Mathematical Contributions
Quillen's most celebrated contribution (mentioned specifically in his Fields medal citation) was his formulation of higher algebraic K-theory in 1972. This new tool, formulated in terms of homotopy theory, proved to be successful in formulating and solving major problems in algebra, particularly in ring theory and module theory. More generally, Quillen developed tools (especially his theory of model categories) which allowed algebro-topological tools to be applied in other contexts.
Before his ground-breaking work in defining higher algebraic K-theory, Quillen worked on the Adams conjecture, formulated by Frank Adams in homotopy theory. His proof of the conjecture used techniques from the modular representation theory of groups, which he later applied to work on cohomology of groups and algebraic K-theory. He also worked on complex cobordism, showing that its formal group law is essentially the universal one.
In related work, he also supplied a proof of Serre's conjecture about the triviality of algebraic vector bundles on affine space. He was also an architect (along with Dennis Sullivan) of rational homotopy theory.
He introduced the Quillen determinant line bundle and the Mathai–Quillen formalism.
Read more about this topic: Daniel Quillen
Famous quotes containing the word mathematical:
“An accurate charting of the American woman’s progress through history might look more like a corkscrew tilted slightly to one side, its loops inching closer to the line of freedom with the passage of time—but like a mathematical curve approaching infinity, never touching its goal. . . . Each time, the spiral turns her back just short of the finish line.”
—Susan Faludi (20th century)