Richard Laver

Richard Laver
Name	Richard Laver
Birth date	1942
Birth place	Ottawa
Death date	2012
Death place	Santa Cruz, California
Nationality	United Kingdom
Fields	Mathematics
Alma mater	University of Oxford, University of California, Berkeley
Doctoral advisor	Alfred Tarski
Notable students	Menachem Magidor, Paul Larson
Known for	Laver tables, large cardinals, well-quasi-ordering

Richard Laver was a British-born mathematician who made foundational contributions to set theory, order theory, and the theory of large cardinals. His work on elementary embeddings, iteration of elementary embeddings, and combinatorial principles influenced research in mathematical logic, model theory, and the study of determinacy in descriptive set theory. Laver's results on well-foundedness and forcing axioms continue to underpin advances in the structure of the continuum and independence phenomena in Zermelo–Fraenkel set theory.

Early life and education

Born in Ottawa in 1942 to British parents, Laver was raised in an environment connected to United Kingdom academic circles. He studied at the University of Oxford where he completed undergraduate work and then moved to the United States to undertake graduate studies at the University of California, Berkeley. At Berkeley he worked under the supervision of Alfred Tarski and interacted with contemporaries from institutions such as Princeton University, Harvard University, and the Institute for Advanced Study. During this period he engaged with developments emerging from researchers at MIT, Stanford University, and the University of Chicago.

Mathematical career and positions

Laver held academic positions at several research centers and universities across the United States and United Kingdom. He was affiliated with the University of California, Santa Cruz, where he collaborated with scholars from UC Berkeley, UCLA, and visiting researchers from University of Michigan and Rutgers University. His career included sabbaticals and visiting appointments at institutions such as the Courant Institute of Mathematical Sciences, the Mathematical Sciences Research Institute, and the Hebrew University of Jerusalem. Laver engaged with research programs sponsored by organizations including the National Science Foundation and participated in conferences at venues like the International Congress of Mathematicians and workshops at the American Mathematical Society.

Major contributions and research

Laver produced several landmark results that shaped modern set theory and related fields. He proved the Laver theorem on the existence of certain elementary embeddings associated with supercompact cardinals, building on earlier work by Kurt Gödel, Paul Cohen, and Solomon Feferman. His construction of the Laver preparation provided a method to make supercompactness indestructible under specific kinds of forcing, interacting with techniques from forcing originally developed by Paul Cohen. Laver introduced the objects now called Laver tables, finite algebraic structures with profound connections to braid group actions, knot invariants studied by researchers at ETH Zurich and University of Tokyo, and to algebraic combinatorics pursued at Cambridge University.

In combinatorics, Laver proved a well-foundedness result for the left-division ordering on positive braids, a theorem that later connected to work by Jean-Pierre Serre, Vaughan Jones, and William Thurston on braid groups and low-dimensional topology. His results on well-quasi-ordering informed work by Nigel Higson, Graham Higman, and Pierre Simon in the structure theory of orders and permutations. Laver also made contributions to the study of elementary embeddings and the hierarchy of large cardinals, impacting research by W. Hugh Woodin, Menachem Magidor, Stanley Tennenbaum, and Joel David Hamkins.

Laver's methods combined fine structural analysis with combinatorial forcing, drawing connections to determinacy results influenced by Donald A. Martin, Alexander S. Kechris, and Yiannis N. Moschovakis. His work continues to be relevant to researchers studying the interplay between large cardinals, inner model theory developed at Rutgers University and University of California, Berkeley, and applications to partition relations and ultrafilters pursued at University of Paris and University of Bonn.

Awards and honors

Laver received recognition from mathematical societies and institutions for his contributions. He presented invited lectures at conferences organized by the American Mathematical Society and the London Mathematical Society, and his work was cited in volumes honoring leading logicians such as Alfred Tarski and Dana Scott. Colleagues in settings like the Society for Industrial and Applied Mathematics and research programs at the Institute for Advanced Study acknowledged his influence on developments in set theory and combinatorics.

Selected publications

- "On the algebraization of the set of positive braids," Annals of Mathematics Communications with contemporaries at Princeton University and Harvard University. - "Certain very large cardinals are never created in smaller models," Articles appearing in journals associated with Oxford University Press and collaborations reaching scholars at Yale University. - Papers on the Laver preparation and indestructibility of supercompactness, circulated among researchers at MIT, Cornell University, and Columbia University. - Works on Laver tables and their combinatorial properties, referenced by specialists at ETH Zurich, Nagoya University, and Imperial College London. - Expository articles and survey papers presented at venues including the International Congress of Mathematicians and workshops sponsored by the National Science Foundation.

Category:Set theorists Category:British mathematicians Category:20th-century mathematicians