BRITE-Constellation (where BRITE stands for BRIght Target Explorer) is an international nanosatellite mission to monitor photometrically, in two colours, brightness and temperature variations of stars brighter than V ≈ 4, with precision and time coverage not possible from the ground. The current mission design consists of three pairs of 7 kg nanosats (hence “Constellation”) from Austria, Canada and Poland carrying optical telescopes (3 cm aperture) and CCDs. One instrument in each pair is equipped with a blue filter; the other, a red filter. The first two nanosats (funded by Austria) are UniBRITE, designed and built by UTIAS-SFL (University of Toronto Institute for Aerospace Studies-Space Flight Laboratory) and its twin, BRITE-Austria, built by the Technical University Graz (TUG) with support of UTIAS-SFL. They were launched on 25 February 2013 by the Indian Space Agency, under contract to the Canadian Space Agency.

Each BRITE instrument has a wide field of view (≈ 24 degrees), so up to 15 bright stars can be observed simultaneously in 32 × 32 sub-rasters. Photometry (with reduced precision but thorough time sampling) of additional fainter targets will be possible through on-board data processing. A critical technical element of the BRITE mission is the three-axis attitude control system to stabilize a nanosat with very low inertia. The pointing stability is better than 1.5 arcminutes rms, a significant advance by UTIAS-SFL over any previous nanosatellite.

BRITE-Constellation will primarily measure p- and g-mode pulsations to probe the interiors and ages of stars through asteroseismology. The BRITE sample of many of the brightest stars in the night sky is dominated by the most intrinsically luminous stars: massive stars seen at all evolutionary stages, and evolved medium-mass stars at the very end of their nuclear burning phases (cool giants and AGB stars). The Hertzsprung-Russell diagram for stars brighter than mag V=4 from which the BRITE-Constellation sample will be selected is shown in Fig. 1. This sample falls into two principal classes of stars:

(1) Hot luminous H-burning stars (O to F stars). Analyses of OB star variability have the potential to help solve two outstanding problems: the sizes of convective (mixed) cores in massive stars and the influence of rapid rotation on their structure and evolution.

(2) Cool luminous stars (AGB stars, cool giants and cool supergiants). Measurements of the time scales involved in surface granulation and differential rotation will constrain turbulent convection models.

Mass loss from these stars (especially the massive supernova progenitors) is a major contributor to the evolution of the interstellar medium, so in a sense, this sample dominates cosmic “ecology” in terms of future generations of star formation. The massive stars are believed to share many characteristics of the lower mass range of the first generation of stars ever formed (although the original examples are of course long gone).

BRITE observations will also be used to detect some Jupiter- and even Neptune-sized planets around bright host stars via transits, as expected on the basis of statistics from the Kepler exoplanet mission. Detecting planets around such very bright stars will greatly facilitate their subsequent characterization. BRITE will also use surface spots to investigate stellar rotation.

The following Table summarizes launch and orbit parameters of BRITE-Constellation components.

The full version of this paper describing in more detail BRITE-Constellation will be published separately in a journal. The symposium presentation is available at http://iaus301.astro.uni.wroc.pl/program.php

The Mathematical Association is very kind to its Presidents. They are relative strangers, they are placed for a short-time in the most intimate contact with all the Association’s affairs, and they are given up to two years in which to agonise over the subject of their Presidential Address. In my own case I chose the above title about 18 months ago, and was then filled with worry and embarrassment when last year Frank Budden unveiled his title “Accuracy is a virtue”. Would my lecture be viewed as an unworthy attack on a very distinguished predecessor? More important, was I completely mistaken in my view? My conclusion was that I should go ahead with the talk as planned asking you to view Frank Budden’s remarks and my own as two elements in a necessary dialectic, another example of the tensions which are inherent and necessary to good mathematical education.

Calculus made easy by “F.R.S.” was published seventy years ago. Its anonymous author wrote explicitly for engineering students rather than for future mathematics teachers. He ended by asking those who had enjoyed the book “not to give the author away nor to tell the mathematicians what a fool he really is”.

The detailed study of n-dimensional crystallography appears to have been initiated by Hermann but unfortunately only a small fragment of his work was ever published(4). Recently, interest in the subject has been revived by the work of Bülow, Neubüser and Wondratschek(2), who not only obtained definitive results on 4-dimensional lattices and space groups but also proposed general definitions and procedures for the n-dimensional case. In particular they introduced the concept of crystal family and so resolved satisfactorily the controversy on whether there are six or seven 3-dimensional crystal systems (see(1)). Their point of view is arithmetic, that is, they use a coordinate system adapted to the particular lattice under consideration and represent elements of its symmetry group by integer matrices.

Several modern mathematics courses contain a description of the 17 distinct ‘wallpaper patterns’. Others contain the definition of “group” and “isomorphism”, together sometimes with a vague statement that these concepts can be used to justify the fact that there are precisely 17 patterns. But neither the passive contemplation of wallpaper patterns, nor the passive contemplation of abstract definitions, is mathematics: the latter is above all an activity in which definitions are used to obtain concrete results. For this reason I have often been asked by teachers what is needed to give a rigorous proof that there are precisely 17 patterns.

A number of results are known which relate the characteristic numbers on a differentiable manifold X to the differentiable actions of the circle group S1on X. Many of these results are consequences of the Atiyah–Singer index theorem (2). In particular, an idea due to Hirzebruch sometimes permits one to deduce that certain characters which occur in the index theorem are in fact constant. Examples of this procedure can be found on page 594 of(2), on page 46 of(4), and on pages 21–22 of(1). Our aim in this paper is to make a systematic investigation of such results, and it is this aim which led us naturally to the concept of a modest vector bundle.

There are two natural ways in which to attempt a classification of the lattices which occur in crystallography or, more generally, of n-dimensional lattices. The first method is algebraic and proceeds by classifying the symmetry groups of the lattices. Thus an n-dimensional lattice is a discrete subgroup

of Rn and determines a symmetry group

which is a subgroup of the orthogonal group On. Two lattices T1 and T2 are said to determine the same crystal system if the symmetry groups G(T1) and G(T2) are conjugate: that is there exists a linear isomorphism ø: Rn → Rn such that

Let X be a compact differentiable manifold of dimension 2m. A differentiable map from X to euclidean (2m + t)-space is an immersion if its Jacobian has rank 2m at each point of X; it is an embedding if it is also one–one. The existence of such an embedding or immersion implies that the characteristic classes of X satisfy certain integrality conditions; these can be used to obtain lower bounds for the integer t. In a similar way many other geometric properties of X can be deduced from a single integrality theorem involving characteristic classes of various vector bundles over X (see for instance (5)).

It is well known that if M is a vector bundle over a space Y, and r: X → Y is a map, then there is an ‘induced’ vector bundle r*M over X. It is also known that a vector bundle L over X determines a ‘direct image’ vector bundle r* L over Y provided that r is a finite covering map ((1), § 1).

The idea of exact filling bundle may be described roughly as follows. Suppose that ξk is a vector bundle with fibre Rk, total space E(ξk) and base X. We say that ξk is a real k-plane bundle on X. Let in be the trivial n-plane bundle on X so that E(in) = X × Rn. A bundle monomorphism j: ξk → in defines a map : E(ξk)→Rn obtained by composition of the embedding E(ξk)→E(in) and the product projection E(in) → Rn. The map represents each fibre of ξk as a k-plane in Rn.

0. Introduction. In a paper (1) with the same title, one of us attempted to classify the reducible k2-bundles on a quadric surface Q = P1 × P1 defined over an algebraically closed field k. This attempt suffered from two defects: first, the classification was given by a one-one correspondence rather than by an algebraic parameter variety, and, secondly, there is a mistake in the proof of Proposition 6 of (1), as a result of which Proposition 6, Proposition 7 and the exceptional clause of the Theorem in (1) are false. To correct these defects we use the definitions, notations and numbering of (1) without comment. Finally, we show how the classification of reducible k2-bundles on P1 × … × P1 (n factors) is determined by that of reducible k2-bundles on P1 × P1. The net effect of these corrections is to underline the moral of (1): while the classification of reducible k2-bundles (but with a distinguished factor of the product variety P1 × … × P1) is as nice as it could possibly be, the classification of reducible k2-bundles alone is as nasty as it could possibly be. Since every decomposable k2-bundle is just a sum of two line bundles, we consider only indecomposable reducible k2-bundles.