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Stephen M. Gersten

{{Short description|American mathematician (born 1940)}}

Stephen M. Gersten (born 2 December 1940) is an American mathematician, specializing in finitely presented groups and their geometric properties.{{cite web|title=Stephen M. Gersten|website=Mathematics Department, University of Utah|url=https://www.math.utah.edu/research/brochure/gerton.pdf}}

Gersten graduated in 1961 with an AB from Princeton University and in 1965 with a PhD from Trinity College, Cambridge. His doctoral thesis was Class Groups of Supplemented Algebras written under the supervision of John R. Stallings.{{MathGenealogy|id=19892|title=Stephen M. Gersten}} In the late 1960s and early 1970s he taught at Rice University. In 1972–1973 he was a visiting scholar at the Institute for Advanced Study.{{cite web|title=Stephen M. Gersten|website=Institute for Advanced Study|date=9 December 2019 |url=https://www.ias.edu/scholars/stephen-m-gersten}} In 1973 he became a professor at the University of Illinois at Urbana–Champaign. In 1974 he was an Invited Speaker at the International Congress of Mathematicians in Vancouver.{{cite book|author=Gersten, S. M.|chapter=Class Groups of Supplemented Algebras|title=Proceedings of the International Congress of Mathematicians, Vancouver, 1974|volume=1|year=1975|pages=309–314}} At the University of Utah he became a professor in 1975 and is now semi-retired there. His PhD students include Roger C. Alperin, R. Keith Dennis and Edward W. Formanek.

Gersten's conjecture has motivated considerable research.{{cite arXiv|author=Mochizuki, Satoshi|title=A survey of Gersten's conjecture|year=2016|eprint=1608.08114|class=math.KT}}

Gersten's theorem

If {{math|φ}} is an automorphism of a finitely generated free group {{math|F}} then

{{ math|{ x : x ∈ F and φ(x) = x }}} is finitely generated.{{cite journal|author=Gersten, S. M.|title=Fixed points of automorphisms of free groups|journal=Advances in Mathematics|volume=64|issue=1|year=1987|pages=51–85|url=https://core.ac.uk/download/pdf/82160100.pdf|doi=10.1016/0001-8708(87)90004-1|doi-access=free}}{{cite book|editor1=Gersten, S. M.|editor2=Stallings, John R.|title=Combinatorial Group Theory and Topology|url=https://archive.org/details/combinatorialgro00gers_0|url-access=registration|date=21 May 1987|publisher=Princeton University Press|isbn=0-691-08410-6}}

Selected publications

  • {{cite journal|doi=10.1090/S0002-9904-1972-12924-0|title=On the spectrum of algebraic K-theory|journal=Bulletin of the American Mathematical Society|volume=78|issue=2|pages=216–220|year=1972|last1=Gersten|first1=S. M.|doi-access=free}}
  • {{cite journal|doi=10.1090/S0002-9904-1973-13150-7|title=Higher K-theory for regular schemes|journal=Bulletin of the American Mathematical Society|volume=79|pages=193–197|year=1973|last1=Gersten|first1=S. M.|doi-access=free}}
  • {{cite book|doi=10.1007/BFb0067049|chapter=Higher K-theory of rings|title=Higher K-Theories|volume=341|pages=3–42|series=Lecture Notes in Mathematics|year=1973|last1=Gersten|first1=S. M.|isbn=978-3-540-06434-3}}
  • {{cite book|doi=10.1007/BFb0067062|chapter=Algebraic K-theory as generalized sheaf cohomology|title=Higher K-Theories|volume=341|pages=266–292|series=Lecture Notes in Mathematics|year=1973|last1=Brown|first1=Kenneth S.|authorlink=Kenneth Brown (mathematician)|last2=Gersten|first2=Stephen M.|isbn=978-3-540-06434-3}}
  • {{cite journal|doi=10.1090/S0002-9939-1983-0702313-2|title=A short proof of the algebraic Weierstrass preparation theorem|journal=Proceedings of the American Mathematical Society | volume=88|issue=4|pages=751–752|year=1983|last1=Gersten|first1=S. M.|doi-access=free}} (See Weierstrass preparation theorem.)
  • {{cite journal|doi=10.1090/S0273-0979-1983-15116-9|title=On fixed points of automorphisms of finitely generated free groups|journal=Bulletin of the American Mathematical Society|volume=8|issue=3|pages=451–455|year=1983|last1=Gersten|first1=S. M.|doi-access=free}} (This paper presents a proof of a conjecture made by G. Peter Scott.)
  • {{cite journal|doi=10.1090/S0273-0979-1984-15246-7|title=On Whitehead's algorithm|journal=Bulletin of the American Mathematical Society|volume=10|issue=2|pages=281–285|year=1984|last1=Gersten|first1=S. M.|doi-access=free}}
  • {{cite book|doi=10.1007/978-1-4613-9586-7_2|chapter=Reducible Diagrams and Equations over Groups|title=Essays in Group Theory|volume=8|pages=15–73|series=Mathematical Sciences Research Institute Publications|year=1987|last1=Gersten|first1=S. M.|isbn=978-1-4613-9588-1}}
  • {{cite journal|doi=10.1007/BF01233430|title=Small cancellation theory and automatic groups|journal=Inventiones Mathematicae|volume=102|pages=305–334|year=1990|last1=Gersten|first1=S. M.|last2=Short|first2=Hamish B.|bibcode=1990InMat.102..305G|s2cid=120267906}}
  • {{cite journal|doi=10.1016/0022-4049(91)90139-S|title=Automatic groups and amalgams|journal=Journal of Pure and Applied Algebra|volume=76|issue=3|pages=229–316|year=1991|last1=Baumslag|first1=Gilbert|authorlink=Gilbert Baumslag|last2=Gersten|first2=S.M.|last3=Shapiro|first3=Michael|last4=Short|first4=H.|doi-access=}}
  • {{cite journal|doi=10.2307/2944334|jstor=2944334|title=Rational Subgroups of Biautomatic Groups|journal=Annals of Mathematics|volume=134|issue=1|pages=125–158|year=1991|last1=Gersten|first1=S. M.|last2=Short|first2=H. B.}}
  • {{cite book|last1=Gersten|first1=S. M.|chapter=Dehn Functions and l1-norms of Finite Presentations|year=1992|pages=195–224|issn=0940-4740|doi=10.1007/978-1-4613-9730-4_9|editor=Baumslag G.|editor2=Miller C.F.|title=Algorithms and Classification in Combinatorial Group Theory|volume=23|series=Mathematical Sciences Research Institute Publications|publisher=Springer|location=New York|isbn=978-1-4613-9732-8}}
  • {{cite book|author=Gersten, S.M.|year=1993|chapter=Isoperimetric and isodiametric functions of finite presentations|title=Geometric group theory|volume=1|pages=79–96|publisher=Cambridge University Press |isbn=9780521435291|chapter-url=https://books.google.com/books?id=lK2hpcVAtX0C&pg=PA79}}

See also

References

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Category:1940 births

Category:Living people

Category:Group theorists

Category:American topologists

Category:20th-century American mathematicians

Category:21st-century American mathematicians

Category:Princeton University alumni

Category:Alumni of Trinity College, Cambridge

Category:Rice University faculty

Category:University of Utah faculty

Category:University of Illinois Urbana-Champaign faculty

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