Large Cardinals and Strong Model Theoretic Transfer Properties served as the title for the 1980 dissertation completed by Matthew Foreman at the University of California, Berkeley. Since earning his Ph.D. under the supervision of Robert M. Solovay, Foreman has established a career centered on set theory, ergodic theory, and descriptive dynamical systems.
Contributions to Set Theory
Foreman began his research in set theory, collaborating with Hugh Woodin to demonstrate the consistency of the failure of the generalized continuum hypothesis at every infinite cardinal. Working alongside Menachem Magidor and Saharon Shelah, he formulated Martin's maximum, a maximal form of Martin's axiom, and established its consistency.
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Starting in the late 1980s, Foreman expanded his focus into measure theory and ergodic theory. With Randall Dougherty, he addressed the 1930 Marczewski problem by identifying a Banach-Tarski decomposition of the unit ball where all segments possess the property of Baire. In collaboration with Benjamin Weiss and Daniel Rudolph, he proved that the isomorphism relation on ergodic measure-preserving transformations is not Borel, effectively completing a research program originally proposed by John von Neumann in 1932. Further work with Weiss showed that smooth area-preserving diffeomorphisms of the 2-torus are unclassifiable.
Professional Recognition
Foreman has held a long-standing academic appointment at the University of California, Irvine. His contributions to the mathematical community include serving as an invited speaker at the 1998 International Congress of Mathematicians in Berlin and delivering the 2021 Gödel Lecture, titled Gödel Diffeomorphisms. In 2023, the American Mathematical Society named him a fellow for his work on mathematical axioms, Banach-Tarski phenomena, and descriptive dynamical systems.
Maritime Expeditions
Beyond his mathematical pursuits, Foreman is an active sailor. Between 2000 and 2008, he and his family navigated their C&C yacht, Veritas, from North America to Europe, traversing the Arctic, the Shetland Islands, Scotland, and the Mediterranean. His sailing achievements include winning the Ullman Trophy twice and leading a 2009 expedition to circumnavigate Newfoundland.
Fast facts
- Born: 1957, Los Alamos
- Ph.D. Advisor: Robert M. Solovay
- Alma Mater: University of California, Berkeley
- Primary Field: Set Theory
- Academic Home: University of California, Irvine
- 2021 Recognition: Gödel Lecturer
- 2023 Recognition: Fellow of the American Mathematical Society
- Vessel Name: Veritas
Questions readers ask
What specific problem did Foreman resolve regarding the unit ball?
He and Randall Dougherty showed that there is a Banach-Tarski decomposition of the unit ball where all pieces have the property of Baire.
What was the significance of Foreman's work with Benjamin Weiss and Daniel Rudolph?
They proved that the isomorphism relation on ergodic measure-preserving transformations is not Borel, finishing a program proposed by von Neumann in 1932.
Achievements
- Held posts at University of California, Irvine
- Fields: set theory
