My aim in this lecture will be to try to convince you that the classification of the finite simple groups is nearing its end. This is, of course, a presumptuous statement, since one does not normally announce theorems as "almost proved". But the classification of simple groups is unlike any other single theorem in the history of mathematics, since the final proof will cover at least 5,000 journal pages. Moreover, at the present time, perhaps 80% of these pages exist either in print or in preprint form. One obtains a better perspective of the subject if instead of thinking of the classification as a single theorem, one views it as an entire field of mathematics-the structure of finite groups. Then when I say that there are some 4,000 pages in print, proving many general and specific results about simple groups, it should sound entirely reasonable, since one can make the same claim concerning many areas of mathematics. Thus my task is really to convince you that we have established so many results about simple groups and have developed sufficient techniques for completing the classification. There are other reasons for skepticism besides my premature announcement of the impending completion of the classification. Indeed, to the nonspecialist, simple group theory appears to be in a rather chaotic state. Strange sporadic simple groups dot the landscape-26 at last count; and they appear to be widely unrelated to each other. The five Mathieu groups, 100 years old, examples of highly transitive permutation groups, the four groups of Janko, each arising from the study of centralizers of involutions, the three Conway groups, determined from the automorphisms of a certain integral lattice in 24-dimensional Euclidean space, etc.

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The Sloan Digital Sky Survey has surveyed 14,555 square degrees of the sky, and delivered over a trillion pixels of imaging data. We present the large-scale clustering of 1.6 million quasars between z=0.5 and z=2.5 that have been classified from this imaging, representing the highest density of quasars ever studied for clustering measurements. This data set spans 0∼ 11,00 square degrees and probes a volume of 80 h−3 Gpc3. In principle, such a large volume and medium density of tracers should facilitate high-precision cosmological constraints. We measure the angular clustering of photometrically classified quasars using an optimal quadratic estimator in four redshift slices with an accuracy of ∼ 25% over a bin width of δl ∼ 10−15 on scales corresponding to matter-radiation equality and larger (0ℓ ∼ 2−3).Observational systematics can strongly bias clustering measurements on large scales, which can mimic cosmologically relevant signals such as deviations from Gaussianity in the spectrum of primordial perturbations. We account for systematics by employing a new method recently proposed by Agarwal et al. (2014) to the clustering of photometrically classified quasars. We carefully apply our methodology to mitigate known observational systematics and further remove angular bins that are contaminated by unknown systematics. Combining quasar data with the photometric luminous red galaxy (LRG) sample of Ross et al. (2011) and Ho et al. (2012), and marginalizing over all bias and shot noise-like parameters, we obtain a constraint on local primordial non-Gaussianity of fNL = −113+154−154 (1σ error). We next assume that the bias of quasar and galaxy distributions can be obtained independently from quasar/galaxy-CMB lensing cross-correlation measurements (such as those in Sherwin et al. (2013)). This can be facilitated by spectroscopic observations of the sources, enabling the redshift distribution to be completely determined, and allowing precise estimates of the bias parameters. In this paper, if the bias and shot noise parameters are fixed to their known values (which we model by fixing them to their best-fit Gaussian values), we find that the error bar reduces to 1σ ≃ 65. We expect this error bar to reduce further by at least another factor of five if the data is free of any observational systematics. We therefore emphasize that in order to make best use of large scale structure data we need an accurate modeling of known systematics, a method to mitigate unknown systematics, and additionally independent theoretical models or observations to probe the bias of dark matter halos.

The integral group ring of a finite group determines the isomorphism type of the chief factors of the group. Two proofs are given, one of which applies Cameron's and Teague's generalisation of Artin's theorem on the orders of finite simple groups to the orders of characteristically simple groups. The generalisation states that a direct power of a finite simple group is determined by its order with the same two types of exception which Artin found. Its proof, given here in detail, adapts and makes explicit certain functions of a natural number variable which Artin used implicitly. These functions contribute to the argument through a series of tables which supply their values for the orders of finite simple groups.