User:Russ Woodroofe/kbrown

Kenneth Stephen Brown (born November 1945) is an American mathematician, and an emeritus professor at Cornell University. Brown has broad mathematical interests, centered around topology and group theory.

Education and career
Brown earned his Ph.D. in 1971 from the Massachusetts Institute of Technology, under the supervision of Daniel Quillen. His thesis was titled Abstract homotopy theory and generalized sheaf cohomology. He was hired directly after his Ph.D as an assistant professor at Cornell University, where he remained until his retirement in 2014. He served as chair of the department from 2002 to 2006.

Brown has broad interests around topology and group theory, also extending to combinatorics, and probability theory. He is also known for his work in category theory, cohomology theory, and geometric group theory. He has published over 30 scientific articles and 2 books, and advised 16 Ph.D. students.

Significant contributions
Brown's early work is in category theory and K-theory. He is known for Ken Brown's Lemma in the theory of model categories, which gives a sufficient condition for a functor to send weak equivalences to weak equivalences. Brown is one of the authors for which the Brown-Gersten-Quillen spectral sequence is named.

Brown discovered the main idea used for practical applications of discrete Morse theory, that of performing repeated simplicial collapse operations in various skeleta of a CW complex to replace the complex with a simpler complex. This work generalizes methods used earlier in joint work with Ross Geoghegan; the idea was later rediscovered (with the connection to the discrete Morse Theory of Robin Forman) by Manoj Chari.

Later in his career, Brown worked on random walks in hyperplane arrangements and semigroups. Based partly on work with Persi Diaconis, he developed a theory of random walks on left regular bands.

Brown has written two graduate textbooks on group cohomology and on buildings. The book on buildings was (together with the book of Mark Ronan) one of the first textbook-level expositions of the topic; an expanded second edition of this textbook was coauthored with Peter Abramenko.

Awards and honors

 * Brown was named as a fellow of the American Mathematical Society in the inaugural class of 2013.
 * A conference titled "Approaches to Group Theory" was held in honor of Brown's 65th birthday in 2010.
 * Brown was an invited speaker at the International Congress of Mathematicians in 1978 at Helsinki.

Books

 * (Reprinted in 1998.)
 * (Reprinted with corrections in 1994.)
 * (Reprinted with corrections in 1994.)