arXiv research

Study of Martin boundary for rank 1 manifolds with nonpositive curvature.

Study diffusions and random walks on hyperbolic spaces, focusing on their Martin boundaries.

Characterizes Martin boundaries of certain hyperbolic groups.

The Martin boundary of certain groups is stable under specific conditions.

Extends Martin boundary to Floyd boundary for random walks.

The Martin boundary of a Cartan-Hadamard manifold describes a fine geometric structure at infinity, which is a sub-space of positive harmonic functions. We describe conditions which ensure that some points of the sphere at infinity belong to the Martin boundary as well. In the case of the universal cover of a compact m…

The study provides criteria for solving the Dirichlet problem at infinity on Cartan-Hadamard surfaces.

Study transforms singular area minimizing hypersurfaces into hyperbolic spaces.

Develops potential theory on singular almost minimizers.

We give elementary constructions for Satake-Furstenberg, Martin and Karpelevich boundaries of symmetric spaces. We also consruct some "new" boundaries

Potential theory extended to Gromov hyperbolic spaces.

In this paper we examine the Laplacian on the product of two asymptotically hyperbolic (or conformally compact, as they are often called) spaces from the point of view of geometric scattering theory. In particular, we describe the asymptotic behavior of the resolvent applied to Schwartz functions and that of the resolv…

The study quantifies geodesic divergence on Riemannian planes with bounded geometry.

New Gehring-Martin-Tan groups with elliptic generators found.

In this paper we consider non-compact non-flat simply connected harmonic manifolds. In particular, we show that the Martin boundary and Busemann boundary coincide for such manifolds. For any finite volume quotient we show that (up to scaling) there is a unique Patterson-Sullivan measure and this measure coincides with …

The study connects hyperbolicity and isoperimetric inequality in manifolds and graphs.

Proves convergence groups on a 2-sphere are Kleinian groups.

The present paper describes a way to relate Martin boundaries on spaces of varying topology. This enables us to approach some detailed inductive analysis of the eigenfunctions of conformal Laplacians on minimal hypersurfaces near their singularities. This can directly be used resp. translated to understand the way how …

In this paper we consider the Martin compactification, associated with the operator $\mathcal{L} = Δ-1$, of a complete non-compact surface $(Σ^2, ds^2)$ with negative curvature. In particular, we investigate positive eigenfunctions with eigenvalue one of the Laplace operator $Δ$ of $(Σ^2, ds^2)$ and prove a uniqueness …

The paper generalizes free boundary min-max theory to equivariant settings.

Elton P. Hsu used probabilistic method to show that the asymptotic Dirichlet problem is uniquely solvable under the curvature conditions $-C e^{2-η}r(x) \leq K_M(x)\leq -1$ with $η>0$. We give an analytical proof of the same statement. In addition, using this new approach we are able to establish two boundary Harnack i…

Study classifies translators for mean curvature flow in 3D.

Combines relatively hyperbolic groups over a complex.

We present two proofs of the fact, originally due to Reiner Martin, that any fully irreducible hyperbolic element of $Out(F_N)$ acts on the projectivized space of geodesic currents $\mathbb{P}Curr(F_N)$ with uniform north-south dynamics. The first proof, using purely train-track methods, provides an elaborated and corr…

Geoffrey Martin's theorem proves normal forms for Lagrangian submanifolds in multisymplectic geometry.

The purpose of this article is to describe all possible beliefs of market participants on objective measures under Markovian environments when a risk-neutral measure is given. To achieve this, we employ the Martin integral representation of Markovian pricing kernels. Then, we offer economic and financial implications o…

Study ratio-limit boundaries for random walks on hyperbolic groups.

Polyconvex energies with conformal invariance have smooth stationary points outside a discrete set.

The article studies random infinite ideal hyperbolic polyhedra and their dual graphs, establishing new boundary theories.

Transform drift of diffusions without knowing if measure change is a martingale.

In the first part of this dissertation, we give a new definition of a Laplace operator for Finsler metric as an average, with regard to an angle measure, of the second directional derivatives. This operator is elliptic, symmetric with respect to the Holmes-Thompson volume, and coincides with the usual Laplace--Beltrami…

New basis and Schur-Weyl duality for loop Hecke algebra defined.

We construct open domains in Euclidean 3-space which do not admit complete properly immersed minimal surfaces with an annular end. These domains can not be smooth by a recent result of Martin and Morales

Functional inequality proves quasi-invariance in infinite dimensions.

Estimates on Einstein manifolds improve Brownian motion behavior and curvature limits.

Starting from works by Scherk (1835) and by Enneper-Weierstraß\ (1863), new minimal surfaces with Scherk ends were found only in 1988 by Karcher (see \cite{Karcher1,Karcher}). In the singly periodic case, Karcher's examples of positive genera had been unique until Traizet obtained new ones in 1996 (see \cite{Traizet}).…

New knot found with unique property.

This paper is concerned with the axiomatic foundation and explicit construction of a general class of optimality criteria that can be used for investment problems with multiple time horizons, or when the time horizon is not known in advance. Both the investment criterion and the optimal strategy are characterized by th…

Proves small cancellation free products have geometric actions on CAT(0) cube complexes.

In his 1974 thesis, Martin Scharlemann constructed a fake homotopy equivalence from a closed smooth manifold f:Q -> S^3 x S^1 # S^2 x S^2 and asked whether the manifold Q itself is diffeomorphic to S^3 x S^1 # S^2 x S^2. Here we answer this question affirmatively.

Abstract: Generalizes SGMs to infinite-dimensional Hilbertian setting.

Develops confidence intervals for unique elements in data streams.

This paper studies Brownian motion and heat kernel measure on a class of infinite dimensional Lie groups. We prove a Cameron-Martin type quasi-invariance theorem for the heat kernel measure and give estimates on the $L^p$ norms of the Radon-Nikodym derivatives. We also prove that a logarithmic Sobolev inequality holds …

We generalize a class of groups introduced by Herbert Abels to produce examples of virtually torsion free groups that have Bredon-finiteness length m-1 and classical finiteness length n-1 for all 0 < m <= n. The proof illustrates how Bredon-finiteness properties can be verified using geometric methods and a version of …

ICML workshop on making machine learning models more understandable.

We show that the spectrum of a complete submanifold properly immersed into a ball of a Riemannian manifold is discrete, provided the norm of the mean curvature vector is sufficiently small. In particular, the spectrum of a complete minimal surface properly immersed into a ball of $\mathbb{R}^{3}$ is discrete. This give…

We study an integration theory in circle equivariant cohomology in order to prove a theorem relating the cohomology ring of a hyperkahler quotient to the cohomology ring of the quotient by a maximal abelian subgroup, analogous to a theorem of Martin for symplectic quotients. We discuss applications of this theorem to q…

Extends HMM to topological spaces for modeling complex data.