arXiv research

Study $L^2$-cohomology in unbounded geometry manifolds.

We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body $C\subset \mathbb{R}^{n+1}$, without assuming any further regularity on the boundary of $C$. Motivated by an example of an unbounded convex body with null isoperimetric profile, we introduce the concept of…

Introduces new spectral triples for parabolic geometry.

Symplectic method solves infinite-dimensional Schrödinger equations.

We prove a version of Gromov's compactness theorem for pseudo-holomorphic curves which holds locally in the target symplectic manifold. This result applies to sequences of curves with an unbounded number of free boundary components, and in families of degenerating target manifolds which have unbounded geometry (e.g. no…

We will discuss theoretical and experimental results concerning comparison of entropy of pseudo-Anosov maps and volume of their mapping tori. Recent study of Weil-Petersson geometry of the Teichmüller space tells us that they admit linear inequalities for both sides under some bounded geometry condition. We construct a…

This paper solves the normalizability crisis in sequential inference by introducing bounded information geometry.

We use ending laminations for Weil-Petersson geodesics to establish that bounded geometry is equivalent to bounded combinatorics for Weil-Petersson geodesic segments, rays, and lines. Further, a more general notion of non-annular bounded combinatorics, which allows arbitrarily large Dehn-twisting, corresponds to an equ…

A 2008 general overview on Weil-Petersson geometry is offered. A preliminary plan for the subsequent CBMS lectures at Central Connecticut State University is included. Mirzakhani's solution of Witten-Kontsevich is not included - this work essentially requires its own lectures. Lectures on Mirzakhani's Witten-Kontsevich…

We present a definition of Riemannian manifold in noncommutative geometry. Using products of unbounded Kasparov modules, we show one can obtain such Riemannian manifolds from noncommutative spin^c manifolds; and conversely, in the presence of a spin^c structure. We also show how to obtain an analogue of Kasparov's fund…

Study group actions in metric spaces, proving convergence of lens spaces.

Classifies homeomorphism groups of countable Stone spaces up to coarse equivalence.

The Serre-Swan theorem in differential geometry establishes an equivalence between the category of smooth vector bundles over a smooth compact manifold and the category of finitely generated projective modules over the unital ring of smooth functions. This theorem is here generalized to manifolds of bounded geometry. I…

Study metrics on half plane with specific curvature properties.

The norm of Cartan torsion plays an important role for studying of immersion theory in Finsler geometry. Indeed, Finsler manifold with unbounded Cartan torsion can not be isometrically imbedded into any Minkowski space. In this paper, we find two subclasses of (?, ?)-metrics which have bounded Cartan torsion. Then, we …

Extends Alexandrov's result to unbounded convex domains in hyperbolic 3-space.

Two new algorithms improve performance in adversarial bandits with unbounded losses.

We consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show the existence on such a manifold of a distance-like function with bounded gradi…

Unbounded convex domains have zero mean curvature on disconnected boundaries.

The Riemann sphere of a C*-algebra is a geometric structure derived from a specific projector.

The paper proves homotopy equivalences for spaces of unbounded Fredholm operators.

Golden L surface has unbounded bunching of saddle connections

We prove that the vector space R^d of any finite dimension d with the standard metric embeds in a bi-Lipschitz way into the group of area-preserving diffeomorphisms G of the two-sphere endowed with the L^p-metric for p>2. Along the way we show that the L^p-metric on the group G is unbounded for p>2 by elementary method…

We construct an unbounded representative for the shriek class associated to the embeddings of spheres into Euclidean space. We equip this unbounded Kasparov cycle with a connection and compute the unbounded Kasparov product with the Dirac operator on $\mathbb R^{n+1}$. We find that the resulting spectral triple for the…

We classify the tube domains in C^4 with affinely homogeneous base whose boundary contains a non-degenerate affinely homogeneous hypersurface. It follows that these domains are holomorphically homogeneous and amongst them there are four new examples of unbounded homogeneous domains (that do not have bounded realisation…

This memoir presents a systematic study of the utility maximization problem of an investor in a constrained and unbounded financial market. Building upon the work of Hu et al. (2005) [Ann. Appl. Probab., 15, 1691--1712] in a bounded framework, we extend our analysis to the more challenging unbounded case. Our methodolo…

It is shown that the compactly supported identity component of the diffeomorphism group of the 2-dimensional punctured torus $\mathbb T^2_p$ is an unbounded group. It follows that the fragmentation norm of $\mathbb T^2_p$ is unbounded.

New inequalities for unbounded functions improve denoising score matching.

Study focal surfaces of wave fronts with unbounded curvatures.

The paper characterizes groups acting on real projective spaces.

We study when the mapping class group of an infinite-type surface $S$ admits an action with unbounded orbits on a connected graph whose vertices are simple closed curves on $S$. We introduce a topological invariant for infinite-type surfaces that determines in many cases whether there is such an action. This allows us …

We consider least energy solutions to the nonlinear equation $-Δ_g u=f(r,u)$ posed on a class of Riemannian models $(M,g)$ of dimension $n\ge 2$ which include the classical hyperbolic space $\mathbb H^n$ as well as manifolds with unbounded sectional geometry. Partial symmetry and existence of least energy solutions is …

Study shows unbounded Pontryagin numbers on curved manifolds.

Paper tackles online control of linear systems with unbounded noise.

Let $(X,ω)$ be a compact Kähler manifold. We prove the existence and uniqueness of solutions to complex Monge-Ampère equations with prescribed singularity type. Compared to previous work, the assumption of small unbounded locus is dropped, and we work with general model type singularities. We state and prove our theore…

We show that domains, that allow for convex functions with unbounded gradient at their boundary, are convex.

Asymptotic subcone of an unbounded metric space is another metric space, capturing the structure of the original space at infinity. In this paper we define a functional metric space S which is an asymptotic subcone of the hyperbolic plane. This space is a real tree branching at every its point. Moreover, it is a homoge…

Paper proves a Penrose inequality in extrinsic geometry.

New approach finds solutions to games with unbounded controls.

The paper provides gradient estimates for Neumann semigroups on manifolds with boundary under unbounded curvature conditions.

New aggregation strategy handles unbounded losses with regret bounds.

Improved sampling from Gaussian distributions with privacy constraints.

New PAC-Bayes bounds for unbounded loss functions.

Geometric model of unbounded sl3 laminations with tropical coordinates.

In this paper, we provided conditions for an entire constant mean curvature Killing graph lying inside a possible unbounded region to be necessarily a slice.

We show that any noncompact Riemann surface admits a complete Ricci flow g(t), t\in[0,\infty), which has unbounded curvature for all t\in[0,\infty).

We construct a complete, embedded minimal surface in euclidean 3-space which has unbounded Gaussian curvature. It has infinite genus, infinitely many catenoidal type ends and one limit end.

In this paper, we derive Li-Yau inequality for unbounded Laplacian on complete weighted graphs with the assumption of the curvature-dimension inequality $CDE'(n,K)$, which can be regarded as a notion of curvature on graphs. Furthermore, we obtain some applications of Li-Yau inequality, including Harnack inequality, hea…