Abstract We show that every locally flat topological embedding of a 3-manifold in a smooth 5-manifold is homotopic, by a small homotopy, to a smooth embedding. We deduce that topologically locally flat concordance implies smooth concordance for smooth surfaces in smooth 4-manifolds.

Abstract We show that for any quadratic extension of number fields $K/F$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> <mml:mo>/</mml:mo> <mml:mi>F</mml:mi> </mml:math> , there exists an abelian variety $A/F$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> <mml:mo>/</mml:mo> <mml:mi>F</mml:mi> </mml:math> of positive rank whose rank does not grow upon base change to $K$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> . This result implies that Hilbert’s tenth problem over the ring of integers of any number field has a negative solution. That is, for the ring ${\mathcal{O}}_{K}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:math> of integers of any number field $K$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> , there does not exist an algorithm that answers the question of whether a polynomial equation in several variables over ${\mathcal{O}}_{K}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:math> has solutions in ${\mathcal{O}}_{K}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:math> .

Abstract We prove that for every $n\in \mathbb{N}$ n ∈ N such that $n\geq 2$ n ≥ 2 , the reduced group $C^{*}$ C ∗ -algebras of the countable free groups $C^{*}_{r}(\mathbb{F}_{n})$ C r ∗ ( F n ) have strict comparison. Our method works in a general setting: for every finitely generated acylindrically hyperbolic group $G$ G with trivial finite radical and the rapid decay property, we have $C^{*}_{r}(G)$ C r ∗ ( G ) have strict comparison. This work also has several applications in the theory of $C^{*}$ C ∗ -algebras including: resolving Leonel Robert’s selflessness problem for $C^{*}_{r}(G)$ C r ∗ ( G ) ; uniqueness of embeddings of the Jiang-Su algebra $\mathcal{Z}$ Z up to approximate unitary equivalence into $C^{*}_{r}(G)$ C r ∗ ( G ) ; full computations of the Cuntz semigroup of $C^{*}_{r}(G)$ C r ∗ ( G ) and future directions in the $C^{*}$ C ∗ -classification program.

Abstract Cutting a hyperbolic surface $X$ X along a simple closed multi-geodesic results in a hyperbolic structure on the complementary subsurface. We study the distribution of the shapes of these subsurfaces in moduli space as boundary lengths go to infinity, showing that they equidistribute to the Kontsevich measure on a corresponding moduli space of metric ribbon graphs. In particular, random subsurfaces look like random ribbon graphs, a law which does not depend on the initial choice of $X$ X . This result strengthens Mirzakhani’s famous simple closed multi-geodesic counting theorems for hyperbolic surfaces.