William Paul Thurston (born October 30, 1946) is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds. He is currently a professor of mathematics and computer science at Cornell University (since 2003).

His early work, in the early 1970s, was mainly in foliation theory, where he had a dramatic impact. His more significant results include:

In fact, Thurston resolved so many outstanding problems in foliation theory in such a short period of time that, according to Thurston, it led to a kind of exodus from the field, where advisors counselled students against going into foliation theory because Thurston was "cleaning out the subject" (see "On Proof and Progress in Mathematics", especially section 6 ).

His later work, starting around the late 1970s, revealed that hyperbolic geometry played a far more important role in the general theory of 3-manifolds than was previously realised. Prior to Thurston, there were only a handful of known examples of hyperbolic 3-manifolds of finite volume, such as the Seifert–Weber space. The independent and distinct approaches of Robert Riley and Troels Jørgensen in the mid-to-late 1970s showed that such examples were less atypical than previously believed; in particular their work showed that the figure eight knot complement was hyperbolic. This was the first example of a hyperbolic knot.

Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician. He is known for work in topology, both algebraic and geometric, and on dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Center, and is a professor at Stony Brook University.

He received his B.A. in 1963 from Rice University and his doctorate in 1966 from Princeton University. His Ph.D. thesis, entitled Triangulating homotopy equivalences, was written under the supervision of William Browder, and was a major contribution to surgery theory. He was a permanent member of the Institut des Hautes Études Scientifiques from 1974 to 1997.

Sullivan is one of the founders of the surgery method of classifying high-dimensional manifolds, along with Browder, Sergei Novikov and C. T. C. Wall. In homotopy theory, Sullivan put forward the radical concept that spaces could directly be localised, a procedure hitherto applied to the algebraic constructs made from them. He founded (along with Daniel Quillen) rational homotopy theory.

Michael Fredric Sipser is a professor of Applied Mathematics in the Theory of Computation Group at the Massachusetts Institute of Technology. He received his Ph.D. in 1980 from the University of California, Berkeley under the direction of Manuel Blum.

His research area is complexity theory, but he is also active in other related areas. In joint work with Farhi, Goldstone, and Gutmann, he introduced Adiabatic quantum computing.

He is the author of Introduction to the Theory of Computation (ISBN 0-534-95097-3), a standard textbook in the study of theoretical computer science.

He is currently (2011) serving as head of the MIT Mathematics department.

Steven Smale a.k.a. Steve Smale, Stephen Smale (born July 15, 1930) is an American mathematician from Flint, Michigan. He was awarded the Fields Medal in 1966, and spent more than three decades on the mathematics faculty of the University of California, Berkeley (1960–61 and 1964–1995).

He entered the University of Michigan in 1948. Initially, Smale was a good student, placing into an honors calculus sequence taught by Bob Thrall and earning himself A's. However, his sophomore and junior years were marred with mediocre grades, mostly Bs, Cs and even an F in nuclear physics. However, with some luck, Smale was accepted as a graduate student at the University of Michigan's mathematics department. Yet again, Smale performed poorly his first years, earning a C average as a graduate student. It was only when the department chair, Hildebrant, threatened to kick out Smale, that he began to work hard^{[citation needed]}. Smale finally earned his Ph.D. in 1957, under Raoul Bott.

Smale began his career as an instructor at the college at the University of Chicago. In 1958, he astounded the mathematical world with a proof of a sphere eversion. He then cemented his reputation with a proof of the Poincaré conjecture for all dimensions greater than or equal to 5; he later generalized the ideas in a 107 page paper that established the h-cobordism theorem.

Tian Gang (Chinese: 田刚; pinyin: Tián Gāng; born November 1958) is a Chinese mathematician and an academician of the Chinese Academy of Sciences. He is known for his contributions to geometric analysis and quantum cohomology, among other fields. He was born in Nanjing, China, was a professor of mathematics at MIT from 1995–2006 (holding the chair of Simons Professor of Mathematics from 1996), but now divides his time between Princeton University and Peking University. His employment at Princeton started from 2003, and now he is entitled Higgins Professor of Mathematics; starting 2005, he has been the director of Beijing International Center for Mathematical Research (BICMR).

Tian graduated from Nanjing University in 1982, and received a master's degree from Peking University in 1984. In 1988, he received a Ph.D. in mathematics from Harvard University, after having studied under Shing-Tung Yau. This work was so exceptional he was invited to present it at the Geometry Festival that year. In 1998, he was appointed as a Cheung Kong Scholar professor at the School of Mathematical Sciences at Peking University, under the "Cheung Kong Scholars Programme" (长江计划) of the Ministry of Education. Later his appointment was changed to Cheung Kong Scholar chair professorship. He was awarded the Alan T. Waterman Award in 1994, and the Veblen Prize in 1996. In 2004 Tian was inducted into the American Academy of Arts and Sciences. In an article on China View (Xinhua) , the world-renowned mathematician Shing-Tung Yau criticized the academic corruption in China, and alleged that Tian committed plagiarism, without naming him directly: "A well-known Harvard professor (Yum-Tong Siu) told Yau that his student (Gang Tian) had plagiarized [Siu's] paper , but, in trying to protect his young student, Yau did not pursue the matter. As a result, matters only went from bad to worse."