# Daniel Grey Quillen

# Daniel Grey Quillen

**1940-**

**American Mathematician**

In 1977 Daniel Quillen and Russian mathematician A. A. Suslin independently discovered two quite distinct proofs of a 20-year-old theory concerning the structure of generalized vector spaces. These proofs established that all abstract spaces of certain common types were constructed in direct analogy with two- and three-dimensional Euclidean space. In the following year, 1978, Quillen received the Fields Medal in recognition of this and other outstanding contributions to mathematical study.

The son of a chemical engineer who went on to become a physics teacher, Daniel Quillen was born in Orange, New Jersey, on June 27, 1940. He attended grade school at the private Newark Academy, then entered Harvard University. Then, after he received his B.A. in 1961, he continued his work at Harvard under the supervision of Raoul Bott. In 1964 Quillen earned his Ph.D. with a thesis on partial differential equations, *Formal Properties of Over-Determined Systems of Linear Partial Differential Equations.*

By this point the 24-year-old Quillen and his wife Jean, a violinist, had two children; eventually they would have five. Quillen went on to join the faculty of the Massachusetts Institute of Technology (MIT), but many of his most pivotal experiences during the 1960s and 1970s took place while serving as a visiting scholar at other universities. In the 1968-69 academic year, Quillen was a Sloan Fellow in Paris, where he came under the influence of Alexander Grothendieck (1928- ). During the following year, as a visiting member of the Institute for Advanced Study at Princeton, he met Michael Atiyah (1929- ), who would have an even more profound effect on Quillen's work. Quillen also spent a year in France as a Guggenheim fellow from 1973 to 1974.

Quillen's principal interests lay in the homology of simplicial objects. A simplicial object, or simplex, is one that has *n* dimensions determined by *n* + 1 points, in a space equal to or greater than *n;* thus a triangle and its interior, determined by three points, is a two-dimensional simplex. Homology is an area of topology concerned with the partitioning of space into geometric components such as points, lines, and triangles, and with the interrelationships of these components, particularly as this relates to group theory.

Quillen had begun in the 1960s by identifying the means of defining the homology of simplicial objects over a variety of categories such as sets; then he turned to developing a conjecture in homotopy theory posited by Frank Adams. Using techniques from algebraic geometry and from the modular representation theory of groups, Quillen and one of his students were able to prove Adams's conjecture. Quillen later applied these concepts to the study of K-theory, which Atiyah had developed as a means of dealing with cohomological questions. He expanded K-theory, establishing a higher algebraic version of it in 1972, and this made possible a variety of discoveries which in turn led to his receiving the Fields Medal six years later.

Quillen has often been lauded by other mathematicians for his fresh approach to challenging questions. He is noted for his rigor and the breadth of his thinking, and in his personal life is known as a quiet family man. During the 1990s Quillen worked at Oxford in England.

**JUDSON KNIGHT**

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**Daniel Grey Quillen**