arXiv research

Paper develops a theory for Patterson-Sullivan measures in higher rank symmetric spaces.

Study shows horofunction compactification's topology matches dual norm's unit ball.

We determine the set of Busemann points of an arbitrary finite-dimensional normed space. These are the points of the horofunction boundary that are the limits of "almost-geodesics". We prove that all points in the horofunction boundary are Busemann points if and only if the set of extreme sets of the dual unit ball is …

Optimal geodesics connect boundary points in Teichmüller space.

We fully describe the horofunction boundary $\partial_h L_2$ with the word metric associated with the generating set $\{t,at\}$ (i.e the metric arising in the Diestel-Leader graph $\text{DL}(2,2)$). The visual boundary $\partial_\infty L_2$ with this metric is a subset of $\partial_h L_2$. Although $\partial_\infty L_2…

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 …

This paper studies convergence of horospheres in CAT(0) spaces.

Extends Teichmüller distance concept to non-distance maps.

The paper studies horofunction compactifications of symmetric cones under Finsler distances.

Study boundary actions on CAT(0) spaces, proving topological freeness.

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.

We show that the horofunction boundary of Teichmüller space with Thurston's Lipschitz metric is the same as the Thurston boundary. We use this to determine the isometry group of the Lipschitz metric, apart from in some exceptional cases. We also show that the Teichmüller spaces of different surfaces, when endowed with …

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

We establish a natural and geometric 1-1 correspondence between projective toric varieties of dimension $n$ and horofunction compactifications of $\mathbb{R}^n$ with respect to rational polyhedral norms. For this purpose, we explain a topological model of toric varieties. Consequently, toric varieties in algebraic geom…

In this paper we answer positively a question raised by Kapovich and Leeb in a paper titled "Finsler bordifications of symmetric and certain locally symmetric spaces". Specifically, we show that for a finite-dimensional vector space with a polyhedral norm, its horofunction compactification is homeomorphic to the dual u…

This work describes compactifications of metric spaces and vector spaces using asymmetric norms.

Researchers describe horofunctions in noncompact Hermitian symmetric spaces.

We determine the asymptotic behaviour of extremal length along arbitrary Teichmüller rays. This allows us to calculate the endpoint in the Gardiner-Masur boundary of any Teichmüller ray. We give a proof that this compactification is the same as the horofunction compactification. An important subset of the latter is the…

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

We give an example of a horocycle in the Teichmüller space of the five-times-punctured sphere that does not converge in the Gardiner--Masur compactification, or equivalently in the horofunction compactification of the Teichmüller metric. As an intermediate step, we exhibit a simple closed curve whose extremal length is…

The paper develops a theory of conformal density at infinity for groups with contracting elements.

As in a symmetric space of noncompact type, one can associate to an oriented geodesic segment in a Euclidean building a vector valued length in the Euclidean Weyl chamber; in addition to the metric length it contains information on the direction of the segment. We study in this paper restrictions on the vector valued s…

Completes the space of vector-valued one-forms on manifolds.

Abstract: Generalizes multisymplectic forms to vector-valued versions.

Study on free boundary problems in RCD spaces, proving existence and regularity.

This work develops discrete Gaussian models for vector-valued data on triangular meshes.

We generalize the natural cross ratio on the ideal boundary of a rank one symmetric spaces, or even $\mathrm{CAT}(-1)$ space, to higher rank symmetric spaces and (non-locally compact) Euclidean buildings - we obtain vector valued cross ratios defined on simplices of the building at infinity. We show several properties …

We give a geometric interpretation of the maximal Satake compactification of symmetric spaces $X=G/K$ of noncompact type, showing that it arises by attaching the horofunction boundary for a suitable $G$-invariant Finsler metric on $X$. As an application, we establish the existence of natural bordifications, as orbifold…

The study examines the asymptotic behavior of extremal length in Teichmüller space.

Transforms uniquely determine Higgs fields on real-analytic manifolds.

The paper extends consistency results for sequential design strategies to vector-valued Gaussian processes.

We discuss sharp Sobolev inequalities for vector valued maps.

Motivated by multi-task machine learning with Banach spaces, we propose the notion of vector-valued reproducing kernel Banach spaces (RKBS). Basic properties of the spaces and the associated reproducing kernels are investigated. We also present feature map constructions and several concrete examples of vector-valued RK…

Optimal rates for vector-valued regression on various norms.

SL(n) covariant valuations on Orlicz spaces are represented and characterized.

Boosting framework for vector-valued prediction with geometric stability.

Study confirms learning rates for vector-valued spectral algorithms, proving consistency.

The space of vector-valued forms on any manifold is a graded Lie algebra with respect to the Frolicher-Nijenhuis bracket. In this paper we consider multiplicative vector-valued forms on Lie groupoids and show that they naturally form a graded Lie subalgebra. Along the way, we discuss various examples and different char…

Vector-valued learning, where the output space admits a vector-valued structure, is an important problem that covers a broad family of important domains, e.g. multi-task learning and transfer learning. Using local Rademacher complexity and unlabeled data, we derive novel semi-supervised excess risk bounds for general v…

The paper shows vector-valued risk measures ignore dependence structures.

The paper proposes methods to find a shared active subspace for multivariate vector-valued functions.

Optimal transport for vector Gaussian mixtures improves efficiency and structure preservation.

We present a framework to derive risk bounds for vector-valued learning with a broad class of feature maps and loss functions. Multi-task learning and one-vs-all multi-category learning are treated as examples. We discuss in detail vector-valued functions with one hidden layer, and demonstrate that the conditions under…

In the first chapter, we give a precise and general description of gerbes valued in arbitrary crossed module and over an arbitrary differential stack. We do it using only Lie groupoids, hence ordinary differential geometry, by considering differential stacks as being Lie groupoids up to Morita equivalence. We prove the…

Study improves self-normalized bounds for vector-valued processes beyond sub-Gaussianity.

The paper develops a new approach to solve vector-valued PDEs on manifolds with minimal regularity.

We approximate derivatives of functions on manifolds by embedding them and applying vector-valued operators.

Paper introduces vector-valued variation spaces for multi-output neural networks.