Singer quadrangles

[no abstract available

Hyperfield extensions, characteristic one and the Connes-Consani plane connection

Inspired by a recent paper of Alain Connes and Catherina Consani which connects the geometric theory surrounding the elusive field with one element to sharply transitive group actions on finite and infinite projective spaces ("Singer actions"), we consider several fudamental problems and conjectures about Singer actions. Among other results, we show that virtually all infinite abelian groups and all (possibly infinitely generated) free groups act as Singer groups on certain projective planes, as a corollary of a general criterion. We investigate for which fields $\mathbb{F}$ the plane $\mathbf{P}^2(\mathbb{F}) = \mathbf{PG}(2,\mathbb{F})$ (and more generally the space $\mathbf{P}^n(\mathbb{F}) = \mathbf{PG}(n,\mathbb{F})$) admits a Singer group, and show, e.g., that for any prime $p$ and any positive integer $n > 1$, $\mathbf{PG}(n,\overline{\mathbb{F}_p})$ cannot admit Singer groups. One of the main results in characteristic $0$, also as a corollary of a criterion which applies to many other fields, is that $\mathbf{PG}(m,\mathbb{R})$ with $m \ne 0$ a positive even integer, cannot admit Singer groups.Comment: 25 pages; submitted (June 2014). arXiv admin note: text overlap with arXiv:1406.544

Nilpotent Singer groups

Let $N$ be a nilpotent group normal in a group $G$. Suppose that $G$ acts transitively upon the points of a finite non-Desarguesian projective plane $\mathcal{P}$. We prove that, if $\mathcal{P}$ has square order, then $N$ must act semi-regularly on $\mathcal{P}$. In addition we prove that if a finite non-Desarguesian projective plane $\mathcal{P}$ admits more than one nilpotent group which is regular on the points of $\mathcal{P}$ then $\mathcal{P}$ has non-square order and the automorphism group of $\mathcal{P}$ has odd order

A new short proof of the local index formula and some of its applications

We give a new short proof of the index formula of Atiyah and Singer based on combining Getzler's rescaling with Greiner's approach of the heat kernel asymptotics. As application we can easily compute the Connes-Moscovici cyclic cocycle of even and odd Dirac spectral triples, and then recover the Atiyah-Singer index formula (even case) and the Atiyah-Patodi-Singer spectral flow formula (odd case).Comment: v5: more typos fixed; 19 page

The Kadison-Singer Problem in Mathematics and Engineering

We will show that the famous, intractible 1959 Kadison-Singer problem in $C^{*}$-algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and Engineering. This gives all these areas common ground on which to interact as well as explaining why each of these areas has volumes of literature on their respective problems without a satisfactory resolution. In each of these areas we will reduce the problem to the minimum which needs to be proved to solve their version of Kadison-Singer. In some areas we will prove what we believe will be the strongest results ever available in the case that Kadison-Singer fails. Finally, we will give some directions for constructing a counter-example to Kadison-Singer

Singer, Singer!

What well-known singer does the definition to cross a river bring to your mind? Why, Tennessee Ernie FORD, of course. similarly, fish country suggests Judy GARLAND (= gar + land) and the clue the evidence: a joint! enigmatically refers to Rosemary CLOONEY (= clue + knee, in pronunciation)