Sad news: Jim Baumgartner passed away. See here.
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Cardinal Invariants Diamond-sharp Weakly compact cardinal indecomposable filter Parameterized proxy principle square incompactness AIM forcing Fodor-type reflection projective Boolean algebra Countryman line weak square Hedetniemi's conjecture Rado's conjecture Prevalent singular cardinals Ascent Path stationary hitting Diamond Almost countably chromatic Forcing with side conditions tensor product graph Generalized Clubs OCA Singular cardinals combinatorics Knaster and friends stationary reflection weak diamond L-space Successor of Regular Cardinal Analytic sets Strong coloring club_AD Was Ulam right? sap free Boolean algebra Prikry-type forcing b-scale Filter reflection Forcing Ramsey theory over partitions nonmeager set xbox Shelah's Strong Hypothesis Rainbow sets specializable Souslin tree Foundations Fat stationary set positive partition relation Amenable C-sequence Singular cofinality strongly bounded groups GMA Fast club Subnormal ideal 54G20 SNR Precaliber Open Access perfectly normal Entangled linear order Local Club Condensation. Subadditive Lipschitz reduction Rock n' Roll Poset O-space Aronszajn tree regressive Souslin tree Selective Ultrafilter Cardinal function Sierpinski's onto mapping principle Sakurai's Bell inequality Non-saturation Singular Density Subtle tree property Absoluteness Closed coloring Reflecting stationary set Souslin Tree ZFC construction Successor of Singular Cardinal PFA Minimal Walks Mandelbrot set Large Cardinals countably metacompact Respecting tree Ineffable cardinal Almost-disjoint family stick Strongly Luzin set Kurepa Hypothesis Monotonically far middle diamond Vanishing levels Uniformly coherent Greatly Mahlo Axiom R Hereditarily Lindelöf space Dushnik-Miller free Souslin tree Dowker space Coherent tree C-sequence Partition Relations Hindman's Theorem Generalized descriptive set theory Subtle cardinal Diamond for trees full tree higher Baire space Chang's conjecture Knaster Cohen real ccc Luzin set approachability ideal super-Souslin tree reflection principles very good scale Almost Souslin polarized partition relation Club Guessing Constructible Universe Intersection model Postprocessing function unbounded function Square-Brackets Partition Relations Strongly compact cardinal PFA(S)[S] transformations Microscopic Approach Ulam matrix Iterated forcing diamond star Erdos Cardinal square principles Reduced Power Jonsson cardinal Small forcing Chromatic number Nonspecial tree Interval topology on trees Erdos-Hajnal graphs Distributive tree Commutative projection system Ascending path Whitehead Problem coloring number Forcing Axioms Universal Sequences Uniformization weak Kurepa tree P-Ideal Dichotomy Well-behaved magma Martin's Axiom Antichain S-Space Uniformly homogeneous Commutative cancellative semigroups Ostaszewski square Slim tree HOD Sigma-Prikry
That’s very sad news indeed.
That is sad.
(*) I will always associate Jim with the phrase “Just Baumgartner it!”. Simon Thomas’s word for bringing in an ultrafilter, seemingly for no reason, and letting it guide your decisions.
(**) I would also like to note that Jim shares his first two names with the actor who portrayed Darth Vader.
This is very sad news indeed. The email sent around Dartmouth said that he passed away while his whole family was visiting for the holidays, so Jim was fortunate to spend his last moments surrounded by family and high spirits.
Baumgartner proved that, in contrast to (weak) square, partial square may hold at a singular cardinal (of any cofinality) above a supercomapct carinal. He also proved that after collapsing a weakly comapct cardinal to $omega_2$, every statinoary subset of $S^2_0$ reflects.
These two gems together with an additional argument shows that $square^*_{aleph_{omega_1}}$ may be introduced by a cofinality-preserving small forcing.