Frontmatter -- 1 Introduction -- 2 Preliminaries -- 3 The nonstationary ideal -- 4 The ℙmax-extension -- 5 Applications -- 6 ℙmax variations. 6.1 2ℙmax -- 6 ℙmax variations. 6.2 Variations for obtaining ω1-dense ideals. 6.2.1 ℚmax -- 6 ℙmax variations. 6.2 Variations for obtaining ω1-dense ideals. 6.2.2 ℚ*max -- 6 ℙmax variations. 6.2 Variations for obtaining ω1-dense ideals. 6.2.3 2ℚmax -- 6 ℙmax variations. 6.2 Variations for obtaining ω1-dense ideals. 6.2.4 Weak Kurepa trees and ℚmax -- 6 ℙmax variations. 6.2 Variations for obtaining ω1-dense ideals. 6.2.5 KTℚmax -- 6 ℙmax variations. 6.2 Variations for obtaining ω1-dense ideals. 6.2.6 Null sets and the nonstationary ideal -- 6 ℙmax variations. 6.3 Nonregular ultrafilters on ω1 -- 7 Conditional variations -- 8 ♣ principles for ω1. 8.1 Condensation Principles -- 8 ♣ principles for ω1. 8.2 ℙ♣NSmax -- 8 ♣ principles for ω1. 8.3 The principles, ♣+NS and ♣++NS -- 9 Extensions of L(Γ, ℝ). 9.1 AD+ -- 9 Extensions of L(Γ, ℝ). 9.2 The ℙmax-extension of L(Γ, ℝ) -- 9 Extensions of L(Γ, ℝ). 9.3 The ℚmax-extension of L(Γ, ℝ) -- 9 Extensions of L(Γ, ℝ). 9.4 Chang's Conjecture -- 9 Extensions of L(Γ, ℝ). 9.5 Weak and Strong Reflection Principles -- 9 Extensions of L(Γ, ℝ). 9.6 Strong Chang's Conjecture -- 9 Extensions of L(Γ, ℝ). 9.7 Ideals on ω2 -- 10 Further results. 10.1 Forcing notions and large cardinals -- 10 Further results. 10.2 Coding into L(P(ω1)) -- 10 Further results. 10.3 Bounded forms of Martin's Maximum -- 10 |