Woodin has demonstrated that, in the presence of large cardinals, there is a single model of ZFC which is maximal for Π₂-sentences over the structure (H(ω₂), ∈, NS_ω1), in the sense that its (H(ω₂), ∈, NS_ω1) satisfies every Π₂-sentence σ for which (H(ω₂), ∈, NS_ω1) ⊨ σ can be forced by set-forcing. In this paper we answer a question of Woodin by showing that there are two Π₂-sentences over the structure (H(ω₂), ∈, ω₁) which can each be forced to hold along with the continuum hypothesis, but whose conjunction implies $ {2^{{{\aleph_0}}}}={2^{{{\aleph_1}}}} $. In the process we establish that there are two preservation theorems for not introducing new real numbers by a countable support iterated forcing which cannot be subsumed into a single preservation theorem.

Let k be an algebraically closed field. We show that the Cremona group of all birational transformations of the projective plane $ \mathbb{P}_{\mathbf{k}}^2 $ is not a simple group. The strategy makes use of hyperbolic geometry, geometric group theory and algebraic geometry to produce elements in the Cremona group that generate non-trivial normal subgroups.

We consider Thurston maps, i.e., branched covering maps f:S² → S² that are post-critically finite. In addition, we assume that f is expanding in a suitable sense. It is shown that each sufficiently high iterate F = fⁿ of f is semi-conjugate to z^d: S¹ → S¹, where d = deg F. More precisely, for such an F we construct a Peano curve γ: S¹ → S² (onto), such that F∘γ(z) = γ(z^d) (for all z∈S¹).

The purpose of this article is to prove that the forcing axiom for completely proper forcings is inconsistent with the continuum hypothesis. This answers a longstanding problem of Shelah.

In 1971, McMullen and Walkup posed the following conjecture, which is called the generalized lower bound conjecture: If P is a simplicial d-polytope then its h-vector (h₀, h₁, …, h_d) satisfies $ {h_0}\leq {h_1}\leq \ldots \leq {h_{{\left\lfloor {{d \left/ {2} \right.}} \right\rfloor }}} $. Moreover, if h_r−1 = h_r for some $ r\leq \frac{1}{2}d $ then P can be triangulated without introducing simplices of dimension ≤d − r.

The first part of the conjecture was solved by Stanley in 1980 using the hard Lefschetz theorem for projective toric varieties. In this paper, we give a proof of the remaining part of the conjecture. In addition, we generalize this result to a certain class of simplicial spheres, namely those admitting the weak Lefschetz property.