Illinois and Colorado

In other fields
34 is also:
 * The traffic code of Istanbul, Turkey
 * "#34", a song by the Dave Matthews Band
 * The number of the French department Hérault
 * +34 is the code for international direct-dial phone calls to Spain
 * The number used to win the 2021 Daytona 500 with Michael McDowell and Front Row Motorsports. The number and team also won at Pocono Raceway with Chris Buescher and Talladega Superspeedway with David Ragan. In 1961, Wendell Scott won with the number to become the first African American to win in NASCAR.

title: 34 (number)