Homotopy Theory:
* Homotopical Algebra: This book, published in 1967, introduced the concept of model categories, which revolutionized algebraic topology and provided a powerful framework for understanding homotopy theory.
* Rational Homotopy Theory: Quillen's work on rational homotopy theory led to a deep understanding of the relationship between topological spaces and their rational homology groups.
Algebraic K-theory:
* Higher Algebraic K-theory: Quillen developed the concept of higher algebraic K-theory, which has become a fundamental tool in algebraic topology and number theory.
* The Quillen-Suslin Theorem: This theorem, proved independently by Quillen and Andrei Suslin, provides a remarkable connection between algebraic K-theory and the theory of polynomial rings.
Other Important Works:
* Projective Modules over Polynomial Rings: Quillen proved a fundamental theorem regarding the structure of projective modules over polynomial rings.
* The Quillen-Lichtenbaum Conjecture: This conjecture, which remains open in general, relates algebraic K-theory to étale cohomology.
In addition to these major works, Quillen has published numerous research papers on a wide range of topics, including algebraic geometry, category theory, and representation theory.
His contributions have earned him many prestigious awards, including the 1978 Fields Medal, the highest honor in mathematics.