144-424: Shiing-Shen Chern ( / tʃ ɜːr n / ; Chinese : 陳省身 , Mandarin: [tʂʰə́n.ɕǐŋ.ʂən] ; 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
288-563: A {\displaystyle a} to object b {\displaystyle b} , and another morphism from object b {\displaystyle b} to object c {\displaystyle c} , then there must also exist one from object a {\displaystyle a} to object c {\displaystyle c} . Composition of morphisms is required to be associative, and there must be an "identity morphism" for every object. Categories are widely used in contemporary mathematics since they provide
432-403: A {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} are usually used for constants and coefficients . The expression 5 x + 3 {\displaystyle 5x+3} is an algebraic expression created by multiplying the number 5 with the variable x {\displaystyle x} and adding
576-746: A 2 x 2 + . . . + a n x n = b {\displaystyle a_{1}x_{1}+a_{2}x_{2}+...+a_{n}x_{n}=b} where a 1 {\displaystyle a_{1}} , a 2 {\displaystyle a_{2}} , ..., a n {\displaystyle a_{n}} and b {\displaystyle b} are constants. Examples are x 1 − 7 x 2 + 3 x 3 = 0 {\displaystyle x_{1}-7x_{2}+3x_{3}=0} and 1 4 x − y = 4 {\textstyle {\frac {1}{4}}x-y=4} . A system of linear equations
720-429: A ∘ a − 1 = a − 1 ∘ a = e {\displaystyle a\circ a^{-1}=a^{-1}\circ a=e} . Every algebraic structure that fulfills these requirements is a group. For example, ⟨ Z , + ⟩ {\displaystyle \langle \mathbb {Z} ,+\rangle } is a group formed by the set of integers together with
864-433: A ∘ b ) ∘ c {\displaystyle (a\circ b)\circ c} is the same as a ∘ ( b ∘ c ) {\displaystyle a\circ (b\circ c)} for all elements. An operation has an identity element or a neutral element if one element e exists that does not change the value of any other element, i.e., if a ∘ e = e ∘
1008-402: A + c a . {\displaystyle (b+c)a=ba+ca.} Moreover, multiplication is associative and has an identity element generally denoted as 1 . Multiplication needs not to be commutative; if it is commutative, one has a commutative ring . The ring of integers ( Z {\displaystyle \mathbb {Z} } ) is one of the simplest commutative rings. A field
1152-437: A = a {\displaystyle a\circ e=e\circ a=a} . An operation has inverse elements if for any element a {\displaystyle a} there exists a reciprocal element a − 1 {\displaystyle a^{-1}} that undoes a {\displaystyle a} . If an element operates on its inverse then the result is the neutral element e , expressed formally as
1296-652: A Lie algebra or an associative algebra . The word algebra comes from the Arabic term الجبر ( al-jabr ), which originally referred to the surgical treatment of bonesetting . In the 9th century, the term received a mathematical meaning when the Persian mathematician Muhammad ibn Musa al-Khwarizmi employed it to describe a method of solving equations and used it in the title of a treatise on algebra, al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah [ The Compendious Book on Calculation by Completion and Balancing ] which
1440-503: A certain extent in South Korea , remain virtually identical to traditional characters, with variations between the two forms largely stylistic. There has historically been a debate on traditional and simplified Chinese characters . Because the simplifications are fairly systematic, it is possible to convert computer-encoded characters between the two sets, with the main issue being ambiguities in simplified representations resulting from
1584-503: 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
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#17328583989041728-860: 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 . Chern's surname (陳/陈, pinyin: Chén)
1872-444: 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
2016-461: A key turning point in the history of algebra and consider what came before it as the prehistory of algebra because it lacked the abstract nature based on symbolic manipulation. In the 17th and 18th centuries, many attempts were made to find general solutions to polynomials of degree five and higher. All of them failed. At the end of the 18th century, the German mathematician Carl Friedrich Gauss proved
2160-467: A large part of linear algebra. A vector space is an algebraic structure formed by a set with an addition that makes it an abelian group and a scalar multiplication that is compatible with addition (see vector space for details). A linear map is a function between vector spaces that is compatible with addition and scalar multiplication. In the case of finite-dimensional vector spaces , vectors and linear maps can be represented by matrices. It follows that
2304-579: 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. 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
2448-569: 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
2592-711: 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 . Traditional Chinese characters Traditional Chinese characters are
2736-412: A positive degree can be factorized into linear polynomials. This theorem was proved at the beginning of the 19th century, but this does not close the problem since the theorem does not provide any way for computing the solutions. Linear algebra starts with the study systems of linear equations . An equation is linear if it can be expressed in the form a 1 x 1 +
2880-428: A second-degree polynomial equation of the form a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} is given by the quadratic formula x = − b ± b 2 − 4 a c 2 a . {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}.} Solutions for
3024-447: A similar way, if one knows the value of one variable one may be able to use it to determine the value of other variables. Algebraic equations can be interpreted geometrically to describe spatial figures in the form of a graph . To do so, the different variables in the equation are understood as coordinates and the values that solve the equation are interpreted as points of a graph. For example, if x {\displaystyle x}
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#17328583989043168-601: A standard set of Chinese character forms used to write Chinese languages . In Taiwan , the set of traditional characters is regulated by the Ministry of Education and standardized in the Standard Form of National Characters . These forms were predominant in written Chinese until the middle of the 20th century, when various countries that use Chinese characters began standardizing simplified sets of characters, often with characters that existed before as well-known variants of
3312-435: A statement formed by comparing two expressions, saying that they are equal. This can be expressed using the equals sign ( = {\displaystyle =} ), as in 5 x 2 + 6 x = 3 y + 4 {\displaystyle 5x^{2}+6x=3y+4} . Inequations involve a different type of comparison, saying that the two sides are different. This can be expressed using symbols such as
3456-527: A unifying framework to describe and analyze many fundamental mathematical concepts. For example, sets can be described with the category of sets , and any group can be regarded as the morphisms of a category with just one object. The origin of algebra lies in attempts to solve mathematical problems involving arithmetic calculations and unknown quantities. These developments happened in the ancient period in Babylonia , Egypt , Greece , China , and India . One of
3600-400: A whole is zero if and only if one of its factors is zero, i.e., if x {\displaystyle x} is either −2 or 5. Before the 19th century, much of algebra was devoted to polynomial equations , that is equations obtained by equating a polynomial to zero. The first attempts for solving polynomial equations was to express the solutions in terms of n th roots . The solution of
3744-850: Is 産 (also the accepted form in Japan and Korea), while in Hong Kong, Macau and Taiwan the accepted form is 產 (also the accepted form in Vietnamese chữ Nôm ). The PRC tends to print material intended for people in Hong Kong, Macau and Taiwan, and overseas Chinese in traditional characters. For example, versions of the People's Daily are printed in traditional characters, and both People's Daily and Xinhua have traditional character versions of their website available, using Big5 encoding. Mainland companies selling products in Hong Kong, Macau and Taiwan use traditional characters in order to communicate with consumers;
3888-397: Is a commutative group under addition: the addition of the ring is associative, commutative, and has an identity element and inverse elements. The multiplication is associative and distributive with respect to addition; that is, a ( b + c ) = a b + a c {\displaystyle a(b+c)=ab+ac} and ( b + c ) a = b
4032-691: 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" ( / tʃ ɜːr n / ). Chern was born in Xiushui, Jiaxing, China in 1911. He graduated from Xiushui Middle School ( 秀水中學 ) 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 [ zh ] . At age 15, Chern entered
4176-438: Is a commutative ring such that 1 ≠ 0 {\displaystyle 1\neq 0} and each nonzero element has a multiplicative inverse . The ring of integers does not form a field because it lacks multiplicative inverses. For example, the multiplicative inverse of 7 {\displaystyle 7} is 1 7 {\displaystyle {\tfrac {1}{7}}} , which
4320-475: Is a function from the underlying set of one algebraic structure to the underlying set of another algebraic structure that preserves certain structural characteristics. If the two algebraic structures use binary operations and have the form ⟨ A , ∘ ⟩ {\displaystyle \langle A,\circ \rangle } and ⟨ B , ⋆ ⟩ {\displaystyle \langle B,\star \rangle } then
4464-504: Is a method used to simplify polynomials, making it easier to analyze them and determine the values for which they evaluate to zero . Factorization consists in rewriting a polynomial as a product of several factors. For example, the polynomial x 2 − 3 x − 10 {\displaystyle x^{2}-3x-10} can be factorized as ( x + 2 ) ( x − 5 ) {\displaystyle (x+2)(x-5)} . The polynomial as
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4608-487: Is a polynomial with one term while two- and three-term polynomials are called binomials and trinomials. The degree of a polynomial is the maximal value (among its terms) of the sum of the exponents of the variables (4 in the above example). Polynomials of degree one are called linear polynomials . Linear algebra studies systems of linear polynomials. A polynomial is said to be univariate or multivariate , depending on whether it uses one or more variables. Factorization
4752-941: Is a set of linear equations for which one is interested in common solutions. Matrices are rectangular arrays of values that have been originally introduced for having a compact and synthetic notation for systems of linear equations For example, the system of equations 9 x 1 + 3 x 2 − 13 x 3 = 0 2.3 x 1 + 7 x 3 = 9 − 5 x 1 − 17 x 2 = − 3 {\displaystyle {\begin{aligned}9x_{1}+3x_{2}-13x_{3}&=0\\2.3x_{1}+7x_{3}&=9\\-5x_{1}-17x_{2}&=-3\end{aligned}}} can be written as A X = B , {\displaystyle AX=B,} where A , B {\displaystyle A,B} and C {\displaystyle C} are
4896-414: Is an expression consisting of one or more terms that are added or subtracted from each other, like x 4 + 3 x y 2 + 5 x 3 − 1 {\displaystyle x^{4}+3xy^{2}+5x^{3}-1} . Each term is either a constant, a variable, or a product of a constant and variables. Each variable can be raised to a positive-integer power. A monomial
5040-629: Is applied to one side of an equation also needs to be done to the other side. For example, if one subtracts 5 from the left side of an equation one also needs to subtract 5 from the right side to balance both sides. The goal of these steps is usually to isolate the variable one is interested in on one side, a process known as solving the equation for that variable. For example, the equation x − 7 = 4 {\displaystyle x-7=4} can be solved for x {\displaystyle x} by adding 7 to both sides, which isolates x {\displaystyle x} on
5184-593: Is equivalent to PhD) degree in February, 1936. He wrote his thesis in German, and it was titled Eine Invariantentheorie der Dreigewebe aus r {\displaystyle r} -dimensionalen Mannigfaltigkeiten im R 2 r {\displaystyle R_{2r}} (English: An invariant theory of 3-webs of r {\displaystyle r} -dimensional manifolds in R 2 r {\displaystyle R_{2r}} ). For his third year, Blaschke recommended Chern to study at
5328-408: Is no solution since they never intersect. If two equations are not independent then they describe the same line, meaning that every solution of one equation is also a solution of the other equation. These relations make it possible to seek solutions graphically by plotting the equations and determining where they intersect. The same principles also apply to systems of equations with more variables, with
5472-482: Is not an integer. The rational numbers , the real numbers , and the complex numbers each form a field with the operations of addition and multiplication. Ring theory is the study of rings, exploring concepts such as subrings , quotient rings , polynomial rings , and ideals as well as theorems such as Hilbert's basis theorem . Field theory is concerned with fields, examining field extensions , algebraic closures , and finite fields . Galois theory explores
5616-400: Is set to zero in the equation y = 0.5 x − 1 {\displaystyle y=0.5x-1} , then y {\displaystyle y} must be −1 for the equation to be true. This means that the ( x , y ) {\displaystyle (x,y)} -pair ( 0 , − 1 ) {\displaystyle (0,-1)} is part of
5760-403: Is the identity matrix . Then, multiplying on the left both members of the above matrix equation by A − 1 , {\displaystyle A^{-1},} one gets the solution of the system of linear equations as X = A − 1 B . {\displaystyle X=A^{-1}B.} Methods of solving systems of linear equations range from
5904-414: Is the application of group theory to analyze graphs and symmetries. The insights of algebra are also relevant to calculus, which uses mathematical expressions to examine rates of change and accumulation . It relies on algebra, for instance, to understand how these expressions can be transformed and what role variables play in them. Algebraic logic employs the methods of algebra to describe and analyze
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6048-423: Is the branch of mathematics that studies algebraic structures and the operations they use. An algebraic structure is a non-empty set of mathematical objects , such as the integers , together with algebraic operations defined on that set, like addition and multiplication . Algebra explores the laws, general characteristics, and types of algebraic structures. Within certain algebraic structures, it examines
6192-425: Is the case because the sum of two even numbers is again an even number. But the set of odd integers together with addition is not a subalgebra because it is not closed: adding two odd numbers produces an even number, which is not part of the chosen subset. Universal algebra is the study of algebraic structures in general. As part of its general perspective, it is not concerned with the specific elements that make up
6336-443: Is the main form of algebra taught in school and examines mathematical statements using variables for unspecified values. It seeks to determine for which values the statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called systems of linear equations . It provides methods to find
6480-421: Is the process of applying algebraic methods and principles to other branches of mathematics , such as geometry , topology , number theory , and calculus . It happens by employing symbols in the form of variables to express mathematical insights on a more general level, allowing mathematicians to develop formal models describing how objects interact and relate to each other. One application, found in geometry,
6624-472: Is the study of numerical operations and investigates how numbers are combined and transformed using the arithmetic operations of addition , subtraction , multiplication , division , exponentiation , extraction of roots , and logarithm . For example, the operation of addition combines two numbers, called the addends, into a third number, called the sum, as in 2 + 5 = 7 {\displaystyle 2+5=7} . Elementary algebra relies on
6768-718: Is the use of algebraic statements to describe geometric figures. For example, the equation y = 3 x − 7 {\displaystyle y=3x-7} describes a line in two-dimensional space while the equation x 2 + y 2 + z 2 = 1 {\displaystyle x^{2}+y^{2}+z^{2}=1} corresponds to a sphere in three-dimensional space. Of special interest to algebraic geometry are algebraic varieties , which are solutions to systems of polynomial equations that can be used to describe more complex geometric figures. Algebraic reasoning can also solve geometric problems. For example, one can determine whether and where
6912-466: Is true for all elements of the underlying set. For example, commutativity is a universal equation that states that a ∘ b {\displaystyle a\circ b} is identical to b ∘ a {\displaystyle b\circ a} for all elements. A variety is a class of all algebraic structures that satisfy certain identities. For example, if two algebraic structures satisfy commutativity then they are both part of
7056-446: Is true if x {\displaystyle x} is either 2 or −2 and false otherwise. Equations with variables can be divided into identity equations and conditional equations. Identity equations are true for all values that can be assigned to the variables, such as the equation 2 x + 5 x = 7 x {\displaystyle 2x+5x=7x} . Conditional equations are only true for some values. For example,
7200-483: The Chinese Commercial News , World News , and United Daily News all use traditional characters, as do some Hong Kong–based magazines such as Yazhou Zhoukan . The Philippine Chinese Daily uses simplified characters. DVDs are usually subtitled using traditional characters, influenced by media from Taiwan as well as by the two countries sharing the same DVD region , 3. With most having immigrated to
7344-587: 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
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#17328583989047488-469: The Kensiu language . Algebra Algebra is the branch of mathematics that studies certain abstract systems , known as algebraic structures , and the manipulation of statements within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations such as addition and multiplication . Elementary algebra
7632-622: The Shanghainese -language character U+20C8E 𠲎 CJK UNIFIED IDEOGRAPH-20C8E —a composition of 伐 with the ⼝ 'MOUTH' radical—used instead of the Standard Chinese 嗎 ; 吗 . Typefaces often use the initialism TC to signify the use of traditional Chinese characters, as well as SC for simplified Chinese characters . In addition, the Noto, Italy family of typefaces, for example, also provides separate fonts for
7776-526: 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
7920-664: 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
8064-547: The difference of two squares method and later in Euclid's Elements . In the 3rd century CE, Diophantus provided a detailed treatment of how to solve algebraic equations in a series of books called Arithmetica . He was the first to experiment with symbolic notation to express polynomials. Diophantus's work influenced Arab development of algebra with many of his methods reflected in the concepts and techniques used in medieval Arabic algebra. In ancient China, The Nine Chapters on
8208-525: The fundamental theorem of algebra , which describes the existence of zeros of polynomials of any degree without providing a general solution. At the beginning of the 19th century, the Italian mathematician Paolo Ruffini and the Norwegian mathematician Niels Henrik Abel were able to show that no general solution exists for polynomials of degree five and higher. In response to and shortly after their findings,
8352-602: The fundamental theorem of finite abelian groups and the Feit–Thompson theorem . The latter was a key early step in one of the most important mathematical achievements of the 20th century: the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 2004, that culminated in a complete classification of finite simple groups . A ring is an algebraic structure with two operations that work similarly to addition and multiplication of numbers and are named and generally denoted similarly. A ring
8496-461: The less-than sign ( < {\displaystyle <} ), the greater-than sign ( > {\displaystyle >} ), and the inequality sign ( ≠ {\displaystyle \neq } ). Unlike other expressions, statements can be true or false and their truth value usually depends on the values of the variables. For example, the statement x 2 = 4 {\displaystyle x^{2}=4}
8640-653: The 12th century further refined Brahmagupta's methods and concepts. In 1247, the Chinese mathematician Qin Jiushao wrote the Mathematical Treatise in Nine Sections , which includes an algorithm for the numerical evaluation of polynomials , including polynomials of higher degrees. The Italian mathematician Fibonacci brought al-Khwarizmi's ideas and techniques to Europe in books including his Liber Abaci . In 1545,
8784-426: The 16th and 17th centuries, when a rigorous symbolic formalism was developed. In the mid-19th century, the scope of algebra broadened beyond a theory of equations to cover diverse types of algebraic operations and structures. Algebra is relevant to many branches of mathematics, such as geometry, topology , number theory , and calculus , and other fields of inquiry, like logic and the empirical sciences . Algebra
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#17328583989048928-528: The 1930s, the American mathematician Garrett Birkhoff expanded these ideas and developed many of the foundational concepts of this field. The invention of universal algebra led to the emergence of various new areas focused on the algebraization of mathematics—that is, the application of algebraic methods to other branches of mathematics. Topological algebra arose in the early 20th century, studying algebraic structures such as topological groups and Lie groups . In
9072-464: The 1940s and 50s, homological algebra emerged, employing algebraic techniques to study homology . Around the same time, category theory was developed and has since played a key role in the foundations of mathematics . Other developments were the formulation of model theory and the study of free algebras . The influence of algebra is wide-reaching, both within mathematics and in its applications to other fields. The algebraization of mathematics
9216-402: The 9th century and the Persian mathematician Omar Khayyam in the 11th and 12th centuries. In India, Brahmagupta investigated how to solve quadratic equations and systems of equations with several variables in the 7th century CE. Among his innovations were the use of zero and negative numbers in algebraic equations. The Indian mathematicians Mahāvīra in the 9th century and Bhāskara II in
9360-575: The American people and the Chinese people, and... all of us shared the desire to promote more exchanges. 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
9504-646: 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
9648-659: 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
9792-466: 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
9936-572: 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
10080-481: The French mathematician Évariste Galois developed what came later to be known as Galois theory , which offered a more in-depth analysis of the solutions of polynomials while also laying the foundation of group theory . Mathematicians soon realized the relevance of group theory to other fields and applied it to disciplines like geometry and number theory. Starting in the mid-19th century, interest in algebra shifted from
10224-598: The German mathematicians David Hilbert , Ernst Steinitz , and Emmy Noether as well as the Austrian mathematician Emil Artin . They researched different forms of algebraic structures and categorized them based on their underlying axioms into types, like groups, rings, and fields. The idea of the even more general approach associated with universal algebra was conceived by the English mathematician Alfred North Whitehead in his 1898 book A Treatise on Universal Algebra . Starting in
10368-554: The IAS. Because of the war, it took him a week to reach Princeton via US military aircraft. 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
10512-567: 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. By
10656-592: The Italian polymath Gerolamo Cardano published his book Ars Magna , which covered many topics in algebra, discussed imaginary numbers , and was the first to present general methods for solving cubic and quartic equations . In the 16th and 17th centuries, the French mathematicians François Viète and René Descartes introduced letters and symbols to denote variables and operations, making it possible to express equations in an abstract and concise manner. Their predecessors had relied on verbal descriptions of problems and solutions. Some historians see this development as
10800-561: The Mathematical Art , a book composed over the period spanning from the 10th century BCE to the 2nd century CE, explored various techniques for solving algebraic equations, including the use of matrix-like constructs. There is no unanimity as to whether these early developments are part of algebra or only precursors. They offered solutions to algebraic problems but did not conceive them in an abstract and general manner, focusing instead on specific cases and applications. This changed with
10944-547: The People's Republic of China, traditional Chinese characters are standardised according to the Table of Comparison between Standard, Traditional and Variant Chinese Characters . Dictionaries published in mainland China generally show both simplified and their traditional counterparts. There are differences between the accepted traditional forms in mainland China and elsewhere, for example the accepted traditional form of 产 in mainland China
11088-454: The Persian mathematician al-Khwarizmi , who published his The Compendious Book on Calculation by Completion and Balancing in 825 CE. It presents the first detailed treatment of general methods that can be used to manipulate linear and quadratic equations by "reducing" and "balancing" both sides. Other influential contributions to algebra came from the Arab mathematician Thābit ibn Qurra also in
11232-577: 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 [ zh ] , 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
11376-465: The United States during the second half of the 19th century, Chinese Americans have long used traditional characters. When not providing both, US public notices and signs in Chinese are generally written in traditional characters, more often than in simplified characters. In the past, traditional Chinese was most often encoded on computers using the Big5 standard, which favored traditional characters. However,
11520-401: The addition of numbers. While elementary algebra and linear algebra work within the confines of particular algebraic structures, abstract algebra takes a more general approach that compares how algebraic structures differ from each other and what types of algebraic structures there are, such as groups , rings , and fields . The key difference between these types of algebraic structures lies in
11664-591: 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. In the summer of 1937, Chern accepted the invitation of Tsinghua University and returned to China. He
11808-443: The characteristics of algebraic structures in general. The term "algebra" is sometimes used in a more narrow sense to refer only to elementary algebra or only to abstract algebra. When used as a countable noun , an algebra is a specific type of algebraic structure that involves a vector space equipped with a certain type of binary operation . Depending on the context, "algebra" can also refer to other algebraic structures, like
11952-416: The corresponding variety. Category theory examines how mathematical objects are related to each other using the concept of categories . A category is a collection of objects together with a collection of so-called morphisms or "arrows" between those objects. These two collections must satisfy certain conditions. For example, morphisms can be joined, or composed : if there exists a morphism from object
12096-430: 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
12240-456: 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.” Chern returned to Shanghai in 1945 to help found
12384-593: The degrees 3 and 4 are given by the cubic and quartic formulas. There are no general solutions for higher degrees, as proven in the 19th century by the so-called Abel–Ruffini theorem . Even when general solutions do not exist, approximate solutions can be found by numerical tools like the Newton–Raphson method . The fundamental theorem of algebra asserts that every univariate polynomial equation of positive degree with real or complex coefficients has at least one complex solution. Consequently, every polynomial of
12528-455: The difference being that the equations do not describe lines but higher dimensional figures. For instance, equations with three variables correspond to planes in three-dimensional space , and the points where all planes intersect solve the system of equations. Abstract algebra, also called modern algebra, is the study of algebraic structures . An algebraic structure is a framework for understanding operations on mathematical objects , like
12672-469: The distributive property. For statements with several variables, substitution is a common technique to replace one variable with an equivalent expression that does not use this variable. For example, if one knows that y = 3 x {\displaystyle y=3x} then one can simplify the expression 7 x y {\displaystyle 7xy} to arrive at 21 x 2 {\displaystyle 21x^{2}} . In
12816-496: The earliest documents on algebraic problems is the Rhind Mathematical Papyrus from ancient Egypt, which was written around 1650 BCE. It discusses solutions to linear equations , as expressed in problems like "A quantity; its fourth is added to it. It becomes fifteen. What is the quantity?" Babylonian clay tablets from around the same time explain methods to solve linear and quadratic polynomial equations , such as
12960-403: The elements of the two algebraic structures. This implies that every element of the first algebraic structure is mapped to one unique element in the second structure without any unmapped elements in the second structure. Another tool of comparison is the relation between an algebraic structure and its subalgebra . The algebraic structure and its subalgebra use the same operations, which follow
13104-429: 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
13248-404: The equation x + 4 = 9 {\displaystyle x+4=9} is only true if x {\displaystyle x} is 5. The main goal of elementary algebra is to determine the values for which a statement is true. This can be achieved by transforming and manipulating statements according to certain rules. A key principle guiding this process is that whatever operation
13392-612: The existence of loops or holes in them. Number theory is concerned with the properties of and relations between integers. Algebraic number theory applies algebraic methods and principles to this field of inquiry. Examples are the use of algebraic expressions to describe general laws, like Fermat's Last Theorem , and of algebraic structures to analyze the behavior of numbers, such as the ring of integers . The related field of combinatorics uses algebraic techniques to solve problems related to counting, arrangement, and combination of discrete objects. An example in algebraic combinatorics
13536-537: The field, but he remained modest about his achievements, preferring to say that he is a man of 'small problems' rather than 'big views.' 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
13680-429: The form of variables in addition to numbers. A higher level of abstraction is found in abstract algebra , which is not limited to a particular domain and examines algebraic structures such as groups and rings . It extends beyond typical arithmetic operations by also covering other types of operations. Universal algebra is still more abstract in that it is not interested in specific algebraic structures but investigates
13824-438: The function h : A → B {\displaystyle h:A\to B} is a homomorphism if it fulfills the following requirement: h ( x ∘ y ) = h ( x ) ⋆ h ( y ) {\displaystyle h(x\circ y)=h(x)\star h(y)} . The existence of a homomorphism reveals that the operation ⋆ {\displaystyle \star } in
13968-413: The graph of the equation. The ( x , y ) {\displaystyle (x,y)} -pair ( 0 , 7 ) {\displaystyle (0,7)} , by contrast, does not solve the equation and is therefore not part of the graph. The graph encompasses the totality of ( x , y ) {\displaystyle (x,y)} -pairs that solve the equation. A polynomial
14112-414: 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
14256-566: 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
14400-495: The introductory, like substitution and elimination, to more advanced techniques using matrices, such as Cramer's rule , the Gaussian elimination , and LU decomposition . Some systems of equations are inconsistent , meaning that no solutions exist because the equations contradict each other. Consistent systems have either one unique solution or an infinite number of solutions. The study of vector spaces and linear maps form
14544-493: The inverse is equally true as well. In digital media, many cultural phenomena imported from Hong Kong and Taiwan into mainland China, such as music videos, karaoke videos, subtitled movies, and subtitled dramas, use traditional Chinese characters. In Hong Kong and Macau , traditional characters were retained during the colonial period, while the mainland adopted simplified characters. Simplified characters are contemporaneously used to accommodate immigrants and tourists, often from
14688-607: The left side and results in the equation x = 11 {\displaystyle x=11} . There are many other techniques used to solve equations. Simplification is employed to replace a complicated expression with an equivalent simpler one. For example, the expression 7 x − 3 x {\displaystyle 7x-3x} can be replaced with the expression 4 x {\displaystyle 4x} since 7 x − 3 x = ( 7 − 3 ) x = 4 x {\displaystyle 7x-3x=(7-3)x=4x} by
14832-620: The line described by y = x + 1 {\displaystyle y=x+1} intersects with the circle described by x 2 + y 2 = 25 {\displaystyle x^{2}+y^{2}=25} by solving the system of equations made up of these two equations. Topology studies the properties of geometric figures or topological spaces that are preserved under operations of continuous deformation . Algebraic topology relies on algebraic theories such as group theory to classify topological spaces. For example, homotopy groups classify topological spaces based on
14976-426: The linear map to the basis vectors. Systems of equations can be interpreted as geometric figures. For systems with two variables, each equation represents a line in two-dimensional space . The point where the two lines intersect is the solution of the full system because this is the only point that solves both the first and the second equation. For inconsistent systems, the two lines run parallel, meaning that there
15120-472: The lowercase letters x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} represent variables. In some cases, subscripts are added to distinguish variables, as in x 1 {\displaystyle x_{1}} , x 2 {\displaystyle x_{2}} , and x 3 {\displaystyle x_{3}} . The lowercase letters
15264-725: The mainland. The increasing use of simplified characters has led to concern among residents regarding protecting what they see as their local heritage. Taiwan has never adopted simplified characters. The use of simplified characters in government documents and educational settings is discouraged by the government of Taiwan. Nevertheless, with sufficient context simplified characters are likely to be successfully read by those used to traditional characters, especially given some previous exposure. Many simplified characters were previously variants that had long been in some use, with systematic stroke simplifications used in folk handwriting since antiquity. Traditional characters were recognized as
15408-682: The majority of Chinese text in mainland China are simplified characters , there is no legislation prohibiting the use of traditional Chinese characters, and often traditional Chinese characters remain in use for stylistic and commercial purposes, such as in shopfront displays and advertising. Traditional Chinese characters remain ubiquitous on buildings that predate the promulgation of the current simplification scheme, such as former government buildings, religious buildings, educational institutions, and historical monuments. Traditional Chinese characters continue to be used for ceremonial, cultural, scholarly/academic research, and artistic/decorative purposes. In
15552-647: The matrices A = [ 9 3 − 13 2.3 0 7 − 5 − 17 0 ] , X = [ x 1 x 2 x 3 ] , B = [ 0 9 − 3 ] . {\displaystyle A={\begin{bmatrix}9&3&-13\\2.3&0&7\\-5&-17&0\end{bmatrix}},\quad X={\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}},\quad B={\begin{bmatrix}0\\9\\-3\end{bmatrix}}.} Under some conditions on
15696-975: The merging of previously distinct character forms. Many Chinese online newspapers allow users to switch between these character sets. Traditional characters are known by different names throughout the Chinese-speaking world. The government of Taiwan officially refers to traditional Chinese characters as 正體字 ; 正体字 ; zhèngtǐzì ; 'orthodox characters'. This term is also used outside Taiwan to distinguish standard characters, including both simplified, and traditional, from other variants and idiomatic characters . Users of traditional characters elsewhere, as well as those using simplified characters, call traditional characters 繁體字 ; 繁体字 ; fántǐzì ; 'complex characters', 老字 ; lǎozì ; 'old characters', or 全體字 ; 全体字 ; quántǐzì ; 'full characters' to distinguish them from simplified characters. Some argue that since traditional characters are often
15840-475: The method of completing the square . Many of these insights found their way to the ancient Greeks. Starting in the 6th century BCE, their main interest was geometry rather than algebra, but they employed algebraic methods to solve geometric problems. For example, they studied geometric figures while taking their lengths and areas as unknown quantities to be determined, as exemplified in Pythagoras ' formulation of
15984-399: The number 3 to the result. Other examples of algebraic expressions are 32 x y z {\displaystyle 32xyz} and 64 x 1 2 + 7 x 2 − c {\displaystyle 64x_{1}^{2}+7x_{2}-c} . Some algebraic expressions take the form of statements that relate two expressions to one another. An equation is
16128-539: The number of operations they use and the laws they follow . Universal algebra and category theory provide general frameworks to investigate abstract patterns that characterize different classes of algebraic structures. Algebraic methods were first studied in the ancient period to solve specific problems in fields like geometry . Subsequent mathematicians examined general techniques to solve equations independent of their specific applications. They described equations and their solutions using words and abbreviations until
16272-470: The number of operations they use and the laws they obey. In mathematics education , abstract algebra refers to an advanced undergraduate course that mathematics majors take after completing courses in linear algebra. On a formal level, an algebraic structure is a set of mathematical objects, called the underlying set, together with one or several operations. Abstract algebra is primarily interested in binary operations , which take any two objects from
16416-511: The number of rows and columns, matrices can be added , multiplied , and sometimes inverted . All methods for solving linear systems may be expressed as matrix manipulations using these operations. For example, solving the above system consists of computing an inverted matrix A − 1 {\displaystyle A^{-1}} such that A − 1 A = I , {\displaystyle A^{-1}A=I,} where I {\displaystyle I}
16560-436: The numbers with variables, it is possible to express a general law that applies to any possible combination of numbers, like the commutative property of multiplication , which is expressed in the equation a × b = b × a {\displaystyle a\times b=b\times a} . Algebraic expressions are formed by using arithmetic operations to combine variables and numbers. By convention,
16704-665: The official script in Singapore until 1969, when the government officially adopted Simplified characters. Traditional characters still are widely used in contexts such as in baby and corporation names, advertisements, decorations, official documents and in newspapers. The Chinese Filipino community continues to be one of the most conservative in Southeast Asia regarding simplification. Although major public universities teach in simplified characters, many well-established Chinese schools still use traditional characters. Publications such as
16848-425: The operation of addition. The neutral element is 0 and the inverse element of any number a {\displaystyle a} is − a {\displaystyle -a} . The natural numbers with addition, by contrast, do not form a group since they contain only positive integers and therefore lack inverse elements. Group theory examines the nature of groups, with basic theorems such as
16992-432: The operations are not restricted to regular arithmetic operations. For instance, the underlying set of the symmetry group of a geometric object is made up of geometric transformations , such as rotations , under which the object remains unchanged . Its binary operation is function composition , which takes two transformations as input and has the transformation resulting from applying the first transformation followed by
17136-700: The original standard forms, they should not be called 'complex'. Conversely, there is a common objection to the description of traditional characters as 'standard', due to them not being used by a large population of Chinese speakers. Additionally, as the process of Chinese character creation often made many characters more elaborate over time, there is sometimes a hesitation to characterize them as 'traditional'. Some people refer to traditional characters as 'proper characters' ( 正字 ; zhèngzì or 正寫 ; zhèngxiě ) and to simplified characters as 簡筆字 ; 简笔字 ; jiǎnbǐzì ; 'simplified-stroke characters' or 減筆字 ; 减笔字 ; jiǎnbǐzì ; 'reduced-stroke characters', as
17280-825: The predominant forms. Simplified characters as codified by the People's Republic of China are predominantly used in mainland China , Malaysia, and Singapore. "Traditional" as such is a retronym applied to non-simplified character sets in the wake of widespread use of simplified characters. Traditional characters are commonly used in Taiwan , Hong Kong , and Macau , as well as in most overseas Chinese communities outside of Southeast Asia. As for non-Chinese languages written using Chinese characters, Japanese kanji include many simplified characters known as shinjitai standardized after World War II, sometimes distinct from their simplified Chinese counterparts . Korean hanja , still used to
17424-449: The relation between field theory and group theory, relying on the fundamental theorem of Galois theory . Besides groups, rings, and fields, there are many other algebraic structures studied by algebra. They include magmas , semigroups , monoids , abelian groups , commutative rings , modules , lattices , vector spaces , algebras over a field , and associative and non-associative algebras . They differ from each other in regard to
17568-462: 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
17712-430: The same axioms. The only difference is that the underlying set of the subalgebra is a subset of the underlying set of the algebraic structure. All operations in the subalgebra are required to be closed in its underlying set, meaning that they only produce elements that belong to this set. For example, the set of even integers together with addition is a subalgebra of the full set of integers together with addition. This
17856-543: The same operations while allowing variables in addition to regular numbers. Variables are symbols for unspecified or unknown quantities. They make it possible to state relationships for which one does not know the exact values and to express general laws that are true, independent of which numbers are used. For example, the equation 2 × 3 = 3 × 2 {\displaystyle 2\times 3=3\times 2} belongs to arithmetic and expresses an equality only for these specific numbers. By replacing
18000-401: The second algebraic structure plays the same role as the operation ∘ {\displaystyle \circ } does in the first algebraic structure. Isomorphisms are a special type of homomorphism that indicates a high degree of similarity between two algebraic structures. An isomorphism is a bijective homomorphism, meaning that it establishes a one-to-one relationship between
18144-442: The second as its output. Abstract algebra classifies algebraic structures based on the laws or axioms that its operations obey and the number of operations it uses. One of the most basic types is a group, which has one operation and requires that this operation is associative and has an identity element and inverse elements . An operation is associative if the order of several applications does not matter, i.e., if (
18288-410: The study of diverse types of algebraic operations and structures together with their underlying axioms , the laws they follow. Elementary algebra, also called school algebra, college algebra, and classical algebra, is the oldest and most basic form of algebra. It is a generalization of arithmetic that relies on variables and examines how mathematical statements may be transformed. Arithmetic
18432-485: The study of polynomials associated with elementary algebra towards a more general inquiry into algebraic structures, marking the emergence of abstract algebra . This approach explored the axiomatic basis of arbitrary algebraic operations. The invention of new algebraic systems based on different operations and elements accompanied this development, such as Boolean algebra , vector algebra , and matrix algebra . Influential early developments in abstract algebra were made by
18576-406: The theories of matrices and finite-dimensional vector spaces are essentially the same. In particular, vector spaces provide a third way for expressing and manipulating systems of linear equations. From this perspective, a matrix is a representation of a linear map: if one chooses a particular basis to describe the vectors being transformed, then the entries in the matrix give the results of applying
18720-627: The traditional character set used in Taiwan ( TC ) and the set used in Hong Kong ( HK ). Most Chinese-language webpages now use Unicode for their text. The World Wide Web Consortium (W3C) recommends the use of the language tag zh-Hant to specify webpage content written with traditional characters. In the Japanese writing system , kyujitai are traditional forms, which were simplified to create shinjitai for standardized Japanese use following World War II. Kyūjitai are mostly congruent with
18864-970: The traditional characters in Chinese, save for minor stylistic variation. Characters that are not included in the jōyō kanji list are generally recommended to be printed in their traditional forms, with a few exceptions. Additionally, there are kokuji , which are kanji wholly created in Japan, rather than originally being borrowed from China. In the Korean writing system , hanja —replaced almost entirely by hangul in South Korea and totally replaced in North Korea —are mostly identical with their traditional counterparts, save minor stylistic variations. As with Japanese, there are autochthonous hanja, known as gukja . Traditional Chinese characters are also used by non-Chinese ethnic groups. The Maniq people living in Thailand and Malaysia use Chinese characters to write
19008-635: 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
19152-413: The types of objects they describe and the requirements that their operations fulfill. Many are related to each other in that a basic structure can be turned into a more advanced structure by adding additional requirements. For example, a magma becomes a semigroup if its operation is associative. Homomorphisms are tools to examine structural features by comparing two algebraic structures. A homomorphism
19296-509: The ubiquitous Unicode standard gives equal weight to simplified and traditional Chinese characters, and has become by far the most popular encoding for Chinese-language text. There are various input method editors (IMEs) available for the input of Chinese characters . Many characters, often dialectical variants, are encoded in Unicode but cannot be inputted using certain IMEs, with one example being
19440-510: The underlying set as inputs and map them to another object from this set as output. For example, the algebraic structure ⟨ N , + ⟩ {\displaystyle \langle \mathbb {N} ,+\rangle } has the natural numbers ( N {\displaystyle \mathbb {N} } ) as the underlying set and addition ( + {\displaystyle +} ) as its binary operation. The underlying set can contain mathematical objects other than numbers and
19584-400: The underlying sets and considers operations with more than two inputs, such as ternary operations . It provides a framework for investigating what structural features different algebraic structures have in common. One of those structural features concerns the identities that are true in different algebraic structures. In this context, an identity is a universal equation or an equation that
19728-516: The use of variables in equations and how to manipulate these equations. Algebra is often understood as a generalization of arithmetic . Arithmetic studies operations like addition, subtraction , multiplication, and division , in a particular domain of numbers, such as the real numbers. Elementary algebra constitutes the first level of abstraction. Like arithmetic, it restricts itself to specific types of numbers and operations. It generalizes these operations by allowing indefinite quantities in
19872-533: The values that solve all equations in the system at the same time, and to study the set of these solutions. Abstract algebra studies algebraic structures, which consist of a set of mathematical objects together with one or several operations defined on that set. It is a generalization of elementary and linear algebra, since it allows mathematical objects other than numbers and non-arithmetic operations. It distinguishes between different types of algebraic structures, such as groups , rings , and fields , based on
20016-571: The words for simplified and reduced are homophonous in Standard Chinese , both pronounced as jiǎn . The modern shapes of traditional Chinese characters first appeared with the emergence of the clerical script during the Han dynasty c. 200 BCE , with the sets of forms and norms more or less stable since the Southern and Northern dynasties period c. the 5th century . Although
20160-511: 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
20304-669: 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 . 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
20448-462: 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
20592-468: 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
20736-403: Was translated into Latin as Liber Algebrae et Almucabola . The word entered the English language in the 16th century from Italian , Spanish , and medieval Latin . Initially, its meaning was restricted to the theory of equations , that is, to the art of manipulating polynomial equations in view of solving them. This changed in the 19th century when the scope of algebra broadened to cover
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