Matthew Foreman
{{short description|American mathematician}}
{{other people|Matt Foreman}}
{{Infobox scientist
| name = Matthew Dean Foreman
| image = Matt_Foreman.jpg
| birth_date =
| birth_place = Los Alamos, New Mexico, US
| death_date =
| death_place =
| nationality = American
| fields = Mathematics
| workplaces = University of California, Irvine
Ohio State University
| alma_mater = University of California, Berkeley
| doctoral_advisor = Robert M. Solovay
| known_for =
| awards =
}}
Matthew Dean Foreman is an American mathematician at
University of California, Irvine. He has made notable contributions in set theory and in ergodic theory.
Biography
Born in Los Alamos, New Mexico, Foreman earned his Ph.D. from the
University of California, Berkeley in 1980 under Robert M. Solovay. His
dissertation title was Large Cardinals and Strong Model Theoretic Transfer Properties.{{cite journal|jstor=1998706|title=Large Cardinals and Strong Model Theoretic Transfer Properties|last1=Foreman|first1=Matthew|journal=Transactions of the American Mathematical Society|year=1982|volume=272|issue=2|pages=427–463|doi=10.1090/S0002-9947-1982-0662045-X|doi-access=free}}
In addition to his mathematical work, Foreman is an avid sailor.
He and his family sailed their sailboat Veritas (a built by C&C Yachts) from North America to Europe in 2000. From 2000–2008 they sailed Veritas to the Arctic, the Shetland Islands, Scotland, Ireland, England, France, Spain, North Africa and Italy.
Notable high points were Fastnet Rock, Irish and Celtic seas and many passages including the
Maelstrom, Stad, Pentland Firth, Loch Ness, the Corryveckan and the Irish Sea.
Further south they sailed through the Chenal du Four and Raz de Sein, across the Bay of Biscay and around Cape Finisterre. After entering Gibraltar, Foreman and his family circumnavigated the Western Mediterranean. Some notable stops included: Barcelona, Morocco, Tunisia, Sicily, Naples, Sardinia and Corsica. In 2009 Foreman, his son with guest members as crew, circumnavigated Newfoundland.Foreman, Zachary (2007) "Under Way", Cruising World Magazine, October 2007
Foreman has been recognized for his sailing by twice winning the Ullman Trophy.Tailwind, Balboa Yacht Club "Annual Awards", 2003, 2011
Work
Foreman began his career in set theory. His early work with Hugh Woodin included showing that it is consistent that the generalized continuum hypothesis (see continuum hypothesis) fails at every infinite cardinal.{{cite journal
| last1=Foreman | first1=Matthew
| last2=Woodin | first2=W. Hugh | authorlink2=W. Hugh Woodin
| title=The generalized continuum hypothesis can fail everywhere
| journal=Annals of Mathematics
| series=Second Series
| volume=133
| date=1991
| issue=1
| pages=1–35
| doi=10.2307/2944324 | jstor=2944324}} In joint work with Menachem Magidor and Saharon Shelah he formulated Martin's maximum, a provably maximal form of Martin's axiom and showed its consistency.{{cite journal
| last1=Foreman | first1=Matthew
| last2=Magidor | first2=Menachem | authorlink2=Menachem Magidor
| last3=Shelah | first3=Saharon | authorlink3=Saharon Shelah
| title=Martin's maximum, saturated ideals, and nonregular ultrafilters. I
| journal=Annals of Mathematics
| series=Second Series
| volume=127
| date=1988
| issue=1
| pages=1–47
| doi=10.2307/1971415
| jstor=1971415}}{{cite journal
| last1=Foreman | first1=Matthew
| last2=Magidor | first2=Menachem | authorlink2=Menachem Magidor
| last3=Shelah | first3=Saharon | authorlink3=Saharon Shelah
| title=Martin's maximum, saturated ideals and nonregular ultrafilters. II
| journal=Annals of Mathematics
| series=Second Series
| volume=127
| date=1988
| issue=3
| pages=521–545
| doi=10.2307/2007004
| jstor=2007004}} Foreman's later work in set theory was primarily concerned with developing the consequences of generic large cardinal axioms.{{cite book
| last1=Foreman | first1=Matthew
| contribution=Ideals and generic elementary embeddings
| title=Handbook of Set Theory
| pages=885–1147
| publisher=Springer
| date=2010
| doi=10.1007/978-1-4020-5764-9_14 | doi-access=free}} He also worked on classical "Hungarian" partition relations, mostly with András Hajnal.{{cite journal
| last1=Foreman | first1=Matthew
| last2=Hajnal | first2=András | authorlink2=András Hajnal
| title=A partition relation for successors of large cardinals
| journal=Mathematische Annalen
| volume=325
| date=2003
| issue=3
| pages=583–623
| doi=10.1007/s00208-002-0323-7 | doi-access=}}
In the late 1980s Foreman became interested in measure theory and ergodic theory. With Randall Dougherty he settled the Marczewski problem (1930) by showing that there is a Banach–Tarski decomposition of the unit ball in which all pieces have the property of Baire (see Banach–Tarski paradox).{{cite journal
| last1=Dougherty | first1=Randall | authorlink1=Randall Dougherty
| last2=Foreman | first2=Matthew
| title=Banach–Tarski decompositions using sets with the property of Baire
| journal=Journal of the American Mathematical Society
| volume=7
| date=1994
| issue=1
| pages=75–124
| doi=10.1090/S0894-0347-1994-1227475-8 | doi-access=free}} A consequence is the existence of a decomposition of an open dense subset of the unit ball into disjoint open sets that can be rearranged by isometries to form two open dense subsets of the unit ball. With Friedrich Wehrung, Foreman showed that the Hahn–Banach theorem implied the existence of a non-Lebesgue measurable set, even in the absence of any other form of the axiom of choice.{{cite journal
| last1=Foreman | first1=Matthew
| last2=Wehrung | first2=Friedrich
| title=The Hahn–Banach theorem implies the existence of a non-Lebesgue measurable set
| journal=Fundamenta Mathematicae
| volume=138
| date=1991
| issue=1
| pages=13–19
| doi=10.4064/fm-138-1-13-19 | doi-access=free}}
This naturally led to attempts to apply the tools of descriptive set theory to classification problems in ergodic theory. His first work in this direction, with Ferenc Beleznay,{{cite journal
| last1=Beleznay | first1=Ferenc
| last2=Foreman | first2=Matthew
| title=The collection of distal flows is not Borel
| journal=American Journal of Mathematics
| volume=117
| date=1995
| issue=1
| pages=203–239
| doi=10.2307/2375041
| jstor=2375041}} showed that classical collections were beyond the Borel hierarchy in complexity. This was followed shortly by a proof of the analogous results for measure-preserving transformations with generalized discrete spectrum. In a collaboration with Benjamin Weiss{{cite journal
| last1=Foreman | first1=Matthew
| last2=Weiss | first2=Benjamin | authorlink2=Benjamin Weiss
| title=An anti-classification theorem for ergodic measure-preserving transformations
| journal=Journal of the European Mathematical Society
| volume=6
| date=2004
| issue=3
| pages=277–292
| doi=10.4171/JEMS/10 | doi-access=free}} and Daniel Rudolph{{cite journal | last1=Foreman | first1=Matthew | last2=Rudolph | first2=Daniel | last3=Weiss | first3=Benjamin | author-link2=Daniel Rudolph |author-link3=Benjamin Weiss |title=The conjugacy problem in ergodic theory | journal=Annals of Mathematics | series=Second Series | volume=173 | issue=3 | date=May 1, 2011 | issn=0003-486X | doi=10.4007/annals.2011.173.3.7 | pages=1529–1586| doi-access=free }} Foreman showed that no residual class of measure-preserving transformations can have algebraic invariants and that the isomorphism relation on ergodic measure-preserving transformations is not Borel. This negative result finished a program proposed by von Neumann in 1932.{{cite journal
| last1=von Neumann | first1=J. | authorlink1=John von Neumann
| title=Zur Operatorenmethode in der klassischen Mechanik
| journal=Annals of Mathematics
| series=Second Series
| volume=33
| issue=3
| pages=587–642
| date=1932
| doi=10.2307/1968537
| jstor=1968537}} This result was extended by Foreman and Weiss to show that smooth area-preserving diffeomorphisms of the 2-torus are unclassifiable.
Foreman's work in set theory continued during this period. He co-edited (with Kanamori) the Handbook of Set Theory and showed that various combinatorial properties of ω2 and ω3 are equiconsistent with huge cardinals.{{cite journal
| last1=Foreman | first1=Matthew
| title=Smoke and mirrors: combinatorial properties of small cardinals equiconsistent with huge cardinals
| journal=Advances in Mathematics
| volume=222
| date=2009
| issue=2
| pages=565–595
| doi=10.1016/j.aim.2009.05.006 | doi-access=free}}
Recognition
In 1998 Foreman was an Invited Speaker of the International Congress of Mathematicians in Berlin.{{cite book
| last1=Foreman | first1=Matthew
| chapter=Generic large cardinals: New axioms for mathematics?
| title=Documenta Mathematica (Bielefeld), Extra Volume ICM Berlin
| date=1998
| volume=II
| pages=11–21
| chapter-url=https://www.elibm.org/ft/10011637000}}
In 2021, he gave the Gödel Lecture titled Gödel Diffeomorphisms.
He was named to the 2023 class of Fellows of the American Mathematical Society, "for contributions to axioms of mathematics, Banach-Tarski phenomena, and descriptive dynamical systems".{{cite web|url=http://www.ams.org/cgi-bin/fellows/fellows_by_year.cgi|title=2023 Class of Fellows|publisher=American Mathematical Society|access-date=2022-11-09}}
References
{{Reflist}}
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Category:20th-century American mathematicians
Category:21st-century American mathematicians
Category:University of California, Berkeley alumni
Category:University of California, Irvine faculty