arXiv research

A manifold's discreteness is tied to its number of ends.

A topology on a set $X$ is the same as a projection (i.e. an idempotent linear operator) $cl:2^X\to 2^X$ satisfying $A\subset cl(A)$ for all $A\subset X$. That's a good way to summarize Kuratowski's closure operator. Basic geometry on a set $X$ is a dot product $\cdot:2^X\times 2^X\to 2^Y$. Its equivalent form is an or…

We present an idea of unifying small scale (topology, proximity spaces, uniform spaces) and large scale (coarse spaces, large scale spaces). It relies on an analog of multilinear forms from Linear Algebra. Each form has a large scale compactification and those include all well-known compactifications: Higson corona, Gr…

We connect the algebraic geometry and representation theory associated to Freudenthal's magic square. We give unified geometric descriptions of several classes of orbit closures, describing their hyperplane sections and desingularizations, and interpreting them in terms of composition algebras. In particular, we show h…

Let $G$ be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface $X \subseteq \mathbb{S}^2$. We prove that $G$ admits such an action that is in addition co-compact, provided we can replace $X$ by another surface $Y \subseteq \mathbb{S}^2$. We also prove that if …

We investigate nonintegrable Riemannian geometries modelled after certain symmetric spaces related to the Freudenthal-Tits Magic Square. The collection of four such structures found by Nurowski is extended by further eight. A focus is given to those admitting a compatible connection with completely skew torsion.

Let J be the exceptional Jordan algebra over R and J^C its complexification. Then the simply connected compact exceptional Lie group F_4 acts on J and F_4 has three orbit types which are F_4/F_4, F_4/Spin(9), F_4/Spin(8). Similarly the simply connected compact exceptional Lie group E_6 acts on J^C and E_6 has five orbi…

The orbit decomposition is given under the automorphism group on the real split Jordan algebra of all hermitian matrices of order three corresponding to any real split composition algebra, or the automorphism group on the complexification, explicitly, in terms of the cross product of H. Freudenthal and the characterist…

We summarize the main results of our investigation of B-type topological Landau-Ginzburg models whose target is an arbitrary open Riemann surface. Such a Riemann surface need not be affine algebraic and in particular it may have infinite genus or an infinite number of Freudenthal ends. Under mild conditions on the Land…

We construct a triangulation of a compactification of the Moduli space of a surface with at least one puncture that is closely related to the Deligne-Mumford compactification. Specifically, there is a surjective map from the compactification we construct to the Deligne-Mumford compactification so that the inverse image…

Method calculates spectra of Rarita-Schwinger operator on symmetric spaces.

Extends harmonic maps compactification to punctured Riemann surfaces.

Paper relates new compactification to classical moduli space.

We consider horofunction compactifications of symmetric spaces with respect to invariant Finsler metrics. We show that any (generalized) Satake compactification can be realized as a horofunction compactification with respect to a polyhedral Finsler metric.

Existence of Kähler-Einstein metrics on compactifications of Lie groups.

The horoboundary of Teichmüller space is path connected and has non-dense Busemann points.

Characterizes toroidal and semi-toric compactifications as log minimal models and applies to weak K-moduli.

Satake has constructed compactifications of symmetric spaces D=G/K which (under a condition called geometric rationality by Casselman) yield compactifications of the corresponding locally symmetric spaces. The different compactifications depend on the choice of a representation of G. One example is the Baily-Borel-Sata…

The paper studies compactifications of SL(2,C) character varieties for punctured surfaces.

Our main aim is to provide a uniform geometric characterization of the analogues over arbitrary fields of the four complex Severi varieties, i.e.~the quadric Veronese varieties in 5-dimensional projective spaces, the Segre varieties in 8-di\-men\-sional projective spaces, the line Grassmannians in 14-dimensional projec…

We give a computer free proof of the Deligne, Cohen and deMan formulas for the dimensions of the irreducible $g$-modules appearing in the tensor powers of $g$, where $g$ ranges over the exceptional complex simple Lie algebras. We give additional dimension formulas for the exceptional series, as well as uniform dimensio…

New coordinates for Teichmüller space compactification.

New compactification for character varieties with good topological properties.

Embeds Higson compactification into adelic solenoids.

We discuss the `hd-compactification' of a semi-simple Lie group to a manifold with corners; it is the real analog of the wonderful compactification of deConcini and Procesi. There is a 1-1 correspondence between the boundary faces of the compactification and conjugacy classes of parabolic subgroups with the boundary fa…

We show that the horofunction compactification of Teichmüller space with the Teichmüller metric is homeomorphic to the Gardiner-Masur compactification.

We define a compactification of symmetric spaces of noncompact type, seen as spaces of isometry classes of marked lattices, analogous to the Thurston compactification of the Teichmüller space, and we show that it is equivariantly isomorphic to a Satake compactification. We then use it to define a new compactification o…

We construct several examples of compactifications of Einstein metrics. We show that the Eguchi--Hanson instanton admits a projective compactification which is non--metric, and that a metric cone over any (pseudo)--Riemannian manifolds admits a metric projective compactification. We construct a para--$c$--projective co…

Researchers create a new compactification of character varieties using geometric and algebraic methods.

Paper studies compactifications of Higgs bundles and self-duality equations.

No natural topological compactification for Fulton-MacPherson.

Paper shows geometric properties preserved by compactifications in relation to coarse structures and group actions.

The paper classifies and computes limits of equivariant compactifications of groups.

The study finds only finitely many Kähler-Einstein compactifications for semisimple groups.

The paper extends end concepts to arbitrary groups and spaces.

In this paper we present a topological way of building a compactification of a symmetric space from a compactification of a Weyl Chamber.

To study a noncompact Riemannian manifold, it is often useful to find a compactification. We discuss several common compactifications and survey some recent results.

The group of direct isometries of the real n-dimensional hyperbolic space is G=SOo(n,1). This isometric action admits many differentiable compactifications into an action on the closed ball. We prove that all such compactifications are topologically conjugate but not necessarily differentiably conjugate. We give the cl…

Consider a finite dimensional (generally reducible) polynomial representation ρof GL_n. A projective compactification of GL_n is the closure of ρ(GL_n) in the space of all operators defined up to a factor (this class of spaces can be characterized as equivariant projective normal compactifications of GL_n). We give an …

Schwartz functions smoothly extend to real projective spaces.

We define a new compactification of outer space $CV_N$ (the \emph{Pacman compactification}) which is an absolute retract, for which the boundary is a $Z$-set. The classical compactification $\overline{CV_N}$ made of very small $F_N$-actions on $\mathbb{R}$-trees, however, fails to be locally $4$-connected as soon as $N…

The paper studies foliations on homogeneous spaces and identifies specific foliations.

Study on curvature properties and Shafarevich conjecture for complex hyperbolic manifolds.

The split version of the Freudenthal-Tits magic square stems from Lie theory and constructs a Lie algebra starting from two split composition algebras [3, 17, 18]. The geometries appearing in the second row are Severi-Brauer varieties [20]. We provide an easy uniform axiomatization of these geometries and related ones,…

The arc metric is an asymmetric metric on the Teichm{ü}ller space T(S) of a surface S with nonempty boundary. In this paper we study the relation between Thurston's compactification and the horofunction compactification of T(S) endowed with the arc metric. We prove that there is a natural homeomorphism between the two …

An Alexander self-dual complex gives rise to a compactification of $M_{0,n}$, called ASD compactification, which is a smooth algebraic variety. ASD compactifications include (but are not exhausted by) the polygon spaces, or the moduli spaces of flexible polygons. We present an explicit description of the Chow rings of …

The paper solves a geometric P=W conjecture for SL(2,C) using Thurston's compactification.

Any nonpositively curved symmetric space admits a topological compactification, namely the Hadamard compactification. For rank one spaces, this topological compactification can be endowed with a differentiable structure such that the action of the isometry group is differentiable. Moreover, the restriction of the actio…