Niveau and coniveau filtrations on cohomology groups and Chow groups

The Bloch-Beilinson-Murre conjectures predict the existence of a descending filtration on Chow groups of smooth projective varieties which is functorial with respect to the action of correspondences and whose graded parts depend solely on the topology -- i.e. the cohomology -- of $X$. In this paper, we wish to explore, at the cost of having to assume general conjectures about algebraic cycles, how the coniveau filtration on the cohomology of $X$ has an incidence on the Chow groups of $X$. However, by keeping such assumptions minimal, we are able to prove some of these conjectures either in low-dimensional cases or when a variety is known to have small Chow groups. For instance, we give a new example of a fourfold of general type with trivial Chow group of zero-cycles and we prove Murre's conjectures for threefolds dominated by a product of curves, for threefolds rationally dominated by the product of three curves, for rationally connected fourfolds and for complete intersections of low degree. The BBM conjectures are closely related to Kimura-O'Sullivan's notion of finite-dimensionality. Assuming the standard conjectures on algebraic cycles the former is known to imply the latter. We show that the missing ingredient for finite-dimensionality to imply the BBM conjectures is the coincidence of a certain niveau filtration with the coniveau filtration on Chow groups.Comment: Final versio

Pure motives with representable Chow groups

Let $k$ be an algebraically closed field. We show using Kahn's and Sujatha's theory of birational motives that a Chow motive over $k$ whose Chow groups are all representable belongs to the full and thick subcategory of motives generated by the twisted motives of curves. -- Motifs purs dont les groupes de Chow sont repr\'esentables. Soit $k$ un corps alg\'ebriquement clos. Nous prouvons, en nous servant de la th\'eorie des motifs birationnels d\'evelopp\'ee par Kahn et Sujatha, qu'un motif de Chow d\'efini sur $k$ dont les groupes de Chow sont tous repr\'esentables appartient \`a la sous-cat\'egorie pleine et \'epaisse des motifs engendr\'ee par les motifs de courbes tordus.Comment: 7 page

Operations in Milnor K-theory

We show that operations in Milnor K-theory mod $p$ of a field are spanned by divided power operations. After giving an explicit formula for divided power operations and extending them to some new cases, we determine for all fields $k$ and all prime numbers $p$, all the operations $K^M_i/p \to K^M_j/p$ commuting with field extensions over the base field $k$. Moreover, the integral case is discussed and we determine the operations $K^M_i/p \to K^M_j/p$ for smooth schemes over a field.Comment: to appear in the Journal of Pure and Applied Algebr

Local non-Bayesian social learning with stubborn agents

We study a social learning model in which agents iteratively update their beliefs about the true state of the world using private signals and the beliefs of other agents in a non-Bayesian manner. Some agents are stubborn, meaning they attempt to convince others of an erroneous true state (modeling fake news). We show that while agents learn the true state on short timescales, they "forget" it and believe the erroneous state to be true on longer timescales. Using these results, we devise strategies for seeding stubborn agents so as to disrupt learning, which outperform intuitive heuristics and give novel insights regarding vulnerabilities in social learning

Assessing extrema of empirical principal component functions

The difficulties of estimating and representing the distributions of functional data mean that principal component methods play a substantially greater role in functional data analysis than in more conventional finite-dimensional settings. Local maxima and minima in principal component functions are of direct importance; they indicate places in the domain of a random function where influence on the function value tends to be relatively strong but of opposite sign. We explore statistical properties of the relationship between extrema of empirical principal component functions, and their counterparts for the true principal component functions. It is shown that empirical principal component funcions have relatively little trouble capturing conventional extrema, but can experience difficulty distinguishing a ``shoulder'' in a curve from a small bump. For example, when the true principal component function has a shoulder, the probability that the empirical principal component function has instead a bump is approximately equal to 1/2. We suggest and describe the performance of bootstrap methods for assessing the strength of extrema. It is shown that the subsample bootstrap is more effective than the standard bootstrap in this regard. A ``bootstrap likelihood'' is proposed for measuring extremum strength. Exploratory numerical methods are suggested.Comment: Published at http://dx.doi.org/10.1214/009053606000000371 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org