Related papers: Convex real projective structures and Weil's local…

We study a class of semialgebraic convex bodies called discotopes. These are instances of zonoids, objects of interest in real algebraic geometry and random geometry. We focus on the face structure and on the boundary hypersurface of…

Let $(M, \dr M)$ be a 3-manifold with incompressible boundary that admits a convex co-compact hyperbolic metric. We consider the hyperbolic metrics on $M$ such that $\dr M$ looks locally like a hyperideal polyhedron, and we characterize the…

We study the geometry and spectral theory of Weil-Petersson random surfaces with genus-$g$ and $n$ cusps in the large-$n$ limit. We show that for a random hyperbolic surface in $\mathcal{M}_{g,n}$ with $n$ large, the number of small…

We prove that, if $W \subset \mathbb{R}^n$ is a locally strongly convex body (not necessarily compact), then for any open set $V \supset \partial W$ and $\varepsilon>0$, and $V \supset \partial W$ is open, then there exists a $C^2$ locally…

In 1970, Kobayashi conjectured that general hypersurfaces of sufficiently large degree in $P^n$ are hyperbolic. In this paper we prove that a general sufficiently ample hypersurface in a smooth projective variety is hyperbolic. To prove…

Let $R$ be a hypersurface in an equicharacteristic or unramified regular local ring. For a pair of modules $(M,N)$ over $R$ we study applications of rigidity of $\Tor^R(M,N)$, based on ideas by Huneke, Wiegand and Jorgensen. We then focus…

Cutting a hyperbolic surface 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…

In this paper, we develop the theory of singular hermitian metrics on vector bundles. As an application, we give a structure theorem of a projective manifold $X$ with pseudo-effective tangent bundle: $X$ admits a smooth fibration $X \to Y$…

We consider the self-dual vortex equations on a positive line bundle L --> M over a compact Kaehler manifold of arbitrary dimension. When M is simply connected, the moduli space of vortex solutions is a projective space. When M is an…

We investigate a question of Cooper adjacent to the Virtual Haken Conjecture. Assuming certain conjectures in number theory, we show that there exist hyperbolic rational homology 3-spheres with arbitrarily large injectivity radius. These…

We show local and cocycle rigidity for $\R^k \times \Z^l$ partially hyperbolic translation actions on homogeneous spaces $\mc G/ \Lambda$. We consider a large class of actions whose geometric properties are more complicated than previously…

We prove various results on the cohomology of arithmetic lattices arising from quaternion algebras over a number field with at least one complex place, including a strong restriction on the allowable weights of cuspidal cohomological…

Allcock and Freitag recently showed that the moduli space of marked cubic surfaces is a subvariety of a nine dimensional projective space which is defined by cubic equations. They used the theory of automorphic forms on ball quotients to…

We develop a notion of projections between sets of probability measures using the geometric properties of the 2-Wasserstein space. It is designed for general multivariate probability measures, is computationally efficient to implement, and…

A purely combinatorial compactification of the configuration space of n (>4) distinct points with equal weights in the real projective line was introduced by M. Yoshida. We geometrize it so that it will be a real hyperbolic cone-manifold of…

A real projective orbifold is an $n$-dimensional orbifold modeled on $\mathbb{RP}^n$ with the group $PGL(n+1, \mathbb{R})$. We concentrate on an orbifold that contains a compact codimension $0$ submanifold whose complement is a union of…

A commutative associative algebra A with an identity over the field of real numbers which has a basis, where all elements are invertible, is considered in the work. Moreover, among matrixes consisting of the structure constants of A, there…

We consider the space $\mathcal M$ of ordered quadruples of distinct points in the boundary of complex hyperbolic $n$-space, $\ch{n},$ up to its holomorphic isometry group ${\rm PU}(n,1).$ One of the important problems in complex hyperbolic…

In this work we perform a general study of properties of a class of locally symmetric embedded hypersurfaces in spacetimes admitting a $1+1+2$ spacetime decomposition. The hypersurfaces are given by specifying the form of the Ricci tensor…

We prove a new kind of estimate that holds on any manifold with lower Ricci bounds. It relates the geometry of two small balls with the same radius, potentially far apart, but centered in the interior of a common minimizing geodesic. It…