arXiv subjects

Nicholas Miller

Publications and source records attributed to Nicholas Miller.

23 records · Page 2Linked to original sources

arXiv2018

Finiteness of Maximal Geodesic Submanifolds in Hyperbolic Hybrids

We show that large classes of non-arithmetic hyperbolic $n$-manifolds, including the hybrids introduced by Gromov and Piatetski-Shapiro and many of their generalizations, have only finitely many finite-volume immersed totally geodesic hypersurfaces. In higher codimension, we prove finiteness for geodesic submanifolds of dimension at least $2$ that are maximal, i.e., not properly contained in a proper geodesic submanifold of the ambient $n$-manifold. The proof is a mix of structure theory for arithmetic groups, dynamics, and geometry in negative curvature.

math.GT↗

arXiv2017

Areas of totally geodesic surfaces of hyperbolic 3-orbifolds

The geodesic length spectrum of a complete, finite volume, hyperbolic 3-orbifold M is a fundamental invariant of the topology of M via Mostow-Prasad Rigidity. Motivated by this, the second author and Reid defined a two-dimensional analogue of the geodesic length spectrum given by the multiset of isometry types of totally geodesic, immersed, finite-area surfaces of M called the geometric genus spectrum. They showed that if $M$ is arithmetic and contains a totally geodesic surface, then the geometric genus spectrum of M determines its commensurability class. In this paper we define a coarser invariant called the totally geodesic area set given by the set of areas of surfaces in the geometric genus spectrum. We prove a number of results quantifying the extent to which non-commensurable arithmetic hyperbolic 3-orbifolds can have arbitrarily large overlaps in their totally geodesic area sets.

math.GT↗

arXiv2016

Locally Equivalent Correspondences

Given a pair of number fields with isomorphic rings of adeles, we construct bijections between objects associated to the pair. For instance we construct an isomorphism of Brauer groups that commutes with restriction. We additionally construct bijections between central simple algebras, maximal orders, various Galois cohomology sets, and commensurability classes of arithmetic lattices in simple, inner algebraic groups. We show that under certain conditions, lattices corresponding to one another under our bijections have the same covolume and pro-congruence completion. We also make effective a finiteness result of Prasad and Rapinchuk.

math.GR↗

arXiv2016

Arithmetic Progressions in the Primitive Length Spectrum

In this article, we prove that every arithmetic locally symmetric orbifold of classical type without Euclidean or compact factors has arbitrarily long arithmetic progressions in its primitive length spectrum. Moreover, we show the stronger property that every primitive length occurs in arbitrarily long arithmetic progressions in its primitive length spectrum. This confirms one direction of a conjecture of Lafont--McReynolds, which states that the property of having every primitive length occur in arbitrarily long arithmetic progressions characterizes the arithmeticity of such spaces.

math.GT↗

arXiv2015

Diverse Large-Scale ITS Dataset Created from Continuous Learning for Real-Time Vehicle Detection

In traffic engineering, vehicle detectors are trained on limited datasets resulting in poor accuracy when deployed in real world applications. Annotating large-scale high quality datasets is challenging. Typically, these datasets have limited diversity; they do not reflect the real-world operating environment. There is a need for a large-scale, cloud based positive and negative mining (PNM) process and a large-scale learning and evaluation system for the application of traffic event detection. The proposed positive and negative mining process addresses the quality of crowd sourced ground truth data through machine learning review and human feedback mechanisms. The proposed learning and evaluation system uses a distributed cloud computing framework to handle data-scaling issues associated with large numbers of samples and a high-dimensional feature space. The system is trained using AdaBoost on $1,000,000$ Haar-like features extracted from $70,000$ annotated video frames. The trained real-time vehicle detector achieves an accuracy of at least $95\%$ for $1/2$ and about $78\%$ for $19/20$ of the time when tested on approximately $7,500,000$ video frames. At the end of 2015, the dataset is expect to have over one billion annotated video frames.

cs.CV↗