Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
and the William Caspar Graustein Professor of Mathematics at
Harvard University
Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of high ...
. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathematics at
Tsinghua University
Tsinghua University (; abbr. THU) is a national public research university in Beijing, China. The university is funded by the Ministry of Education.
The university is a member of the C9 League, Double First Class University Plan, Projec ...
.
Yau was born in Shantou, China, moved to Hong Kong at a young age, and to the United States in 1969. He was awarded the
Fields Medal in 1982, in recognition of his contributions to
partial differential equations, the
Calabi conjecture
In the mathematical field of differential geometry, the Calabi conjecture was a conjecture about the existence of certain kinds of Riemannian metrics on certain complex manifolds, made by . It was proved by , who received the Fields Medal and Oswal ...
, the
positive energy theorem
The positive energy theorem (also known as the positive mass theorem) refers to a collection of foundational results in general relativity and differential geometry. Its standard form, broadly speaking, asserts that the gravitational energy of a ...
, and the
Monge–Ampère equation
In mathematics, a (real) Monge–Ampère equation is a nonlinear second-order partial differential equation of special kind. A second-order equation for the unknown function ''u'' of two variables ''x'',''y'' is of Monge–Ampère type if it is li ...
. Yau is considered one of the major contributors to the development of modern
differential geometry and
geometric analysis
Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of ...
.
The impact of Yau's work can be seen in the mathematical and physical fields of differential geometry,
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
,
convex geometry
In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of num ...
,
algebraic geometry,
enumerative geometry
In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory.
History
The problem of Apollonius is one of the earliest examp ...
,
mirror symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.
In 2D ther ...
,
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, and
string theory, while his work has also touched upon
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
,
engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...
, and
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...
.
Biography
Yau was born in Shantou, China in 1949 to Chinese Hakka parents. Professor Yau's ancestral hometown is Jiaoling county, China. His mother, Yeuk Lam Leung, was from
Meixian District
Meixian (, Hakka: Moiyen), formerly Meihsien, is a district of Meizhou City, in northeastern Guangdong Province, China. The county is an important Hakka settlement and is the ancestral home of many Hakka descendants living in Taiwan.
History ...
; his father, Chen Ying Chiu, was a Chinese scholar of philosophy, history, literature, and economics. He was the fifth of eight children, with
Hakka
The Hakka (), sometimes also referred to as Hakka Han, or Hakka Chinese, or Hakkas are a Han Chinese subgroup whose ancestral homes are chiefly in the Hakka-speaking provincial areas of Guangdong, Fujian, Jiangxi, Guangxi, Sichuan, Hunan, Zhej ...
ancestry.
During the Communist takeover of mainland China, when he was only a few months old, his family moved to
Hong Kong
Hong Kong ( (US) or (UK); , ), officially the Hong Kong Special Administrative Region of the People's Republic of China (abbr. Hong Kong SAR or HKSAR), is a city and special administrative region of China on the eastern Pearl River Delta i ...
where he was forced to learn to speak the Cantonese language as well as speak Chinese Hakka. He was not able to revisit until 1979, at the invitation of
Hua Luogeng
Hua Luogeng or Hua Loo-Keng (; 12 November 1910 – 12 June 1985) was a Chinese mathematician and politician famous for his important contributions to number theory and for his role as the leader of mathematics research and education in the Peop ...
, when mainland China entered the
reform and opening era.. They had financial troubles from having lost all of their possessions, and his father and second-oldest sister died when he was thirteen. Yau began to read and appreciate his father's books, and became more devoted to schoolwork. After graduating from
Pui Ching Middle School, he studied mathematics at the
Chinese University of Hong Kong
The Chinese University of Hong Kong (CUHK) is a public research university in Ma Liu Shui, Hong Kong, formally established in 1963 by a charter granted by the Legislative Council of Hong Kong. It is the territory's second-oldest university and ...
from 1966 to 1969, without receiving a degree due to graduating early. He left his textbooks with his younger brother,
Stephen Shing-Toung Yau
Stephen Shing-Toung Yau (; born 1952) is a Chinese-American mathematician. He is a Distinguished Professor Emeritus at the University of Illinois at Chicago, and currently teaches at Tsinghua University. He is a Fellow of the Institute of Electri ...
, who then decided to major in mathematics as well.
Yau left for the Ph.D. program in mathematics at
University of California, Berkeley
The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...
in the fall of 1969. Over the winter break, he read the first issues of the
Journal of Differential Geometry
The ''Journal of Differential Geometry'' is a peer-reviewed scientific journal of mathematics published by International Press on behalf of Lehigh University in 3 volumes of 3 issues each per year. The journal publishes an annual supplement in b ...
, and was deeply inspired by
John Milnor
John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Univ ...
's papers on
geometric group theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such group (mathematics), groups and topology, topological and geometry, geometric pro ...
. Subsequently he formulated a generalization of
Preissman's theorem
In Riemannian geometry, a field of mathematics, Preissmann's theorem is a statement that restricts the possible topology of a sectional curvature, negatively curved compact space, compact Riemannian manifold. It is named for Alexandre Preissmann, w ...
, and developed his ideas further with
Blaine Lawson
Herbert Blaine Lawson, Jr. is a mathematician best known for his work in minimal surfaces, calibrated geometry, and algebraic cycles. He is currently a Distinguished Professor of Mathematics at Stony Brook University. He received his PhD fr ...
over the next semester.
[Page at ''Center of Mathematical Sciences at Zhejiang University''](_blank)
/ref> Using this work, he received his Ph.D. the following year, in 1971, under the supervision of 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 geome ...
.[Shing-Tung Yau]
Mathematics Genealogy.
He spent a year as a member of the Institute for Advanced Study
The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent schola ...
at Princeton
Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ni ...
before joining Stony Brook University in 1972 as an assistant professor. In 1974, he became an associate professor at Stanford University. In 1976 he took a visiting faculty position with UCLA and married physicist Yu-Yun Kuo, who he knew from his time as a graduate student at Berkeley. From 1984 to 1987 he worked at University of California, San Diego
The University of California, San Diego (UC San Diego or colloquially, UCSD) is a public land-grant research university in San Diego, California. Established in 1960 near the pre-existing Scripps Institution of Oceanography, UC San Diego is t ...
. Since 1987, he has been at Harvard University
Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of high ...
. In April 2022, Yau announced a forthcoming move from Harvard to Tsinghua University
Tsinghua University (; abbr. THU) is a national public research university in Beijing, China. The university is funded by the Ministry of Education.
The university is a member of the C9 League, Double First Class University Plan, Projec ...
.
In 1978, Yau became "stateless" after the British Consulate revoked his Hong Kong residency due to his United States permanent residency status.["Stephen Hawking invited me to discuss he proofwith him at Cambridge University in late August 1978. I gladly accepted.... Travel was difficult, however, because the British Consulate had recently taken my Hong Kong resident card, maintaining that I could not keep it now that I had a U.S. green card. In the process, I had become stateless. I was no longer a citizen of any country.... until I became a U.S. citizen in 1990."] Regarding his status when receiving his Fields Medal in 1982, Yau stated "I am proud to say that when I was awarded the Fields Medal in mathematics, I held no passport of any country and should certainly be considered Chinese." Yau remained "stateless" until 1990, when he obtained United States citizenship.
With science journalist
Science journalism conveys reporting about science to the public. The field typically involves interactions between scientists, journalists, and the public.
Origins
Modern science journalism dates back to '' Digdarshan'' (means showing the d ...
Steve Nadis, Yau has written a non-technical account of Calabi-Yau manifolds and string theory, a history of Harvard's mathematics department, and an autobiography.
Academic activities
Yau has made major contributions to the development of modern differential geometry and geometric analysis
Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of ...
. As said by William Thurston
William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds.
Thursto ...
in 1981:
His most widely celebrated results include the resolution (with Shiu-Yuen Cheng
Shiu-Yuen Cheng (鄭紹遠) is a Hong Kong mathematician. He is currently the Chair Professor of Mathematics at the Hong Kong University of Science and Technology. Cheng received his Ph.D. in 1974, under the supervision of Shiing-Shen Chern, from ...
) of the boundary-value problem for the Monge-Ampère equation, the positive mass theorem
The positive energy theorem (also known as the positive mass theorem) refers to a collection of foundational results in general relativity and differential geometry. Its standard form, broadly speaking, asserts that the gravitational energy of an ...
in the mathematical analysis of general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
(achieved with Richard Schoen
Richard Melvin Schoen (born October 23, 1950) is an American mathematician known for his work in differential geometry and geometric analysis. He is best known for the resolution of the Yamabe problem in 1984.
Career
Born in Celina, Ohio, and a ...
), the resolution of the Calabi conjecture
In the mathematical field of differential geometry, the Calabi conjecture was a conjecture about the existence of certain kinds of Riemannian metrics on certain complex manifolds, made by . It was proved by , who received the Fields Medal and Oswal ...
, the topological
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
theory of minimal surfaces
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
(with William Meeks), the Donaldson-Uhlenbeck-Yau theorem (done with Karen Uhlenbeck
Karen Keskulla Uhlenbeck (born August 24, 1942) is an American mathematician and one of the founders of modern geometric analysis. She is a professor emeritus of mathematics at the University of Texas at Austin, where she held the Sid W. Richard ...
), and the Cheng−Yau and Li−Yau gradient estimates for partial differential equations (found with Shiu-Yuen Cheng and Peter Li). Many of Yau's results (in addition to those of others) were written into textbooks co-authored with Schoen.
In addition to his research, Yau is the founder and director of several mathematical institutes, mostly in China. John Coates has commented that "no other mathematician of our times has come close" to Yau's success at fundraising for mathematical activities in China and Hong Kong. During a sabbatical year at National Tsinghua University in Taiwan
Taiwan, officially the Republic of China (ROC), is a country in East Asia, at the junction of the East and South China Seas in the northwestern Pacific Ocean, with the People's Republic of China (PRC) to the northwest, Japan to the nort ...
, Yau was asked by Charles Kao
Sir Charles Kao Kuen [Charles K. Kao was elected in 1990](_blank)
as a memb ...
to start a mathematical institute at the Chinese University of Hong Kong
The Chinese University of Hong Kong (CUHK) is a public research university in Ma Liu Shui, Hong Kong, formally established in 1963 by a charter granted by the Legislative Council of Hong Kong. It is the territory's second-oldest university and ...
. After a few years of fundraising efforts, Yau established the multi-disciplinary Institute of Mathematical Sciences in 1993, with his frequent co-author Shiu-Yuen Cheng
Shiu-Yuen Cheng (鄭紹遠) is a Hong Kong mathematician. He is currently the Chair Professor of Mathematics at the Hong Kong University of Science and Technology. Cheng received his Ph.D. in 1974, under the supervision of Shiing-Shen Chern, from ...
as associate director. In 1995, Yau assisted Yongxiang Lu with raising money from Ronnie Chan
Ronnie Chan Chi-chung (; born 1949) is a Hong Kong businessman.
Education
Chan earned bachelor's and master's degrees in biology from a California State University. He received an MBA from the University of Southern California in 1976.
Chan ...
and Gerald Chan
Gerald L. Chan (; born 1950/1951) is an American billionaire and the brother of fellow billionaire Ronnie Chan. They run the Hang Lung Group.
Early life and education
Gerald Chan is the son of T.H. Chan.
He received a bachelor of science and ...
's Morningside Group for the new Morningside Center of Mathematics at the Chinese Academy of Sciences
The Chinese Academy of Sciences (CAS); ), known by Academia Sinica in English until the 1980s, is the national academy of the People's Republic of China for natural sciences. It has historical origins in the Academia Sinica during the Republi ...
. Yau has also been involved with the Center of Mathematical Sciences at Zhejiang University
Zhejiang University, abbreviated as ZJU or Zheda and formerly romanized as Chekiang University, is a national public research university based in Hangzhou, Zhejiang, China. It is a member of the prestigious C9 League and is selected into the n ...
, at Tsinghua University
Tsinghua University (; abbr. THU) is a national public research university in Beijing, China. The university is funded by the Ministry of Education.
The university is a member of the C9 League, Double First Class University Plan, Projec ...
, at National Taiwan University
National Taiwan University (NTU; ) is a public research university in Taipei, Taiwan.
The university was founded in 1928 during Japanese rule as the seventh of the Imperial Universities. It was named Taihoku Imperial University and served d ...
, and in Sanya
Sanya (; also spelled Samah) is the southernmost city on Hainan Island, and one of the four prefecture-level cities of Hainan Province in South China.
According to the 2020 census, the total population of Sanya was 1,031,396 inhabitants, li ...
. More recently, in 2014, Yau raised money to establish the Center of Mathematical Sciences and Applications (of which he is the director), the Center for Green Buildings and Cities, and the Center for Immunological Research, all at Harvard University.
Modeled on an earlier physics conference organized by Tsung-Dao Lee
Tsung-Dao Lee (; born November 24, 1926) is a Chinese-American physicist, known for his work on parity violation, the Lee–Yang theorem, particle physics, relativistic heavy ion (RHIC) physics, nontopological solitons, and soliton star ...
and Chen-Ning Yang
Yang Chen-Ning or Chen-Ning Yang (; born 1 October 1922), also known as C. N. Yang or by the English name Frank Yang, is a Chinese theoretical physicist who made significant contributions to statistical mechanics, integrable systems, gauge t ...
, Yau proposed the International Congress of Chinese Mathematicians, which is now held every three years. The first congress was held at the Morningside Center from December 12 to 18, 1998. He co-organizes the annual "Journal of Differential Geometry" and "Current Developments in Mathematics" conferences. Yau is an editor-in-chief of the Journal of Differential Geometry
The ''Journal of Differential Geometry'' is a peer-reviewed scientific journal of mathematics published by International Press on behalf of Lehigh University in 3 volumes of 3 issues each per year. The journal publishes an annual supplement in b ...
, Asian Journal of Mathematics, and Advances in Theoretical and Mathematical Physics
'' Advances in Theoretical and Mathematical Physics'' ''(ATMP)'' is a peer-reviewed, mathematics journal, published by International Press.
Established in 1997, the journal publishes articles on theoretical physics and mathematics.
The current ...
. As of 2021, he has advised over seventy Ph.D. students.
In Hong Kong, with the support of Ronnie Chan
Ronnie Chan Chi-chung (; born 1949) is a Hong Kong businessman.
Education
Chan earned bachelor's and master's degrees in biology from a California State University. He received an MBA from the University of Southern California in 1976.
Chan ...
, Yau set up the Hang Lung Award for high school students. He has also organized and participated in meetings for high school and college students, such as the panel discussions ''Why Math? Ask Masters!'' in Hangzhou
Hangzhou ( or , ; , , Standard Mandarin pronunciation: ), also romanized as Hangchow, is the capital and most populous city of Zhejiang, China. It is located in the northwestern part of the province, sitting at the head of Hangzhou Bay, whic ...
, July 2004, and ''The Wonder of Mathematics'' in Hong Kong, December 2004. Yau also co-initiated a series of books on popular mathematics, "Mathematics and Mathematical People".
In 2002 and 2003, Grigori Perelman
Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...
posted preprints to the arXiv
arXiv (pronounced "archive"—the X represents the Greek letter chi ⟨χ⟩) is an open-access repository of electronic preprints and postprints (known as e-prints) approved for posting after moderation, but not peer review. It consists of ...
claiming to prove the Thurston geometrization conjecture
In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimens ...
and, as a special case, the renowned Poincaré conjecture
In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.
Originally conjectured ...
. Although his work contained many new ideas and results, his proofs lacked detail on a number of technical arguments. Over the next few years, several mathematicians devoted their time to fill in details and provide expositions of Perelman's work to the mathematical community. A well-known August 2006 article in the ''New Yorker
New Yorker or ''variant'' primarily refers to:
* A resident of the State of New York
** Demographics of New York (state)
* A resident of New York City
** List of people from New York City
* ''The New Yorker'', a magazine founded in 1925
* '' The ...
'' written by Sylvia Nasar and David Gruber about the situation brought some professional disputes involving Yau to public attention.
* Alexander Givental
Alexander Givental (russian: Александр Борисович Гивенталь) is a Russian-American mathematician working in symplectic topology and singularity theory, as well as their relation to topological string theories. He graduat ...
alleged that Bong Lian, Kefeng Liu
Kefeng Liu ( Chinese: 刘克峰; born 12 December 1965), is a Chinese-American mathematician who is known for his contributions to geometric analysis, particularly the geometry, topology and analysis of moduli spaces of Riemann surfaces and Cala ...
, and Yau illegitimately took credit from him for resolving a well-known conjecture in the field of mirror symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry.
In 2D ther ...
. Although it is undisputed that Lian−Liu−Yau's article appeared after Givental's, they claim that his work contained gaps that were only filled in following work in their own publication; Givental claims that his original work was complete. Nasar and Gruber quote an anonymous mathematician as agreeing with Givental.[For both sides of the dispute, see:
*
and Footnote 17 in
*]
* In the 1980s, Yau's colleague Yum-Tong Siu
Yum-Tong Siu (; born May 6, 1943 in Guangzhou, China) is the William Elwood Byerly Professor of Mathematics at Harvard University.
Siu is a prominent figure in the study of functions of several complex variables. His research interests invol ...
accused Yau's Ph.D. student Gang Tian of plagiarizing some of his work. At the time, Yau defended Tian against Siu's accusations. In the 2000s, Yau began to amplify Siu's allegations, saying that he found Tian's dual position at Princeton University
Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ...
and Peking University to be highly unethical due to his high salary from Peking University compared to other professors and students who made more active contributions to the university. Science Magazine
''Science'', also widely referred to as ''Science Magazine'', is the peer-reviewed academic journal of the American Association for the Advancement of Science (AAAS) and one of the world's top academic journals.
It was first published in 1880, ...
covered the broader phenomena of such positions in China, with Tian and Yau as central figures.
* Nasar and Gruber say that, having allegedly not done any notable work since the middle of the 1980s, Yau tried to regain prominence by claiming that Xi-Ping Zhu and Yau's former student Huai-Dong Cao had solved the Thurston and Poincaré conjectures, only partially based on some of Perelman's ideas. Nasar and Gruber quoted Yau as agreeing with the acting director of one of Yau's mathematical centers, who at a press conference assigned Cao and Zhu thirty percent of the credit for resolving the conjectures, with Perelman receiving only twenty-five (with the rest going to Richard Hamilton). A few months later, a segment of NPR
National Public Radio (NPR, stylized in all lowercase) is an American privately and state funded nonprofit media organization headquartered in Washington, D.C., with its NPR West headquarters in Culver City, California. It differs from other ...
's ''All Things Considered
''All Things Considered'' (''ATC'') is the flagship news program on the American network National Public Radio (NPR). It was the first news program on NPR, premiering on May 3, 1971. It is broadcast live on NPR affiliated stations in the United ...
'' covering the situation reviewed an audio recording of the press conference and found no such statement made by either Yau or the acting director.
Yau claimed that Nasar and Gruber's article was defamatory
Defamation is the act of communicating to a third party false statements about a person, place or thing that results in damage to its reputation. It can be spoken (slander) or written (libel). It constitutes a tort or a crime. The legal defini ...
and contained several falsehoods, and that they did not give him the opportunity to represent his own side of the disputes. He considered filing a lawsuit against the magazine, claiming professional damage, but says he decided that it wasn't sufficiently clear what such an action would achieve. He established a public relations website, with letters responding to the ''New Yorker'' article from several mathematicians, including himself and two others quoted in the article.
In his autobiography, Yau said that his statements in 2006 such as that Cao and Zhu gave "the first complete and detailed account of the proof of the Poincaré conjecture" should have been phrased more carefully. Although he does believe Cao and Zhu's work to be the first and most rigorously detailed account of Perelman's work, he says he should have clarified that they had "not surpassed Perelman's work in any way." He has also maintained the view that (as of 2019) the final parts of Perelman's proof should be better understood by the mathematical community, with the corresponding possibility that there remain some unnoticed errors.
Technical contributions to mathematics
Yau has made a number of major research contributions, centered on differential geometry and its appearance in other fields of mathematics and science. In addition to his research, Yau has compiled influential sets of open problems In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (i.e., no solution for it is kno ...
in differential geometry, including both well-known old conjectures with new proposals and problems. Two of Yau's most widely cited problem lists from the 1980s have been updated with notes on progress as of 2014. Particularly well-known are a conjecture on existence of minimal hypersurfaces and on the spectral geometry of minimal hypersurfaces.
Calabi conjecture
In 1978, by studying the complex Monge–Ampère equation
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
, Yau resolved the Calabi conjecture
In the mathematical field of differential geometry, the Calabi conjecture was a conjecture about the existence of certain kinds of Riemannian metrics on certain complex manifolds, made by . It was proved by , who received the Fields Medal and Oswal ...
, which had been posed by Eugenio Calabi
Eugenio Calabi (born 11 May 1923) is an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics, Emeritus, at the University of Pennsylvania, specializing in differential geometry, partial differential equations and ...
in 1954. As a special case, this showed that Kähler-Einstein metrics exist on any closed Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arn ...
whose first Chern class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ...
is nonpositive. Yau's method adapted earlier work of Calabi, Jürgen Moser
Jürgen Kurt Moser (July 4, 1928 – December 17, 1999) was a German-American mathematician, honored for work spanning over four decades, including Hamiltonian dynamical systems and partial differential equations.
Life
Moser's mother Ilse Strehl ...
, and Aleksei Pogorelov
Aleksei Vasil'evich Pogorelov (russian: Алексе́й Васи́льевич Погоре́лов, ua, Олексі́й Васи́льович Погорє́лов; March 2, 1919 – December 17, 2002), was a Soviet and Ukrainian ...
, developed for quasilinear elliptic partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
and the real Monge–Ampère equation
In mathematics, a (real) Monge–Ampère equation is a nonlinear second-order partial differential equation of special kind. A second-order equation for the unknown function ''u'' of two variables ''x'',''y'' is of Monge–Ampère type if it is li ...
, to the setting of the complex Monge–Ampère equation.
*In differential geometry, Yau's theorem is significant in proving the general existence of closed manifolds of special holonomy; any simply-connected closed Kähler manifold which is Ricci flat must have its holonomy group contained in the special unitary group
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
, according to the Ambrose–Singer theorem. Examples of compact Riemannian manifolds with other special holonomy groups have been found by Dominic Joyce and Peter Kronheimer
Peter Benedict Kronheimer (born 1963) is a British mathematician, known for his work on gauge theory and its applications to 3- and 4-dimensional topology. He is William Caspar Graustein Professor of Mathematics at Harvard University and forme ...
, although no proposals for general existence results, analogous to Calabi's conjecture, have been successfully identified in the case of the other groups.
* In algebraic geometry, the existence of canonical metrics as proposed by Calabi allows one to give equally canonical representatives of characteristic classes by differential forms. Due to Yau's initial efforts at disproving the Calabi conjecture by showing that it would lead to contradictions in such contexts, he was able to draw striking corollaries to the conjecture itself. In particular, the Calabi conjecture implies the Miyaoka–Yau inequality on Chern number
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...
s of surfaces, in addition to homotopical characterizations of the complex structures of the complex projective plane
In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates
:(Z_1,Z_2,Z_3) \in \mathbf^3,\qquad (Z_1, ...
and of quotients of the two-dimensional complex unit ball.
* A special case of the Calabi conjecture asserts that a Kähler metric of zero Ricci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...
must exist on any Kähler manifold whose first Chern class is zero. In string theory, it was discovered in 1985 by Philip Candelas, Gary Horowitz
Gary T. Horowitz (born April 14, 1955 in Washington, D.C.) is an American theoretical physicist who works on string theory and quantum gravity.
Biography
Horowitz studied at Princeton University (Bachelor 1976) and obtained his Ph.D. in 1979 at t ...
, Andrew Strominger
Andrew Eben Strominger (; born 1955) is an American theoretical physicist who is the director of Harvard's Center for the Fundamental Laws of Nature. He has made significant contributions to quantum gravity and string theory. These include his ...
, and Edward Witten
Edward Witten (born August 26, 1951) is an American mathematical and theoretical physicist. He is a Professor Emeritus in the School of Natural Sciences at the Institute for Advanced Study in Princeton. Witten is a researcher in string theory, q ...
that these ''Calabi–Yau manifolds'', due to their special holonomy, are the appropriate configuration spaces for superstrings. For this reason, Yau's resolution of the Calabi conjecture is considered to be of fundamental importance in modern string theory.
The understanding of the Calabi conjecture in the noncompact setting is less definitive. Gang Tian and Yau extended Yau's analysis of the complex Monge−Ampère equation to the noncompact setting, where the use of cutoff functions and corresponding integral estimates necessitated the conditional assumption of certain controlled geometry near infinity. This reduces the problem to the question of existence of Kähler metrics with such asymptotic properties; they obtained such metrics for certain smooth quasi-projective complex varieties. They later extended their work to allow orbifold singularities. With Brian Greene
Brian Randolph Greene (born February 9, 1963) is a American theoretical physicist, mathematician, and string theorist. Greene was a physics professor at Cornell University from 19901995, and has been a professor at Columbia University since 1 ...
, Alfred Shapere, and Cumrun Vafa
Cumrun Vafa ( fa, کامران وفا ; born 1 August 1960) is an Iranian-American theoretical physicist and the Hollis Professor of Mathematics and Natural Philosophy at Harvard University.
Early life and education
Cumrun Vafa was born in Tehra ...
, Yau introduced an ansatz
In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural Ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the ...
for a Kähler metric on the set of regular points of certain surjective holomorphic maps, with Ricci curvature approximately zero. They were able to apply the Tian−Yau existence theorem to construct a Kähler metric which is exactly Ricci-flat. The Greene−Shapere−Vafa−Yau ansatz and its natural generalization, now known as a ''semi-flat metric'', has become important in several analyses of problems in Kähler geometry.
Scalar curvature and general relativity
The positive energy theorem, obtained by Yau in collaboration with his former doctoral student Richard Schoen
Richard Melvin Schoen (born October 23, 1950) is an American mathematician known for his work in differential geometry and geometric analysis. He is best known for the resolution of the Yamabe problem in 1984.
Career
Born in Celina, Ohio, and a ...
, can be described in physical terms:
However, it is a precise theorem of differential geometry and geometric analysis
Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of ...
, in which physical systems are modeled by Riemannian manifolds with nonnegativity of a certain generalized scalar curvature
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometr ...
. As such, Schoen and Yau's approach originated in their study of Riemannian manifolds of positive scalar curvature, which is of interest in and of itself. The starting point of Schoen and Yau's analysis is their identification of a simple but novel way of inserting the Gauss–Codazzi equations
In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi Formulas) are fundamental formulas which link together the induced ...
into the second variation formula for the area of a stable minimal hypersurface of a three-dimensional Riemannian manifold. The Gauss–Bonnet theorem
In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology.
In the simplest application, the case of a t ...
then highly constrains the possible topology of such a surface when the ambient manifold has positive scalar curvature.
Schoen and Yau exploited this observation by finding novel constructions of stable minimal hypersurfaces with various controlled properties. Some of their existence results were developed simultaneously with similar results of Jonathan Sacks and Karen Uhlenbeck
Karen Keskulla Uhlenbeck (born August 24, 1942) is an American mathematician and one of the founders of modern geometric analysis. She is a professor emeritus of mathematics at the University of Texas at Austin, where she held the Sid W. Richard ...
, using different techniques. Their fundamental result is on the existence of minimal immersions with prescribed topological behavior. As a consequence of their calculation with the Gauss–Bonnet theorem, they were able to conclude that certain topologically distinguished three-dimensional manifolds cannot have any Riemannian metric of nonnegative scalar curvature.
Schoen and Yau then adapted their work to the setting of certain Riemannian asymptotically flat initial data sets in general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. They proved that negativity of the mass would allow one to invoke the Plateau problem
In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who experimented with soap films. The problem is ...
to construct stable minimal surfaces which are geodesically complete. A noncompact analogue of their calculation with the Gauss–Bonnet theorem then provides a logical contradiction to the negativity of mass. As such, they were able to prove the positive mass theorem in the special case of their Riemannian initial data sets.
Schoen and Yau extended this to the full Lorentzian formulation of the positive mass theorem by studying a partial differential equation proposed by Pong-Soo Jang. They proved that solutions to the Jang equation exist away from the apparent horizons of black holes, at which solutions can diverge to infinity. By relating the geometry of a Lorentzian initial data set to the geometry of the graph of such a solution to the Jang equation, interpreting the latter as a Riemannian initial data set, Schoen and Yau proved the full positive energy theorem. Furthermore, by reverse-engineering their analysis of the Jang equation, they were able to establish that any sufficient concentration of energy in general relativity must be accompanied by an apparent horizon.
Due to the use of the Gauss–Bonnet theorem, these results were originally restricted to the case of three-dimensional Riemannian manifolds and four-dimensional Lorentzian manifolds. Schoen and Yau established an induction on dimension by constructing Riemannian metrics of positive scalar curvature on minimal hypersurfaces of Riemannian manifolds which have positive scalar curvature. Such minimal hypersurfaces, which were constructed by means of geometric measure theory
In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfa ...
by Frederick Almgren
Frederick Justin Almgren Jr. (July 3, 1933 – February 5, 1997) was an American mathematician working in geometric measure theory. He was born in Birmingham, Alabama.
Almgren received a Guggenheim Fellowship in 1974. Between 1963 and 1992 he wa ...
and Herbert Federer
Herbert Federer (July 23, 1920 – April 21, 2010) was an American mathematician. He is one of the creators of geometric measure theory, at the meeting point of differential geometry and mathematical analysis.Parks, H. (2012''Remembering Herbert ...
, are generally not smooth in large dimensions, so these methods only directly apply up for Riemannian manifolds of dimension less than eight. Without any dimensional restriction, Schoen and Yau proved the positive mass theorem in the class of locally conformally flat manifolds. In 2017, Schoen and Yau published a preprint claiming to resolve these difficulties, thereby proving the induction without dimensional restriction and verifying the Riemannian positive mass theorem in arbitrary dimension.
Gerhard Huisken
Gerhard Huisken (born 20 May 1958) is a German mathematician whose research concerns differential geometry and partial differential equations. He is known for foundational contributions to the theory of the mean curvature flow, including Hui ...
and Yau made a further study of the asymptotic region of Riemannian manifolds with strictly positive mass. Huisken had earlier initiated the study of volume-preserving mean curvature flow of hypersurfaces of Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
. Huisken and Yau adapted his work to the Riemannian setting, proving a long-time existence and convergence theorem for the flow. As a corollary, they established a new geometric feature of positive-mass manifolds, which is that their asymptotic regions are foliated by surfaces of constant mean curvature.
Omori−Yau maximum principle
Traditionally, the maximum principle
In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations.
...
technique is only applied directly on compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", ...
s, as maxima are then guaranteed to exist. In 1967, Hideki Omori found a novel maximum principle which applies on noncompact Riemannian manifolds whose sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...
s are bounded below. It is trivial that ''approximate'' maxima exist; Omori additionally proved the existence of approximate maxima where the values of the gradient and second derivatives are suitably controlled. Yau partially extended Omori's result to require only a lower bound on Ricci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...
; the result is known as the Omori−Yau maximum principle. Such generality is useful due to the appearance of Ricci curvature in the Bochner formula
In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold (M, g) to the Ricci curvature. The formula is named after the American mathematician Salomon Bochner.
Formal statement
If u \colon M \righ ...
, where a lower bound is also typically used in algebraic manipulations. In addition to giving a very simple proof of the principle itself, Shiu-Yuen Cheng
Shiu-Yuen Cheng (鄭紹遠) is a Hong Kong mathematician. He is currently the Chair Professor of Mathematics at the Hong Kong University of Science and Technology. Cheng received his Ph.D. in 1974, under the supervision of Shiing-Shen Chern, from ...
and Yau were able to show that the Ricci curvature assumption in the Omori−Yau maximum principle can be replaced by the assumption of the existence of cutoff functions with certain controllable geometry.
Yau was able to directly apply the Omori−Yau principle to generalize the classical Schwarz−Pick lemma of complex analysis. Lars Ahlfors
Lars Valerian Ahlfors (18 April 1907 – 11 October 1996) was a Finnish mathematician, remembered for his work in the field of Riemann surfaces and his text on complex analysis.
Background
Ahlfors was born in Helsinki, Finland. His mother, Sie ...
, among others, had previously generalized the lemma to the setting of Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
s. With his methods, Yau was able to consider the setting of a mapping from a complete Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arn ...
(with a lower bound on Ricci curvature) to a Hermitian manifold
In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on ea ...
with holomorphic bisectional curvature bounded above by a negative number.
Cheng and Yau extensively used their variant of the Omori−Yau principle to find Kähler−Einstein metrics on noncompact Kähler manifolds, under an ansatz
In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural Ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the ...
developed by Charles Fefferman
Charles Louis Fefferman (born April 18, 1949) is an American mathematician at Princeton University, where he is currently the Herbert E. Jones, Jr. '43 University Professor of Mathematics. He was awarded the Fields Medal in 1978 for his contri ...
. The estimates involved in the method of continuity were not as difficult as in Yau's earlier work on the Calabi conjecture, due to the fact that Cheng and Yau only considered Kähler−Einstein metrics with negative scalar curvature. The more subtle question, where Fefferman's earlier work became important, is to do with geodesic completeness. In particular, Cheng and Yau were able to find complete Kähler-Einstein metrics of negative scalar curvature on any bounded, smooth, and strictly pseudoconvex subset of complex Euclidean space. These can be thought of as complex geometric analogues of the Poincaré ball model of hyperbolic space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. The ...
.
Differential Harnack inequalities
Yau's original application of the Omori−Yau maximum principle was to establish gradient estimates for a number of second-order elliptic partial differential equations. Given a function on a complete and smooth Riemannian manifold which satisfies various conditions relating the Laplacian to the function and gradient values, Yau applied the maximum principle to various complicated composite expressions to control the size of the gradient. Although the algebraic manipulations involved are complex, the conceptual form of Yau's proof is strikingly simple.
Yau's novel gradient estimates have come to be called "differential Harnack inequalities" since they can be integrated along arbitrary paths in to recover inequalities which are of the form of the classical Harnack inequalities, directly comparing the values of a solution to a differential equation at two different input points. By making use of Calabi's study of the distance function on a Riemannian manifold, Yau and Shiu-Yuen Cheng
Shiu-Yuen Cheng (鄭紹遠) is a Hong Kong mathematician. He is currently the Chair Professor of Mathematics at the Hong Kong University of Science and Technology. Cheng received his Ph.D. in 1974, under the supervision of Shiing-Shen Chern, from ...
gave a powerful localization of Yau's gradient estimates, using the same methods to simplify the proof of the Omori−Yau maximum principle. Such estimates are widely quoted in the particular case of harmonic functions on a Riemannian manifold, although Yau and Cheng−Yau's original results cover more general scenarios.
In 1986, Yau and Peter Li made use of the same methods to study parabolic partial differential equations on Riemannian manifolds. Richard Hamilton generalized their results in certain geometric settings to matrix inequalities. Analogues of the Li−Yau and Hamilton−Li−Yau inequalities are of great importance in the theory of Ricci flow
In the mathematical fields of differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be ana ...
, where Hamilton proved a matrix differential Harnack inequality for the curvature operator of certain Ricci flows, and Grigori Perelman
Grigori Yakovlevich Perelman ( rus, links=no, Григорий Яковлевич Перельман, p=ɡrʲɪˈɡorʲɪj ˈjakəvlʲɪvʲɪtɕ pʲɪrʲɪlʲˈman, a=Ru-Grigori Yakovlevich Perelman.oga; born 13 June 1966) is a Russian mathemati ...
proved a differential Harnack inequality for the solutions of a backwards heat equation coupled with a Ricci flow.
Cheng and Yau were able to use their differential Harnack estimates to show that, under certain geometric conditions, closed submanifolds of complete Riemannian or pseudo-Riemannian spaces are themselves complete. For instance, they showed that if is a spacelike hypersurface of Minkowski space which is topologically closed and has constant mean curvature, then the induced Riemannian metric on is complete. Analogously, they showed that if is an affine hypersphere of affine space which is topologically closed, then the induced affine metric on is complete. Such results are achieved by deriving a differential Harnack inequality for the (squared) distance function to a given point and integrating along intrinsically defined paths.
Donaldson−Uhlenbeck−Yau theorem
In 1985, Simon Donaldson
Sir Simon Kirwan Donaldson (born 20 August 1957) is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähler geometry. H ...
showed that, over a nonsingular projective variety of complex dimension two, a holomorphic vector bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a ...
admits a hermitian Yang–Mills connection In mathematics, and in particular gauge theory and complex geometry, a Hermitian Yang–Mills connection (or Hermite-Einstein connection) is a Chern connection associated to an inner product on a holomorphic vector bundle over a Kähler manifold t ...
if and only if the bundle is stable. A result of Yau and Karen Uhlenbeck
Karen Keskulla Uhlenbeck (born August 24, 1942) is an American mathematician and one of the founders of modern geometric analysis. She is a professor emeritus of mathematics at the University of Texas at Austin, where she held the Sid W. Richard ...
generalized Donaldson's result to allow a compact Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arn ...
of any dimension. The Uhlenbeck–Yau method relied upon elliptic partial differential equations while Donaldson's used parabolic partial differential equations, roughly in parallel to Eells and Sampson's epochal work on harmonic map
In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy a certain nonlinear partial differential equation. This partial differential equation for ...
s. The results of Donaldson and Uhlenbeck–Yau have since been extended by other authors. Uhlenbeck and Yau's article is important in giving a clear reason that stability of the holomorphic vector bundle can be related to the analytic methods used in constructing a hermitian Yang–Mills connection. The essential mechanism is that if an approximating sequence of hermitian connections fails to converge to the required Yang–Mills connection, then they can be rescaled to converge to a subsheaf which can be verified to be destabilizing by Chern–Weil theory.
Like the Calabi–Yau theorem, the Donaldson–Uhlenbeck–Yau theorem is of interest in theoretical physics. In the interest of an appropriately general formulation of supersymmetry, Andrew Strominger
Andrew Eben Strominger (; born 1955) is an American theoretical physicist who is the director of Harvard's Center for the Fundamental Laws of Nature. He has made significant contributions to quantum gravity and string theory. These include his ...
included the hermitian Yang–Mills condition as part of his Strominger system, a proposal for the extension of the Calabi−Yau condition to non-Kähler manifolds. Ji-Xiang Fu and Yau introduced an ansatz
In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural Ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of the ...
for the solution of Strominger's system on certain three-dimensional complex manifolds, reducing the problem to a complex Monge−Ampère equation, which they solved.
Yau's solution of the Calabi conjecture had given a reasonably complete answer to the question of how Kähler metrics on compact complex manifolds of nonpositive first Chern class can be deformed into Kähler–Einstein metrics. Akito Futaki showed that the existence of holomorphic vector fields can act as an obstruction to the direct extension of these results to the case when the complex manifold has positive first Chern class. A proposal of Calabi's suggested that Kähler–Einstein metrics exist on any compact Kähler manifolds with positive first Chern class which admit no holomorphic vector fields. During the 1980s, Yau and others came to understand that this criterion could not be sufficient. Inspired by the Donaldson−Uhlenbeck−Yau theorem, Yau proposed that the existence of Kähler–Einstein metrics must be linked to stability of the complex manifold in the sense of geometric invariant theory
In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in clas ...
, with the idea of studying holomorphic vector fields along projective embeddings, rather than holomorphic vector fields on the manifold itself. Subsequent research of Gang Tian and Simon Donaldson
Sir Simon Kirwan Donaldson (born 20 August 1957) is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähler geometry. H ...
refined this conjecture, which became known as the Yau–Tian–Donaldson conjecture relating Kähler–Einstein metrics and K-stability. In 2019, Xiuxiong Chen, Donaldson, and Song Sun were awarded the Oswald Veblen prize for resolution of the conjecture.
Geometric variational problems
In 1982, Li and Yau resolved the Willmore conjecture in the non-embedded case. More precisely, they established that, given any smooth immersion of a closed surface in the 3-sphere which fails to be an embedding, the Willmore energy
In differential geometry, the Willmore energy is a quantitative measure of how much a given surface deviates from a round sphere. Mathematically, the Willmore energy of a smooth closed surface embedded in three-dimensional Euclidean space is def ...
is bounded below by 8π. This is complemented by a 2012 result of Fernando Marques and André Neves
André da Silva Graça Arroja Neves (born 1975, Lisbon) is a Portuguese mathematician and a professor at the University of Chicago. He joined the faculty of the University of Chicago in 2016. In 2012, jointly with Fernando Codá Marques, he sol ...
, which says that in the alternative case of a smooth embedding of the 2-dimensional torus , the Willmore energy is bounded below by 2π2. Together, these results comprise the full Willmore conjecture, as originally formulated by Thomas Willmore in 1965. Although their assumptions and conclusions are quite similar, the methods of Li−Yau and Marques−Neves are distinct. Nonetheless, they both rely on structurally similar minimax schemes. Marques and Neves made novel use of the Almgren–Pitts min-max theory In mathematics, the Almgren–Pitts min-max theory (named after Frederick J. Almgren, Jr. and his student Jon T. Pitts) is an analogue of Morse theory for hypersurfaces.
The theory started with the efforts for generalizing George David Birkhoff's ...
of the area functional from geometric measure theory
In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfa ...
; Li and Yau's approach depended on their new "conformal invariant", which is a min-max quantity based on the Dirichlet energy
In mathematics, the Dirichlet energy is a measure of how ''variable'' a function is. More abstractly, it is a quadratic functional on the Sobolev space . The Dirichlet energy is intimately connected to Laplace's equation and is named after the ...
. The main work of their article is devoted to relating their conformal invariant to other geometric quantities.
William Meeks and Yau produced some foundational results on minimal surfaces in three-dimensional manifolds, revisiting points left open by older work of Jesse Douglas
Jesse Douglas (3 July 1897 – 7 September 1965) was an American mathematician and Fields Medalist known for his general solution to Plateau's problem.
Life and career
He was born to a Jewish family in New York City, the son of Sarah (née ...
and Charles Morrey. Following these foundations, Meeks, Leon Simon, and Yau gave a number of fundamental results on surfaces in three-dimensional Riemannian manifolds which minimize area within their homology class. They were able to give a number of striking applications. For example, they showed that if is an orientable 3-manifold such that every smooth embedding of a 2-sphere can be extended to a smooth embedding of the unit ball, then the same is true of any covering space of . Interestingly, Meeks-Simon-Yau's paper and Hamilton's foundational paper on Ricci flow
In the mathematical fields of differential geometry and geometric analysis, the Ricci flow ( , ), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential equation for a Riemannian metric. It is often said to be ana ...
, published in the same year, have a result in common, obtained by very distinct methods: any simply-connected compact 3-dimensional Riemannian manifold with positive Ricci curvature is diffeomorphic to the 3-sphere.
Geometric rigidity theorems
In the geometry of submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
s, both the extrinsic and intrinsic geometries are significant. These are reflected by the intrinsic Riemannian metric
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
and the second fundamental form
In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by \mathrm (read "two"). Together with the first fundame ...
. Many geometers have considered the phenomena which arise from restricting these data to some form of constancy. This includes as special cases the problems of minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below).
The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...
s, constant mean curvature, and submanifolds whose metric has constant scalar curvature
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometr ...
.
* The archetypical example of such questions is Bernstein's problem
In differential geometry, Bernstein's problem is as follows: if the graph of a function on R''n''−1 is a minimal surface in R''n'', does this imply that the function is linear?
This is true in dimensions ''n'' at most 8, but false in dimens ...
, as completely settled in famous work of James Simons
James Harris Simons (; born 25 April 1938) is an American mathematician, billionaire hedge fund manager, and philanthropist. He is the founder of Renaissance Technologies, a quantitative hedge fund based in East Setauket, New York. He and his ...
, Enrico Bombieri
Enrico Bombieri (born 26 November 1940, Milan) is an Italian mathematician, known for his work in analytic number theory, Diophantine geometry, complex analysis, and group theory. Bombieri is currently Professor Emeritus in the School of Mathem ...
, Ennio De Giorgi, and Enrico Giusti
Enrico Giusti (born Priverno, 1940), is an Italian mathematician mainly known for his contributions to the fields of calculus of variations, regularity theory of partial differential equations, minimal surfaces and history of mathematics. He ha ...
in the 1960s. Their work asserts that a minimal hypersurface which is a graph over Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
must be a plane in low dimensions, with counterexamples in high dimensions. The key point of the proof of planarity is the non-existence of conical and non-planar stable minimal hypersurfaces of Euclidean spaces of low dimension; this was given a simple proof by Richard Schoen
Richard Melvin Schoen (born October 23, 1950) is an American mathematician known for his work in differential geometry and geometric analysis. He is best known for the resolution of the Yamabe problem in 1984.
Career
Born in Celina, Ohio, and a ...
, Leon Simon, and Yau. Their technique of combining the Simons inequality with the formula for second variation of area has subsequently been used many times in the literature.
* Given the "threshold" dimension phenomena in the standard Bernstein problem, it is a somewhat surprising fact, due to Shiu-Yuen Cheng
Shiu-Yuen Cheng (鄭紹遠) is a Hong Kong mathematician. He is currently the Chair Professor of Mathematics at the Hong Kong University of Science and Technology. Cheng received his Ph.D. in 1974, under the supervision of Shiing-Shen Chern, from ...
and Yau, that there is no dimensional restriction in the Lorentzian analogue: any spacelike hypersurface of multidimensional Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
which is a graph over Euclidean space and has zero mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space.
The ...
must be a plane. Their proof makes use of the maximum principle techniques which they had previously used to prove differential Harnack estimates. Later they made use of similar techniques to give a new proof of the classification of complete parabolic or elliptic affine hyperspheres in affine geometry
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle.
As the notion of '' parallel lines'' is one of the main properties that is ...
.
* In one of his earliest papers, Yau considered the extension of the constant mean curvature condition to higher codimension, where the condition can be interpreted either as the mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space.
The ...
being parallel as a section of the normal bundle
In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion).
Definition
Riemannian manifold
Let (M,g) be a Riemannian m ...
, or as the constancy of the length of the mean curvature. Under the former interpretation, he fully characterized the case of two-dimensional surfaces in Riemannian space form
Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consi ...
s, and found partial results under the (weaker) second interpretation. Some of his results were independently found by Bang-Yen Chen
Chen Bang-yen is a Taiwanese mathematician who works mainly on differential geometry and related subjects. He was a University Distinguished Professor of Michigan State University from 1990 to 2012. After 2012 he became University Distinguished ...
.
* Extending Philip Hartman and Louis Nirenberg
Louis Nirenberg (February 28, 1925 – January 26, 2020) was a Canadian-American mathematician, considered one of the most outstanding mathematicians of the 20th century.
Nearly all of his work was in the field of partial differential equat ...
's earlier work on intrinsically flat hypersurfaces of Euclidean space, Cheng and Yau considered hypersurfaces of space forms which have constant scalar curvature
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometr ...
. The key tool in their analysis was an extension of Hermann Weyl's differential identity used in the solution of the Weyl isometric embedding problem.
Outside of the setting of submanifold rigidity problems, Yau was able to adapt Jürgen Moser
Jürgen Kurt Moser (July 4, 1928 – December 17, 1999) was a German-American mathematician, honored for work spanning over four decades, including Hamiltonian dynamical systems and partial differential equations.
Life
Moser's mother Ilse Strehl ...
's method of proving Caccioppoli inequalities, thereby proving new rigidity results for functions on complete Riemannian manifolds. A particularly famous result of his says that a subharmonic function
In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory.
Intuitively, subharmonic functions are related to convex functio ...
cannot be both positive and L''p'' integrable unless it is constant. Similarly, on a complete Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arn ...
, a holomorphic function cannot be L''p'' integrable unless it is constant.
Minkowski problem and Monge–Ampère equation
The Minkowski problem
In differential geometry, the Minkowski problem, named after Hermann Minkowski, asks for the construction of a strictly convex compact surface ''S'' whose Gaussian curvature is specified. More precisely, the input to the problem is a strictly posit ...
of classical differential geometry can be viewed as the problem of prescribing Gaussian curvature. In the 1950s, Louis Nirenberg
Louis Nirenberg (February 28, 1925 – January 26, 2020) was a Canadian-American mathematician, considered one of the most outstanding mathematicians of the 20th century.
Nearly all of his work was in the field of partial differential equat ...
and Aleksei Pogorelov
Aleksei Vasil'evich Pogorelov (russian: Алексе́й Васи́льевич Погоре́лов, ua, Олексі́й Васи́льович Погорє́лов; March 2, 1919 – December 17, 2002), was a Soviet and Ukrainian ...
resolved the problem for two-dimensional surfaces, making use of recent progress on the Monge–Ampère equation
In mathematics, a (real) Monge–Ampère equation is a nonlinear second-order partial differential equation of special kind. A second-order equation for the unknown function ''u'' of two variables ''x'',''y'' is of Monge–Ampère type if it is li ...
for two-dimensional domains. By the 1970s, higher-dimensional understanding of the Monge–Ampère equation was still lacking. In 1976, Shiu-Yuen Cheng
Shiu-Yuen Cheng (鄭紹遠) is a Hong Kong mathematician. He is currently the Chair Professor of Mathematics at the Hong Kong University of Science and Technology. Cheng received his Ph.D. in 1974, under the supervision of Shiing-Shen Chern, from ...
and Yau resolved the Minkowski problem in general dimensions via the method of continuity, making use of fully geometric estimates instead of the theory of the Monge–Ampère equation.
As a consequence of their resolution of the Minkowski problem, Cheng and Yau were able to make progress on the understanding of the Monge–Ampère equation. The key observation is that the Legendre transform
In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions ...
of a solution of the Monge–Ampère equation has its graph's Gaussian curvature prescribed by a simple formula depending on the "right-hand side" of the Monge–Ampère equation. As a consequence, they were able to prove the general solvability of the Dirichlet problem
In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.
The Dirichlet prob ...
for the Monge–Ampère equation, which at the time had been a major open question except for two-dimensional domains.
Cheng and Yau's papers followed some ideas presented in 1971 by Pogorelov, although his publicly available works (at the time of Cheng and Yau's work) had lacked some significant detail. Pogorelov also published a more detailed version of his original ideas, and the resolutions of the problems are commonly attributed to both Cheng–Yau and Pogorelov. The approaches of Cheng−Yau and Pogorelov are no longer commonly seen in the literature on the Monge–Ampère equation, as other authors, notably Luis Caffarelli
Luis Angel Caffarelli (born December 8, 1948) is an Argentine mathematician and luminary in the field of partial differential equations and their applications.
Career
Caffarelli was born and grew up in Buenos Aires. He obtained his Masters of S ...
, Nirenberg, and Joel Spruck, have developed direct techniques which yield more powerful results, and which do not require the auxiliary use of the Minkowski problem.
Affine sphere In mathematics, and especially differential geometry, an affine sphere is a hypersurface for which the affine normals all intersect in a single point. The term affine sphere is used because they play an analogous role in affine differential geometr ...
s are naturally described by solutions of certain Monge–Ampère equations, so that their full understanding is significantly more complicated than that of Euclidean spheres, the latter not being based on partial differential equations. In the ''parabolic'' case, affine spheres were completely classified as paraboloid
In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry.
Every plan ...
s by successive work of Konrad Jörgens, Eugenio Calabi
Eugenio Calabi (born 11 May 1923) is an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics, Emeritus, at the University of Pennsylvania, specializing in differential geometry, partial differential equations and ...
, and Pogorelov. The ''elliptic'' affine spheres were identified as ellipsoids by Calabi. The ''hyperbolic'' affine spheres exhibit more complicated phenomena. Cheng and Yau proved that they are asymptotic to convex cones, and conversely that every (uniformly) convex cone corresponds in such a way to some hyperbolic affine sphere. They were also able to provide new proofs of the previous classifications of Calabi and Jörgens–Calabi–Pogorelov.
Mirror symmetry
A Calabi–Yau manifold
In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstri ...
is a compact Kähler manifold which is Ricci-flat; as a special case of Yau's verification of the Calabi conjecture, such manifolds are known to exist. Mirror symmetry, which is a proposal developed by theoretical physicists dating from the late 1980s, postulates that Calabi−Yau manifolds of complex dimension three can be grouped into pairs which share certain characteristics, such as Euler and Hodge numbers. Based on this conjectural picture, the physicists Philip Candelas, Xenia de la Ossa
Xenia de la Ossa Osegueda (born 30 June 1958, San José, Costa Rica) is a theoretical physicist whose research focuses on mathematical structures that arise in string theory. She is a professor at Oxford's Mathematical Institute.
Academic car ...
, Paul Green, and Linda Parkes proposed a formula of enumerative geometry
In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory.
History
The problem of Apollonius is one of the earliest examp ...
which encodes the number of rational curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane c ...
s of any fixed degree in a general quintic hypersurface of four-dimensional complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
. Bong Lian, Kefeng Liu
Kefeng Liu ( Chinese: 刘克峰; born 12 December 1965), is a Chinese-American mathematician who is known for his contributions to geometric analysis, particularly the geometry, topology and analysis of moduli spaces of Riemann surfaces and Cala ...
, and Yau gave a rigorous proof that this formula holds. A year earlier, Alexander Givental
Alexander Givental (russian: Александр Борисович Гивенталь) is a Russian-American mathematician working in symplectic topology and singularity theory, as well as their relation to topological string theories. He graduat ...
had published a proof of the mirror formulas; according to Lian, Liu, and Yau, the details of his proof were only successfully filled in following their own publication. The proofs of Givental and Lian–Liu–Yau have some overlap but are distinct approaches to the problem, and each have since been given textbook expositions.
The works of Givental and of Lian−Liu−Yau confirm a prediction made by the more fundamental mirror symmetry conjecture of how three-dimensional Calabi−Yau manifolds can be paired off. However, their works do not logically depend on the conjecture itself, and so have no immediate bearing on its validity. With Andrew Strominger
Andrew Eben Strominger (; born 1955) is an American theoretical physicist who is the director of Harvard's Center for the Fundamental Laws of Nature. He has made significant contributions to quantum gravity and string theory. These include his ...
and Eric Zaslow
Eric Zaslow is an American mathematical physicist at Northwestern University.
Biography
Zaslow attended Harvard University, earning his Ph.D. in physics in 1995,
with thesis "Kinks, twists, and folds : exploring the geometric musculature of qua ...
, Yau proposed a geometric picture of how mirror symmetry might be systematically understood and proved to be true. Their idea is that a Calabi−Yau manifold with complex dimension three should be foliated by ''special Lagrangian'' tori, which are certain types of three-dimensional minimal submanifolds of the six-dimensional Riemannian manifold underlying the Calabi−Yau structure. Mirror manifolds would then be characterized, in terms of this conjectural structure, by having ''dual'' foliations. The Strominger−Yau−Zaslow (SYZ) proposal has been modified and developed in various ways since 1996. The conceptual picture that it provides has had a significant influence in the study of mirror symmetry, and research on its various aspects is currently an active field. It can be contrasted with the alternative homological mirror symmetry
Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory.
History
In an address ...
proposal by Maxim Kontsevich
Maxim Lvovich Kontsevich (russian: Макси́м Льво́вич Конце́вич, ; born 25 August 1964) is a Russian and French mathematician and mathematical physicist. He is a professor at the Institut des Hautes Études Scientifiques an ...
. The viewpoint of the SYZ conjecture is on geometric phenomena in Calabi–Yau spaces, while Kontsevich's conjecture abstracts the problem to deal with purely algebraic structures and category theory.
Comparison geometry
In one of Yau's earliest papers, written with Blaine Lawson
Herbert Blaine Lawson, Jr. is a mathematician best known for his work in minimal surfaces, calibrated geometry, and algebraic cycles. He is currently a Distinguished Professor of Mathematics at Stony Brook University. He received his PhD fr ...
, a number of fundamental results were found on the topology of closed Riemannian manifolds with nonpositive curvature. Their ''flat torus theorem'' characterizes the existence of a flat and totally geodesic
This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.
The following articles may also be useful; they either contain specialised vocabulary or provi ...
immersed torus in terms of the algebra of the fundamental group. The ''splitting theorem'' says that the splitting of the fundamental group as a maximally noncommutative direct product implies the isometric splitting of the manifold itself. Similar results were obtained at the same time by Detlef Gromoll
Detlef Gromoll (13 May 1938 – 31 May 2008) was a mathematician who worked in Differential geometry.
Biography
Gromoll was born in Berlin in 1938, and was a classically trained violinist.
After living and attending school in Rosdorf and gra ...
and Joseph Wolf
Joseph Wolf (22 January 1820 – 20 April 1899) was a German artist who specialized in natural history illustration. He moved to the British Museum in 1848 and became the preferred illustrator for explorers and naturalists including David Livi ...
. Their results have been extended to the broader context of isometric group actions on metric spaces of nonpositive curvature.
Jeff Cheeger
Jeff Cheeger (born December 1, 1943, Brooklyn, New York City) is a mathematician. Cheeger is professor at the Courant Institute of Mathematical Sciences at New York University in New York City. His main interests are differential geometry and ...
and Yau studied the heat kernel on a Riemannian manifold. They established the special case of Riemannian metrics for which geodesic spheres have constant mean curvature, which they proved to be characterized by radial symmetry of the heat kernel. Specializing to rotationally symmetric metrics, they used the exponential map to transplant the heat kernel to a geodesic ball on a general Riemannian manifold. Under the assumption that the symmetric "model" space under-estimates the Ricci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...
of the manifold itself, they carried out a direct calculation showing that the resulting function is a subsolution of the heat equation. As a consequence, they obtained a lower estimate of the heat kernel on a general Riemannian manifold in terms of lower bounds on its Ricci curvature. In the special case of nonnegative Ricci curvature, Peter Li and Yau were able to use their gradient estimates to amplify and improve the Cheeger−Yau estimate.
A well-known result of Yau's, obtained independently by Calabi, shows that any noncompact Riemannian manifold of nonnegative Ricci curvature must have volume growth of at least a linear rate. A second proof, using the Bishop–Gromov inequality
In mathematics, the Bishop–Gromov inequality is a comparison theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov. It is closely related to Myers' theorem, and is the key point in the proof of Gromov's compactness ...
instead of function theory, was later found by Cheeger, Mikhael Gromov, and Michael Taylor.
Spectral geometry
Given a smooth compact Riemannian manifold, with or without boundary, spectral geometry studies the eigenvalues of the Laplace–Beltrami operator, which in the case that the manifold has a boundary is coupled with a choice of boundary condition, usually Dirichlet or Neumann conditions. Paul Yang and Yau showed that in the case of a closed two-dimensional manifold, the first eigenvalue is bounded above by an explicit formula depending only on the genus and volume of the manifold. Earlier, Yau had modified Jeff Cheeger
Jeff Cheeger (born December 1, 1943, Brooklyn, New York City) is a mathematician. Cheeger is professor at the Courant Institute of Mathematical Sciences at New York University in New York City. His main interests are differential geometry and ...
's analysis of the Cheeger constant
In Riemannian geometry, the Cheeger isoperimetric constant of a compact Riemannian manifold ''M'' is a positive real number ''h''(''M'') defined in terms of the minimal area of a hypersurface that divides ''M'' into two disjoint pieces. In 1970, J ...
so as to be able to estimate the first eigenvalue from below in terms of geometric data.
In the 1910s, Hermann Weyl showed that, in the case of Dirichlet boundary conditions on a smooth and bounded open subset of the plane, the eigenvalues have an asymptotic behavior which is dictated entirely by the area contained in the region. His result is known as Weyl's law In mathematics, especially spectral theory, Weyl's law describes the asymptotic behavior of eigenvalues of the Laplace–Beltrami operator. This description was discovered in 1911 (in the d=2,3 case) by Hermann Weyl for eigenvalues for the Laplace ...
. In 1960, George Pólya
George Pólya (; hu, Pólya György, ; December 13, 1887 – September 7, 1985) was a Hungarian mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamenta ...
conjectured that the Weyl law actually gives control of each individual eigenvalue, and not only of their asymptotic distribution. Li and Yau proved a weakened version of Pólya's conjecture, obtaining control of the ''averages'' of the eigenvalues by the expression in the Weyl law.
In 1980, Li and Yau identified a number of new inequalities for Laplace–Beltrami eigenvalues, all based on the maximum principle and the differential Harnack estimates as pioneered five years earlier by Yau and Cheng−Yau. Their result on lower bounds based on geometric data is particularly well-known, and was the first of its kind to not require any conditional assumptions. Around the same time, a similar inequality was obtained by isoperimetric
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
methods by Mikhael Gromov, although his result is weaker than Li and Yau's. In collaboration with Isadore Singer
Isadore Manuel Singer (May 3, 1924 – February 11, 2021) was an American mathematician. He was an Emeritus Institute Professor in the Department of Mathematics at the Massachusetts Institute of Technology and a Professor Emeritus of Mathemat ...
, Bun Wong, and Shing-Toung Yau, Yau used the Li–Yau methodology to establish a gradient estimate for the quotient of the first two eigenfunctions. Analogously to Yau's integration of gradient estimates to find Harnack inequalities, they were able to integrate their gradient estimate to obtain control of the ''fundamental gap,'' which is the difference between the first two eigenvalues. The work of Singer–Wong–Yau–Yau initiated a series of works by various authors in which new estimates on the fundamental gap were found and improved.
In 1982, Yau identified fourteen problems of interest in spectral geometry, including the above Pólya conjecture. A particular conjecture of Yau's, on the control of the size of level sets of eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
s by the value of the corresponding eigenvalue, was resolved by Alexander Logunov and Eugenia Malinnikova, who were awarded the 2017 Clay Research Award __NOTOC__
The Clay Research Award is an annual award given by the Oxford-based Clay Mathematics Institute to mathematicians to recognize their achievement in mathematical research. The following mathematicians have received the award:
{, class=" ...
in part for their work.
Discrete and computational geometry
Xianfeng Gu and Yau considered the numerical computation of conformal maps between two-dimensional manifolds (presented as discretized meshes), and in particular the computation of uniformizing maps as predicted by the uniformization theorem
In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The theorem is a generalization ...
. In the case of genus-zero surfaces, a map is conformal if and only if it is harmonic, and so Gu and Yau are able to compute conformal maps by direct minimization of a discretized Dirichlet energy
In mathematics, the Dirichlet energy is a measure of how ''variable'' a function is. More abstractly, it is a quadratic functional on the Sobolev space . The Dirichlet energy is intimately connected to Laplace's equation and is named after the ...
. In the case of higher genus, the uniformizing maps are computed from their gradients, as determined from the Hodge theory of closed and harmonic 1-forms. The main work is thus to identify numerically effective discretizations of the classical theory. Their approach is sufficiently flexible to deal with general surfaces with boundary. With Tony Chan, Paul Thompson, and Yalin Wang, Gu and Yau applied their work to the problem of matching two brain surfaces, which is an important issue in medical imaging. In the most-relevant genus-zero case, conformal maps are only well-defined up to the action of the Möbius group
Moebius, Möbius or Mobius may refer to:
People
* August Ferdinand Möbius (1790–1868), German mathematician and astronomer
* Theodor Möbius (1821–1890), German philologist
* Karl Möbius (1825–1908), German zoologist and ecologist
* Paul ...
. By further optimizing a Dirichlet-type energy which measures the mismatch of brain landmarks such as the central sulcus, they obtained mappings which are well-defined by such neurological features.
In the field of graph theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, Fan Chung
Fan-Rong King Chung Graham (; born October 9, 1949), known professionally as Fan Chung, is a Taiwanese-born American mathematician who works mainly in the areas of spectral graph theory, extremal graph theory and random graphs, in particular in g ...
and Yau extensively developed analogues of notions and results from Riemannian geometry. These results on differential Harnack inequalities, Sobolev inequalities
In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the R ...
, and heat kernel analysis, found partly in collaboration with Ronald Graham
Ronald Lewis Graham (October 31, 1935July 6, 2020) was an American mathematician credited by the American Mathematical Society as "one of the principal architects of the rapid development worldwide of discrete mathematics in recent years". He ...
and Alexander Grigor'yan, were later written into textbook form as the last few chapters of her well-known book "Spectral Graph Theory". Later, they introduced a Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differenti ...
as defined for graphs, amounting to a pseudo-inverse
In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element ''x'' is an element ''y'' that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized in ...
of the graph Laplacian
In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Named after Pierre-Simon Laplace, the graph Lapl ...
. Their work is naturally applicable to the study of hitting time In the study of stochastic processes in mathematics, a hitting time (or first hit time) is the first time at which a given process "hits" a given subset of the state space. Exit times and return times are also examples of hitting times.
Definitions ...
s for random walk
In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space.
An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...
s and related topics.
In the interest of finding general graph-theoretic contexts for their results, Chung and Yau introduced a notion of Ricci-flatness of a graph. A more flexible notion of Ricci curvature, dealing with Markov chains on metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
s, was later introduced by Yann Ollivier. Yong Lin, Linyuan Lu, and Yau developed some of the basic theory of Ollivier's definition in the special context of graph theory, considering for instance the Ricci curvature of Erdös–Rényi random graphs. Lin and Yau also considered the ''curvature–dimension inequalities'' introduced earlier by Dominique Bakry
Roger-Dominique Bakry (born 12 December 1954), known as Dominique Bakry, is a French mathematician, a professor at the Université Paul-Sabatier in Toulouse, and a senior member of Institut Universitaire de France.
Bakry graduated from , and pre ...
and Michel Émery, relating it and Ollivier's curvature to Chung–Yau's notion of Ricci-flatness. They were further able to prove general lower bounds on Bakry–Émery and Ollivier's curvatures in the case of locally finite graphs.
Honors and awards
Yau has received honorary professorships from many Chinese universities, including Hunan Normal University
Hunan Normal University (), founded in 1938, is a public university in Changsha, Hunan Province. The university is the 211 Project university, one of the country's 100 national key universities in the 21st century that enjoy priority in obtain ...
, Peking University, Nankai University, and Tsinghua University
Tsinghua University (; abbr. THU) is a national public research university in Beijing, China. The university is funded by the Ministry of Education.
The university is a member of the C9 League, Double First Class University Plan, Projec ...
. He has honorary degrees from many international universities, including Harvard University
Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of high ...
, Chinese University of Hong Kong
The Chinese University of Hong Kong (CUHK) is a public research university in Ma Liu Shui, Hong Kong, formally established in 1963 by a charter granted by the Legislative Council of Hong Kong. It is the territory's second-oldest university and ...
, and University of Waterloo
The University of Waterloo (UWaterloo, UW, or Waterloo) is a public research university with a main campus in Waterloo, Ontario, Canada. The main campus is on of land adjacent to "Uptown" Waterloo and Waterloo Park. The university also operates ...
. He is a foreign member of the National Academies of Sciences of China, India, and Russia.
His awards include:
* 1975–1976, Sloan Fellow
The Sloan Fellows program is the world's first mid-career and senior career master's degree in general management and leadership. It was initially supported by a grant from Alfred P. Sloan, the late CEO of General Motors, to his alma mater, MI ...
.
* 1981, Oswald Veblen Prize in Geometry
__NOTOC__
The Oswald Veblen Prize in Geometry is an award granted by the American Mathematical Society for notable research in geometry or topology. It was founded in 1961 in memory of Oswald Veblen. The Veblen Prize is now worth US$5000, and is ...
.
* 1981, John J. Carty Award for the Advancement of Science
The John J. Carty Award for the Advancement of Science is awarded by the U.S. National Academy of Sciences "for noteworthy and distinguished accomplishments in any field of science within the charter of the Academy". Established by the America ...
, United States National Academy of Sciences
The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nati ...
.
* 1982, Fields Medal, for "his contributions to partial differential equations, to the Calabi conjecture in algebraic geometry, to the positive mass conjecture of general relativity theory, and to real and complex Monge–Ampère equations."
* 1982, elected to the American Academy of Arts and Sciences
The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, a ...
* 1982, Guggenheim Fellowship.
* 1984–1985, MacArthur Fellow
The MacArthur Fellows Program, also known as the MacArthur Fellowship and commonly but unofficially known as the "Genius Grant", is a prize awarded annually by the John D. and Catherine T. MacArthur Foundation typically to between 20 and 30 indi ...
.
* 1991, Humboldt Research Award
The Humboldt Prize, the Humboldt-Forschungspreis in German, also known as the Humboldt Research Award, is an award given by the Alexander von Humboldt Foundation of Germany to internationally renowned scientists and scholars who work outside of G ...
, Alexander von Humboldt Foundation
The Alexander von Humboldt Foundation (german: Alexander von Humboldt-Stiftung) is a foundation established by the government of the Federal Republic of Germany and funded by the Federal Foreign Office, the Federal Ministry of Education and Rese ...
, Germany.
* 1993, elected to the United States National Academy of Sciences
* 1994, Crafoord Prize
The Crafoord Prize is an annual science prize established in 1980 by Holger Crafoord, a Swedish industrialist, and his wife Anna-Greta Crafoord. The Prize is awarded in partnership between the Royal Swedish Academy of Sciences and the Crafoord Foun ...
.
* 1997, United States
The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territori ...
National Medal of Science
The National Medal of Science is an honor bestowed by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge in the fields of behavioral and social scienc ...
.
* 2003, China International Scientific and Technological Cooperation Award, for "his outstanding contribution to PRC in aspects of making progress in sciences and technology, training researchers."
* 2010, Wolf Prize in Mathematics, for "his work in geometric analysis and mathematical physics".
* 2018, Marcel Grossmann Awards, "for the proof of the positivity of total mass in the theory of general relativity and perfecting as well the concept of quasi-local mass, for his proof of the Calabi conjecture, for his continuous inspiring role in the study of black holes physics."[Marcel Grossmann]
15th Marcel Grossmann Meeting
/ref>
Major publications
Research articles.
Yau is the author of over five hundred articles. The following, among the most cited, are surveyed above:
Survey articles and publications of collected works.
Textbooks and technical monographs.
Popular books.
References
External links
''Center of Mathematical Sciences at Zhejiang University'': commentary by various mathematicians on Yau
Discover Magazine Interview, June 2010 issue
Interview
(11 pages long in Traditional Chinese
A tradition is a belief or behavior (folk custom) passed down within a group or society with symbolic meaning or special significance with origins in the past. A component of cultural expressions and folklore, common examples include holidays ...
)
Yau's autobiographical account
(mostly English, some Chinese)
*
*
UC Irvine courting Yau with a $2.5 million professorship
{{DEFAULTSORT:Yau, Shing-Tung
1949 births
20th-century American mathematicians
21st-century American mathematicians
Alumni of the Chinese University of Hong Kong
American people of Chinese descent
American relativity theorists
Differential geometers
Educators from Guangdong
Fellows of the American Mathematical Society
Fields Medalists
Foreign Members of the Russian Academy of Sciences
Foreign members of the Chinese Academy of Sciences
Hakka scientists
Harvard University faculty
Hong Kong emigrants to the United States
Hong Kong people of Hakka descent
Institute for Advanced Study faculty
Living people
MacArthur Fellows
Mathematicians from Guangdong
Members of Academia Sinica
Members of Committee of 100
Members of the United States National Academy of Sciences
National Medal of Science laureates
People from Shantou
Stanford University faculty
Stateless people
UC Berkeley College of Letters and Science alumni
University of California, San Diego faculty
Zhejiang University faculty