Matthew Foreman
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Matthew Dean Foreman is an American mathematician at
University of California, Irvine The University of California, Irvine (UCI or UC Irvine) is a public land-grant research university in Irvine, California. One of the ten campuses of the University of California system, UCI offers 87 undergraduate degrees and 129 graduate and p ...
. He has made notable contributions in
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
and in ergodic theory.


Biography

Born in
Los Alamos, New Mexico Los Alamos is an census-designated place in Los Alamos County, New Mexico, United States, that is recognized as the development and creation place of the atomic bomb—the primary objective of the Manhattan Project by Los Alamos National Labo ...
, Foreman earned his
Ph.D. A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is ...
from the University of California, Berkeley in 1980 under Robert M. Solovay. His dissertation title was ''Large Cardinals and Strong Model Theoretic Transfer Properties''. 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 Scotland (, ) is a Countries of the United Kingdom, country that is part of the United Kingdom. Covering the northern third of the island of Great Britain, mainland Scotland has a Anglo-Scottish border, border with England to the southeast ...
,
Ireland Ireland ( ; ga, Éire ; Ulster Scots dialect, Ulster-Scots: ) is an island in the Atlantic Ocean, North Atlantic Ocean, in Northwestern Europe, north-western Europe. It is separated from Great Britain to its east by the North Channel (Grea ...
,
England England is a country that is part of the United Kingdom. It shares land borders with Wales to its west and Scotland to its north. The Irish Sea lies northwest and the Celtic Sea to the southwest. It is separated from continental Europe b ...
,
France France (), officially the French Republic ( ), is a country primarily located in Western Europe. It also comprises of overseas regions and territories in the Americas and the Atlantic, Pacific and Indian Oceans. Its metropolitan area ...
,
Spain , image_flag = Bandera de España.svg , image_coat = Escudo de España (mazonado).svg , national_motto = ''Plus ultra'' (Latin)(English: "Further Beyond") , national_anthem = (English: "Royal March") , i ...
,
North Africa North Africa, or Northern Africa is a region encompassing the northern portion of the African continent. There is no singularly accepted scope for the region, and it is sometimes defined as stretching from the Atlantic shores of Mauritania in ...
and
Italy Italy ( it, Italia ), officially the Italian Republic, ) or the Republic of Italy, is a country in Southern Europe. It is located in the middle of the Mediterranean Sea, and its territory largely coincides with the homonymous geographical ...
. Notable high points were
Fastnet Rock Fastnet Lighthouse is a 54m high lighthouse situated on the remote Fastnet Rock in the Atlantic Ocean. It is the most southerly point of Ireland and lies southwest of Cape Clear Island and from County Cork on the Irish mainland. The current l ...
, Irish and Celtic seas and many passages including the
Maelstrom Maelstrom may refer to: * Maelstrom (whirlpool), a powerful whirlpool ** originally the Moskstraumen in English Amusement rides * Maelstrom (ride), a former log flume dark ride attraction in the Epcot theme park at Walt Disney World Resort ...
, Stad,
Pentland Firth The Pentland Firth ( gd, An Caol Arcach, meaning the Orcadian Strait) is a strait which separates the Orkney Islands from Caithness in the north of Scotland. Despite the name, it is not a firth. Etymology The name is presumed to be a corrup ...
,
Loch Ness Loch Ness (; gd, Loch Nis ) is a large freshwater loch in the Scottish Highlands extending for approximately southwest of Inverness. It takes its name from the River Ness, which flows from the northern end. Loch Ness is best known for claim ...
, the Corryveckan and the Irish Sea. Further south they sailed through the Chenal du Four and
Raz de Sein Raz may refer to: Places France *Pointe du Raz, the western point of the ''commune'' of Plogoff, Finistère, France * Alderney Race (''Raz Blanchard''), a strong tidal current between La Hague and Alderney * Raz de Sein, a stretch of water located ...
, across the Bay of Biscay and around Cape Finisterre. After entering
Gibraltar ) , anthem = " God Save the King" , song = " Gibraltar Anthem" , image_map = Gibraltar location in Europe.svg , map_alt = Location of Gibraltar in Europe , map_caption = United Kingdom shown in pale green , mapsize = , image_map2 = Gib ...
, Foreman and his family circumnavigated the Western Mediterranean. Some notable stops included:
Barcelona Barcelona ( , , ) is a city on the coast of northeastern Spain. It is the capital and largest city of the autonomous community of Catalonia, as well as the second most populous municipality of Spain. With a population of 1.6 million within ci ...
,
Morocco Morocco (),, ) officially the Kingdom of Morocco, is the westernmost country in the Maghreb region of North Africa. It overlooks the Mediterranean Sea to the north and the Atlantic Ocean to the west, and has land borders with Algeria t ...
,
Tunisia ) , image_map = Tunisia location (orthographic projection).svg , map_caption = Location of Tunisia in northern Africa , image_map2 = , capital = Tunis , largest_city = capital , ...
,
Sicily (man) it, Siciliana (woman) , population_note = , population_blank1_title = , population_blank1 = , demographics_type1 = Ethnicity , demographics1_footnotes = , demographi ...
,
Naples Naples (; it, Napoli ; nap, Napule ), from grc, Νεάπολις, Neápolis, lit=new city. is the regional capital of Campania and the third-largest city of Italy, after Rome and Milan, with a population of 909,048 within the city's adminis ...
,
Sardinia Sardinia ( ; it, Sardegna, label=Italian, Corsican and Tabarchino ; sc, Sardigna , sdc, Sardhigna; french: Sardaigne; sdn, Saldigna; ca, Sardenya, label=Algherese and Catalan) is the second-largest island in the Mediterranean Sea, after ...
and Corsica. In 2009 Foreman, his son with guest members as crew, circumnavigated Newfoundland. Foreman has been recognized for his sailing by twice winning the Ullman Trophy.


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 In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that or equivalently, that In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...
) fails at every infinite cardinal. In joint work with
Menachem Magidor Menachem Magidor (Hebrew: מנחם מגידור; born January 24, 1946) 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 t ...
and Saharon Shelah he formulated Martin's maximum, a provably maximal form of
Martin's axiom In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consist ...
and showed its consistency. Foreman's later work in set theory was primarily concerned with developing the consequences of generic large cardinal axioms. He also worked on classical "Hungarian" partition relations, mostly with
András Hajnal András Hajnal (May 13, 1931 – July 30, 2016) was a professor of mathematics at Rutgers University and a member of the Hungarian Academy of Sciences known for his work in set theory and combinatorics. Biography Hajnal was born on 13 May 1931, ...
. 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 A subset A of a topological space X has the property of Baire (Baire property, named after René-Louis Baire), or is called an almost open set, if it differs from an open set by a meager set; that is, if there is an open set U\subseteq X such th ...
(see
Banach–Tarski paradox 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 p ...
). 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 The Hahn–Banach theorem is a central tool in functional analysis. It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
implied the existence of a non-Lebesgue measurable set, even in the absence of any other form of the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
. 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, showed that classical collections were beyond the
Borel hierarchy In mathematical logic, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets. Each Borel set is assigned a unique countable ordinal number called ...
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 and Daniel Rudolph 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. 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 cardinal In mathematics, a cardinal number κ is called huge if there exists an elementary embedding ''j'' : ''V'' → ''M'' from ''V'' into a transitive inner model ''M'' with critical point κ and :^M \subset M.\! Here, ''αM'' is the class of al ...
s.


Recognition

In 1998 Foreman was an Invited Speaker of the International Congress of Mathematicians in Berlin. In 2021, he gave the
Gödel Lecture 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. ...
titled ''Gödel Diffeomorphisms.'' He was named to the 2023 class of Fellows 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, ...
, "for contributions to axioms of mathematics, Banach-Tarski phenomena, and descriptive dynamical systems".


References

{{DEFAULTSORT:Foreman, Matthew American logicians 20th-century American mathematicians 21st-century American mathematicians 1957 births Living people University of California, Berkeley alumni University of California, Irvine faculty Set theorists People from Los Alamos, New Mexico Gödel Lecturers Fellows of the American Mathematical Society