We are sorry to share the sad news that James E. Baumgartner, one of the greatest set theorists, died of a heart attack on December 28. Jim was born in 1943 and started as a student at Caltech and then moved to Berkeley where he received his PhD under the supervision of Robert Vaught in 1970. He spent most of his career at Dartmouth College where he trained numerous PhD students, among them Jean Larson, Tadatoshi Miyamoto, Alan Taylor and Stan Wagon, and postdocs, including Alessandro Andretta, Jorg Brendle, Elizabeth Brown, James Cummings, Jean-Pierre Levinski, Stevo Todorcevic and Jindrich Zapletal.
Baumgartner’s mathematical career started with a wonderful proof of the consistency that every Aronszajn tree is special (independently obtained by Reindhart and Malitz, and published as a joint paper). His later consistency result that all aleph_1 dense sets of reals can be isomorphic initiated a long series of important contributions in this direction. His joint work with Hajnal to prove omega_1 arrows (alpha)^2_n, for all countable alpha and finite n using the methods of
forcing and absolutness was a real breakthrough that has become known as the Baumgartner-Hajnal Theorem.
Together with Laver, Baumgartner studied countable support iterations of Sacks forcing and later isolated an important class of forcing notions which can be iterated without collapsing \aleph_1: Axiom A forcings. Following the work of Shelah on proper forcing, Baumgartner formulated and proved the relative consistency of the Proper Forcing Axiom, which has become one of the most studied additional axioms of set theory. In addition, he developed general techniques for building proper partial orders by using side conditions and obtained numerous consequences of PFA. In all his papers, Jim showed extreme originality and clarity. In addition he wrote many survey papers which have inspired young set theorists: his introduction to iterated forcing is still one of the basic references in this area, as well as his beautiful surveys on uncountable linear orders and on applications of the Proper Forcing Axiom.
In spite of his great talent and achievements he never put his own work in front of that of the others and generously helped all who came across him as students and colleagues. His mathematical genius and humanity were a constant inspiration for young mathematicians following in his footsteps. He will be deeply missed by his friends and colleagues.
[This post prepared by Mirna Dzamonja, Jean Larson and Boban Velickovic].