Open Problems on Central Simple Algebras

We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners.Comment: v2 has some small revisions to the text. Some items are re-numbered, compared to v

On Albanese torsors and the elementary obstruction

We show that the elementary obstruction to the existence of 0-cycles of degree 1 on an arbitrary variety X (over an arbitrary field) can be expressed in terms of the Albanese 1-motives associated with dense open subsets of X. Arithmetic applications are given

Chromatin establishes an immature version of neuronal protocadherin selection during the naive-to-primed conversion of pluripotent stem cells.

In the mammalian genome, the clustered protocadherin (cPCDH) locus provides a paradigm for stochastic gene expression with the potential to generate a unique cPCDH combination in every neuron. Here we report a chromatin-based mechanism that emerges during the transition from the naive to the primed states of cell pluripotency and reduces, by orders of magnitude, the combinatorial potential in the human cPCDH locus. This mechanism selectively increases the frequency of stochastic selection of a small subset of cPCDH genes after neuronal differentiation in monolayers, 10-month-old cortical organoids and engrafted cells in the spinal cords of rats. Signs of these frequent selections can be observed in the brain throughout fetal development and disappear after birth, except in conditions of delayed maturation such as Down's syndrome. We therefore propose that a pattern of limited cPCDH-gene expression diversity is maintained while human neurons still retain fetal-like levels of maturation

Uncertainty quantification in graph-based classification of high dimensional data

Classification of high dimensional data finds wide-ranging applications. In many of these applications equipping the resulting classification with a measure of uncertainty may be as important as the classification itself. In this paper we introduce, develop algorithms for, and investigate the properties of, a variety of Bayesian models for the task of binary classification; via the posterior distribution on the classification labels, these methods automatically give measures of uncertainty. The methods are all based around the graph formulation of semi-supervised learning. We provide a unified framework which brings together a variety of methods which have been introduced in different communities within the mathematical sciences. We study probit classification in the graph-based setting, generalize the level-set method for Bayesian inverse problems to the classification setting, and generalize the Ginzburg-Landau optimization-based classifier to a Bayesian setting; we also show that the probit and level set approaches are natural relaxations of the harmonic function approach introduced in [Zhu et al 2003]. We introduce efficient numerical methods, suited to large data-sets, for both MCMC-based sampling as well as gradient-based MAP estimation. Through numerical experiments we study classification accuracy and uncertainty quantification for our models; these experiments showcase a suite of datasets commonly used to evaluate graph-based semi-supervised learning algorithms.Comment: 33 pages, 14 figure

An MBO scheme for minimizing the graph Ohta-Kawasaki functional

We study a graph based version of the Ohta-Kawasaki functional, which was originally introduced in a continuum setting to model pattern formation in diblock copolymer melts and has been studied extensively as a paradigmatic example of a variational model for pattern formation. Graph based problems inspired by partial differential equations (PDEs) and varational methods have been the subject of many recent papers in the mathematical literature, because of their applications in areas such as image processing and data classification. This paper extends the area of PDE inspired graph based problems to pattern forming models, while continuing in the tradition of recent papers in the field. We introduce a mass conserving Merriman-Bence-Osher (MBO) scheme for minimizing the graph Ohta-Kawasaki functional with a mass constraint. We present three main results: (1) the Lyapunov functionals associated with this MBO scheme Γ-converge to the Ohta-Kawasaki functional (which includes the standard graph based MBO scheme and total variation as a special case); (2) there is a class of graphs on which the Ohta-Kawasaki MBO scheme corresponds to a standard MBO scheme on a transformed graph and for which generalized comparison principles hold; (3) this MBO scheme allows for the numerical computation of (approximate) minimizers of the graph Ohta-Kawasaki functional with a mass constraint

Graph MBO method for multiclass segmentation of hyperspectral stand-off detection video

We consider the challenge of detection of chemical plumes in hyperspectral image data. Segmentation of gas is difficult due to the diffusive nature of the cloud. The use of hyperspectral imagery provides non-visual data for this problem, allowing for the utilization of a richer array of sensing information. In this paper, we present a method to track and classify objects in hyperspectral videos. The method involves the application of a new algorithm recently developed for high dimensional data. It is made efficient by the application of spectral methods and the Nyström extension to calculate the eigenvalues/eigenvectors of the graph Laplacian. Results are shown on plume detection in LWIR standoff detection

Global Binary Optimization on Graphs for Classification of High-Dimensional Data

© 2015 Springer Science+Business Media New York This work develops a global minimization framework for segmentation of high-dimensional data into two classes. It combines recent convex optimization methods from imaging with recent graph- based variational models for data segmentation. Two convex splitting algorithms are proposed, where graph-based PDE techniques are used to solve some of the subproblems. It is shown that global minimizers can be guaranteed for semi-supervised segmentation with two regions. If constraints on the volume of the regions are incorporated, global minimizers cannot be guaranteed, but can often be obtained in practice and otherwise be closely approximated. Experiments on benchmark data sets show that our models produce segmentation results that are comparable with or outperform the state-of-the-art algorithms. In particular, we perform a thorough comparison to recent MBO (Merriman–Bence–Osher, AMS-Selected Lectures in Mathematics Series: Computational Crystal Growers Workshop, 1992) and phase field methods, and show the advantage of the algorithms proposed in this paper