Michael Aschbacher
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Michael George Aschbacher (born April 8, 1944) is an American
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
best known for his work on
finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...
s. He was a leading figure in the completion of the
classification of finite simple groups In mathematics, the classification of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every List of finite simple groups, finite simple group is either cyclic group, cyclic, or alternating gro ...
in the 1970s and 1980s. It later turned out that the classification was incomplete, because the case of
quasithin group In mathematics, a quasithin group is a finite simple group that resembles a group of Lie type of rank at most 2 over a field of characteristic 2. The classification of quasithin groups is a crucial part of the classification of finite simple g ...
s had not been finished. This gap was fixed by Aschbacher and Stephen D. Smith in 2004, in a pair of books comprising about 1300 pages. Aschbacher is currently the Shaler Arthur Hanisch Professor of Mathematics at the
California Institute of Technology The California Institute of Technology (branded as Caltech) is a private research university in Pasadena, California, United States. The university is responsible for many modern scientific advancements and is among a small group of institutes ...
.


Education and career

Aschbacher received his B.S. at the
California Institute of Technology The California Institute of Technology (branded as Caltech) is a private research university in Pasadena, California, United States. The university is responsible for many modern scientific advancements and is among a small group of institutes ...
in 1966 and his Ph.D. at the
University of Wisconsin–Madison The University of Wisconsin–Madison (University of Wisconsin, Wisconsin, UW, UW–Madison, or simply Madison) is a public land-grant research university in Madison, Wisconsin, United States. It was founded in 1848 when Wisconsin achieved st ...
in 1969. He joined the faculty of the California Institute of Technology in 1970 and became a full professor in 1976. He was a visiting scholar at the
Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...
in 1978–79. He was awarded the
Cole Prize The Frank Nelson Cole Prize, or Cole Prize for short, is one of twenty-two prizes awarded to mathematicians by the American Mathematical Society, one for an outstanding contribution to algebra, and the other for an outstanding contribution to numbe ...
in 1980, and was elected to the
National Academy of Sciences The National Academy of Sciences (NAS) is a United States nonprofit, NGO, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the ...
in 1990. In 1992, Aschbacher was elected a Fellow of the
American Academy of Arts and Sciences The American Academy of Arts and Sciences (The Academy) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and other ...
. He was awarded the
Rolf Schock Prize The Rolf Schock Prizes were established and endowed by bequest of philosopher and artist Rolf Schock (1933–1986). The prizes were first awarded in Stockholm, Sweden, in 1993 and, since 2005, are awarded every three years. It is sometimes conside ...
for Mathematics by the
Royal Swedish Academy of Sciences The Royal Swedish Academy of Sciences () is one of the Swedish Royal Academies, royal academies of Sweden. Founded on 2 June 1739, it is an independent, non-governmental scientific organization that takes special responsibility for promoting nat ...
in 2011. In 2012 he received the Leroy P. Steele Prize for Mathematical Exposition and the
Wolf Prize in Mathematics The Wolf Prize in Mathematics is awarded almost annually by the Wolf Foundation in Israel. It is one of the six Wolf Prizes established by the Foundation and awarded since 1978; the others are in Agriculture, Chemistry, Medicine, Physics and Arts. ...
, and became a fellow of the
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
.


Classification of finite simple groups

In 1973, Aschbacher became a leading figure in the
classification of finite simple groups In mathematics, the classification of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every List of finite simple groups, finite simple group is either cyclic group, cyclic, or alternating gro ...
. Aschbacher considered himself somewhat of an outsider in the world of conventional group theory, claiming that he was not "plugged into the system at that point in time". Although he had access to several preprints that were shared among the practitioners of the field, he reproduced many proofs that had already been discovered by other researchers and published them in his early papers. Aschbacher only became interested in finite simple groups as a postdoctorate. He wrote his dissertation in
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
and was able to utilize many techniques developed in this area to make early contributions to the study of finite simple groups, which surprised the community of researchers. In particular,
Daniel Gorenstein Daniel E. Gorenstein (January 1, 1923 – August 26, 1992) was an American mathematician best remembered for his contribution to the classification of finite simple groups. Gorenstein mastered calculus at age 12 and subsequently matriculated at ...
, another leader of the
classification of finite simple groups In mathematics, the classification of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every List of finite simple groups, finite simple group is either cyclic group, cyclic, or alternating gro ...
, said that Aschbacher's entrance was "dramatic". In fact, the rate of Aschbacher's results proved so astounding that many other mathematicians decided to leave the field to pursue other problems. Aschbacher was proving one major result after another and when he announced his progress at the Duluth conference, mathematicians were convinced that the problem was almost solved. This conference represented a turning point for the problem as many mathematicians (in particular those relatively new to the field) decided to leave the field to pursue other problems. However, Aschbacher's entrance into the field did not come without difficulties. His papers, beginning with the first he wrote in the field for publication, were very difficult to read. Some commented that his proofs lacked explanations of very sophisticated
counting Counting is the process of determining the number of elements of a finite set of objects; that is, determining the size of a set. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for ever ...
arguments. As Aschbacher's proofs became longer, it became even more difficult for others to understand his proofs. Even some of his own coauthors had trouble reading their own papers. From that point on, researchers no longer read papers as independent documents, but rather ones that required the context of its author. As a result, responsibility of finding errors in the classification problem was up to the entire community of researchers rather than just peer-reviewers alone. That Aschbacher's proofs were hard to read was not due to a lack of ability, but rather to the astounding complexity of the ideas he was able to produce.


Books

*''Finite group theory'' *''Sporadic groups'' *''3-Transposition groups'' *''The finite simple groups and their classification'' *''Overgroups of Sylow subgroups in sporadic groups'' * * *


References


External links


Aschbacher's webpage at Caltech
{{DEFAULTSORT:Aschbacher, Michael 1944 births Living people Members of the United States National Academy of Sciences Group theorists California Institute of Technology alumni California Institute of Technology faculty University of Wisconsin–Madison alumni Fellows of the American Academy of Arts and Sciences Fellows of the American Mathematical Society Institute for Advanced Study visiting scholars Wolf Prize in Mathematics laureates 20th-century American mathematicians 21st-century American mathematicians Mathematicians from Arkansas People from Little Rock, Arkansas