User:Cubiejoe

=Life Outside of Mathematics=

Kenneth Ira Appel was born in Brooklyn, New York October 8, 1932 and died in Dover, Delaware April 19, 2013 after being diagnosed with esophageal cancer in October 2013. He was an American mathematician, who along with Wolfgang Haken, succeeded in proving that any map in a plane or on a sphere can be colored with only four colors in such a way that no two neighboring countries are of the same color. This problem that they solved was known as the four color theorem.

Kenneth Appel grew up in Queens, New York and was the son of Irwin Appel and Lillian Sender Appel. He worked as an actuary for a brief time and then served in the United States Army for two years, at Fort Benning, Georgia, and in Baumholder, Germany. In 1959 he finished his doctoral program at the University of Michigan and also married Carole S. Stein in Philadelphia. After getting married the newlyweds moved to Princeton, New Jersey where Kenneth worked for the Institute for Defense Analyses from 1959 to 1961. His main work at the Institute of Defense Analysis was doing research in cryptography. Towards the end of his life, in 2012, he was elected a Fellow of the American Mathematical Society.

Kenneth Appel was also the treasurer of the Strafford County Democratic Committee. He played tennis up till his fifties, was a lifelong stamp collector, a player of the game of Go, and a baker of bread. He and Carole have two sons, Andrew W. Appel, a noteworthy computer scientist, and Peter H. Appel, and had a daughter, Laurel F. Appel who passed away March 4, 2013. Kenneth also has five grandchildren namely Avi and Joseph Appel, Rebecca and Nathaniel Weir, and Carmen Appel.

=Schooling and Teaching=

Kenneth Appel received his bachelor's degree from Queens College in 1953. After serving the army he attended the University of Michigan where he earned his M.A. in 1956, and then later his Ph.D. in 1959. Roger Lyndon was his doctoral advisor and is a mathematician whose main mathematical focus was in group theory.

After working for the Institute for Defense Analyses, in 1969 he joined the Mathematics Department faculty at the University of Illinois as an Assistant Professor. While here Kenneth researched on group theory and computability theory. In 1967 he became an Associate Professor and in 1977 obtained the title of Professor. It was while he was at this school that he and Wolfgang Haken proved the four color theorem. From their work and proof of this theorem they were later awarded the Delbert Ray Fulkerson prize, in 1979, by the American Mathematical Society and the Mathematical Programming Society.

While at the University of Illinois Kenneth took on five students during their doctoral program. Each student helped contribute to the work on the Mathematics Genealogy Project.

In 1993 he moved to New Hampshire where he earned the position of the Chairman of the Mathematics Department at the University of New Hampshire. In 2003 Kenneth Appel's teaching career came to an end as he retired as professor emeritus. During his retirement he would volunteer in mathematics enrichment programs in Dover and southern Maine public schools. His belief was "that students should be afforded the opportunity to study mathematics at the level of their ability, even if it is well above their grade level."

=Contributions to Mathematics=

The Four Color Theorem
Kenneth Appel is most well known for his work in topology, which is the branch of mathematics that explores certain properties of geometric figures. Mainly his biggest accomplishment was proving the four color theorem in 1976 with Wolfgang Haken. The New York Times wrote in 1976, "Now the four-color conjecture has been proved by two University of Illinois mathematicians, Kenneth Appel and Wolfgang Haken. They had an invaluable tool that earlier mathematicians lacked--modern computers. Their present proof rests in part on 1,200 hours of computer calculation during which about ten billion logical decisions had to be made. The proof of the four-color conjecture is unlikely to be of applied significance. Nevertheless, what has been accomplished is a major intellectual feat. It gives us an important new insight into the nature of two-dimensional space and of the ways in which such space can be broken into discrete portions." At first most mathematicians were against the fact that Appel and Haken were using computers since it was new during this time, and Appel even mentions, "Most mathematicians, even as late as the 1970s, had no real interest in learning about computers. It was almost as if those of us who enjoyed playing with computers were doing something non-mathematical or suspect." The actual proof was described in an article as long as a typical book titled Every Planar Map is Four Colorable, Contemporary Mathematics, vol. 98, American Mathematical Society, 1989.

Group Theory
Kenneth Appel also co-wrote an article with P.E. Schupp titled Artin Groups and Infinite Coxeter Groups. In this article Appel and Schupp introduce four theorems that are true about Coxeter groups and proves them true for Artin groups. The proofs of these four theorems use "the results and methods of small cancellation theory. This theory had its origins in Dehn's work on Fuchsian groups."

=Other Artin Groups=

We define that an Artin group or a Coxeter group is of large type if mi j ≥ 3 for all i ≠ j. We say that an Artin group or a Coxeter group is of extra-large type if mi j ≥ 4 for all i ≠ j.

Kenneth Appel and P.E. Schupp looked further into Artin groups and the properties that hold true for them. They proved four theorems, which were known to be true for Coxeter groups, and showed that they also held for Artin groups. Appel and Schupp had discovered that they could study extra-large Artin and Coxeter groups through the techniques of small cancellation theory. They also discovered that they could use a "refinement" of these same techniques to work with these groups of large type.

Theorem 1 : Let G be an Artin or Coxeter group of extra-large type. If J ⊆ I then GJ has a presentation defined by the Coxeter matrix MJ and the generalized word problem for GJ in G is solvable. If J, K ⊆ I then GJ ∩ GK = G(J ∩ K).

Theorem 2 : An Artin group of of extra-large type is torsion-free.

Theorem 3 : Let G be an Artin group of extra-large type. Then the set {ai2 : i ∈ I} freely generates a free subgroup of G.

Theorem 4 : An Artin or Coxeter group of extra-large type has solvable conjugacy problem.

=References=