Related papers: The Vaserstein symbol on real smooth affine threef…

We show that the Thurston seminorms of all finite covers of an aspherical 3-manifold determine whether it is a graph manifold, a mixed 3-manifold or hyperbolic.

We study the interactions between toric manifolds and Weinstein handlebodies. We define a partially-centered condition on a Delzant polytope, which we prove ensures that the complement of a corresponding partial smoothing of the toric…

We study the affine quasi-Einstein Equation for homogeneous surfaces. This gives rise through the modified Riemannian extension to new half conformally flat generalized quasi-Einstein neutral signature $(2,2)$ manifolds, to conformally…

Geodesically complete affine manifolds are quotients of the Euclidean space through a properly discontinuous action of a subgroup of affine Euclidean transformations. An equivalent definition is that the tangent bundle of such a manifold…

About a decade ago Thurston proved that a vast collection of 3-manifolds carry metrics of constant negative curvature. These manifolds are thus elements of {\em hyperbolic geometry}, as natural as Euclid's regular polyhedra. For a closed…

We adapt Seifert's algorithm for classical knots and links to the setting of tri-plane diagrams for bridge trisected surfaces in the 4-sphere. Our approach allows for the construction of a Seifert solid that is described by a Heegaard…

Let $X$ be a Gorenstein minimal projective 3-fold with at worst locally factorial terminal singularities. Suppose the canonical map is of fiber type. Denote by $F$ a smooth model of a generic irreducible component in fibers of the canonical…

We show injectivity of the geodesic X-ray transform on piecewise constant functions when the transform is weighted by a continuous matrix weight. The manifold is assumed to be compact and nontrapping of any dimension, and in dimension three…

We prove that a smooth rationally connected projective threefold of Picard number two is toric if and only if it admits an int-amplified endomorphism. As a corollary, we show that a totally invariant smooth curve of a non-isomorphic…

This is the third of a series of papers studying real algebraic threefolds, but the methods are mostly independent from the previous two. Let $f:X\to S$ be a map of a smooth projective real algebraic 3-fold to a surface $S$ whose general…

Let $X$ be a projective minimal Gorenstein 3-fold of general type with canonical singularities. We prove that the 5-canonical map is birational onto its image.

Consider a smooth $4$-manifold $X$ and a diffeomorphism $f : X \to X$. We give an obstruction in the form of an adjunction inequality for an embedded surface in $X$ to be isotopic to its image under $f$. It follows that the minimal genus of…

Affine manifolds are called integral if there is an atlas such that all transition maps are affine transformations with integer matrices of linear parts. In this paper we describe all complete integral affine structures on compact…

Let M be a compact connected special affine manifold equipped with an affine Gauduchon metric. We show that a pair (E, \phi), consisting of a flat vector bundle E over M and a flat nonzero section \phi\ of E, admits a solution to the vortex…

A smooth variety is said to satisfy Condition (A) if every finite abelian subgroup of its automorphism group has a fixed point. We classify smooth Fano 3-folds that satisfy Condition (A).

Given an orientable ideally triangulated $3$--manifold $M$, we define a system of real valued equations and inequalities whose solutions can be used to construct projective structures on $M$. These equations represent a unifying framework…

We study the structure induced on a smooth manifold by a continuous selection of smooth functions. In case such selection is suitably generic, it provides a stratification of the manifold, whose strata are algebraically defined smooth…

Given a smooth, compact four-manifold $X$ viewed as a cobordism from the empty set to its connected boundary, we demonstrate how to use the data of a trisection map $\pi:X^{4} \to \mathbb{D}^{2}$ to compute the induced cobordism maps on…

As a tool to address the equivalence problem in sub-Riemannian geometry, we introduce a canonical choice of grading and compatible affine connection, available on any sub-Riemannian manifold with constant symbol. We completely compute these…

For any asymptotically dynamically convex contact manifold $Y$, we show that $SH_*(W)=0$ is a property independent of the choice of topologically simple (i.e.\ $c_1(W)=0$ and $\pi_{1}(Y)\rightarrow \pi_1(W)$ is injective) Liouville filling…