Matthew Dean Foreman | |
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| Born | |
| Nationality | American |
| Alma mater | University of California, Berkeley |
| Scientific career | |
| Fields | Mathematics |
| Institutions | University of California, Irvine Ohio State University |
| Doctoral advisor | Robert M. Solovay |
Matthew Dean Foreman is an American mathematician at University of California, Irvine. He has made notable contributions in set theory and in ergodic theory.
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. [1]
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. [2]
Foreman has been recognized for his sailing by twice winning the Ullman Trophy. [3]
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. [4] 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. [5] [6] Foreman's later work in set theory was primarily concerned with developing the consequences of generic large cardinal axioms. [7] He also worked on classical "Hungarian" partition relations, mostly with András Hajnal. [8]
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). [9] 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. [10]
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, [11] 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 [12] and Daniel Rudolph [13] 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. [14] 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. [15]
In 1998 Foreman was an Invited Speaker of the International Congress of Mathematicians in Berlin. [16]
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". [17]
Saharon Shelah is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey.
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large". The proposition that such cardinals exist cannot be proved in the most common axiomatization of set theory, namely ZFC, and such propositions can be viewed as ways of measuring how "much", beyond ZFC, one needs to assume to be able to prove certain desired results. In other words, they can be seen, in Dana Scott's phrase, as quantifying the fact "that if you want more you have to assume more".
In mathematics, a non-measurable set is a set which cannot be assigned a meaningful "volume". The mathematical existence of such sets is construed to provide information about the notions of length, area and volume in formal set theory. In Zermelo–Fraenkel set theory, the axiom of choice entails that non-measurable subsets of exist.
In mathematical logic, a theory is categorical if it has exactly one model. Such a theory can be viewed as defining its model, uniquely characterizing the model's structure.
PCF theory is the name of a mathematical theory, introduced by Saharon Shelah (1978), that deals with the cofinality of the ultraproducts of ordered sets. It gives strong upper bounds on the cardinalities of power sets of singular cardinals, and has many more applications as well. The abbreviation "PCF" stands for "possible cofinalities".
Robert Martin Solovay is an American mathematician specializing in set theory.
In set theory, an Aronszajn tree is a tree of uncountable height with no uncountable branches and no uncountable levels. For example, every Suslin tree is an Aronszajn tree. More generally, for a cardinal κ, a κ-Aronszajn tree is a tree of height κ in which all levels have size less than κ and all branches have height less than κ. They are named for Nachman Aronszajn, who constructed an Aronszajn tree in 1934; his construction was described by Kurepa (1935).
In set theory, a branch of mathematical logic, Martin's maximum, introduced by Foreman, Magidor & Shelah (1988) and named after Donald Martin, is a generalization of the proper forcing axiom, itself a generalization of Martin's axiom. It represents the broadest class of forcings for which a forcing axiom is consistent.
Menachem Magidor is an Israeli mathematician who specializes in mathematical logic, in particular set theory. He served as president of the Hebrew University of Jerusalem, was president of the Association for Symbolic Logic from 1996 to 1998, and is currently the president of the Division for Logic, Methodology and Philosophy of Science and Technology of the International Union for History and Philosophy of Science. In 2016 he was elected an honorary foreign member of the American Academy of Arts and Sciences. In 2018 he received the Solomon Bublick Award.
The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces.
Richard Joseph Laver was an American mathematician, working in set theory.
In the mathematical field of set theory, the Solovay model is a model constructed by Robert M. Solovay (1970) in which all of the axioms of Zermelo–Fraenkel set theory (ZF) hold, exclusive of the axiom of choice, but in which all sets of real numbers are Lebesgue measurable. The construction relies on the existence of an inaccessible cardinal.
Moti Gitik is a mathematician, working in set theory, who is professor at the Tel-Aviv University. He was an invited speaker at the 2002 International Congresses of Mathematicians, and became a fellow of the American Mathematical Society in 2012.
In set theory, Jensen's covering theorem states that if 0# does not exist then every uncountable set of ordinals is contained in a constructible set of the same cardinality. Informally this conclusion says that the constructible universe is close to the universe of all sets. The first proof appeared in. Silver later gave a fine-structure-free proof using his machines and finally Magidor (1990) gave an even simpler proof.
In mathematics, iterated forcing is a method for constructing models of set theory by repeating Cohen's forcing method a transfinite number of times. Iterated forcing was introduced by Solovay and Tennenbaum (1971) in their construction of a model of set theory with no Suslin tree. They also showed that iterated forcing can construct models where Martin's axiom holds and the continuum is any given regular cardinal.
The Banach–Tarski Paradox is a book in mathematics on the Banach–Tarski paradox, the fact that a unit ball can be partitioned into a finite number of subsets and reassembled to form two unit balls. It was written by Stan Wagon and published in 1985 by the Cambridge University Press as volume 24 of their Encyclopedia of Mathematics and its Applications book series. A second printing in 1986 added two pages as an addendum, and a 1993 paperback printing added a new preface. In 2016 the Cambridge University Press published a second edition, adding Grzegorz Tomkowicz as a co-author, as volume 163 of the same series. The Basic Library List Committee of the Mathematical Association of America has recommended its inclusion in undergraduate mathematics libraries.
The Gödel Lecture is an honor in mathematical logic given by the Association for Symbolic Logic, associated with an annual lecture at the association's general meeting. The award is named after Kurt Gödel and has been given annually since 1990.
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