Unramified cohomology of classifying varieties for exceptional simply connected groups

Let BG be a classifying variety for an exceptional simple simply connected algebraic group G. We compute the degree 3 unramified Galois cohomology of BG with values in Q/Z(2) over an arbitrary field F. Combined with a paper by Merkurjev, this completes the computation of these cohomology groups for G semisimple simply connected over all fields. These computations provide another example of a simple simply connected group G such that BG is not stably rational

The characteristic polynomial and determinant are not ad hoc constructions

The typical definition of the characteristic polynomial seems totally ad hoc to me. This note gives a canonical construction of the characteristic polynomial as the minimal polynomial of a "generic" matrix. This approach works not just for matrices but also for a very broad class of algebras including the quaternions, all central simple algebras, and Jordan algebras. The main idea of this paper dates back to the late 1800s. (In particular, it is not due to the author.) This note is intended for a broad audience; the only background required is one year of graduate algebra.Comment: v2 is heavily revised and somewhat expanded. The product formula for the determinant on an algebra is prove

Hiring Freeze and Bankruptcy in Unemployment Dynamics

This paper proposes a matching model that distinguishes between job creation by existing firms and job creation by firm entrants. The paper argues that vacancy posting and job destruction on the extensive margin, i.e. from firms that enter and exit the labour market, represents a potentially viable mechanism for understanding the cyclical properties of vacancies and unemployment. The model features both hiring freeze and bankruptcies, where the former represents a sudden shut down of vacancy posting at the firm level with labour downsizing governed by natural turnover. A bankrupt firm, conversely, shut down its vacancies and lay offs its stock of workers. Recent research in macroeconomics has shown that a calibration of the Mortensen and Pissarides matching model account for 10 percent of the cyclical variability of the vacancy unemployment ratio displayed by U.S. data. A calibration of the model that explicitly considers hiring freeze and bankruptcy can account for 20 to 35 percent of the variability displayed by the data.unemployment dynamics, matching models

A nonsmooth two-sex population model

This paper considers a two-dimensional logistic model to study populations with two genders. The growth behavior of a population is guided by two coupled ordinary differential equations given by a non-differentiable vector field whose parameters are the secondary sex ratio (the ratio of males to females at time of birth), inter-, intra- and outer-gender competitions, fertility and mortality rates and a mating function. For the case where there is no inter-gender competition and the mortality rates are negligible with respect to the density-dependent mortality, using geometrical techniques, we analyze the singularities and the basin of attraction of the system, determining the relationships between the parameters for which the system presents an equilibrium point. In particular, we describe conditions on the secondary sex ratio and discuss the role of the average number of female sexual partners of each male for the conservation of a two-sex species.Comment: 18 pages, 6 figures. Section 2, in which the model is presented, was rewritten to better explain the elements of the proposed model. The description of parameter "r" was correcte

Totaro's question for G_2, F_4, and E_6

In a 2004 paper, Totaro asked whether a G-torsor X that has a zero-cycle of degree d > 0 will necessarily have a closed etale point of degree dividing d, where G is a connected algebraic group. This question is closely related to several conjectures regarding exceptional algebraic groups. Totaro gave a positive answer to his question in the following cases: G simple, split, and of type G_2, type F_4, or simply connected of type E_6. We extend the list of cases where the answer is "yes" to all groups of type G_2 and some nonsplit groups of type F_4 and E_6. No assumption on the characteristic of the base field is made. The key tool is a lemma regarding linkage of Pfister forms.Comment: 15 page

The gamma-filtration and the Rost invariant

Let X be the variety of Borel subgroups of a simple and strongly inner linear algebraic group G over a field k. We prove that the torsion part of the second quotient of Grothendieck's gamma-filtration on X is a cyclic group of order the Dynkin index of G. As a byproduct of the proof we obtain an explicit cycle that generates this cyclic group; we provide an upper bound for the torsion of the Chow group of codimension-3 cycles on X; we relate the generating cycle with the Rost invariant and the torsion of the respective generalized Rost motives; we use this cycle to obtain a uniform lower bound for the essential dimension of (almost) all simple linear algebraic groups.Comment: 19 pages; this is an essentially extended version of the previous preprint. Applications to cohomological invariants and essential dimensions of linear algebraic groups are provide