We introduce the notion of ${\cal C}$-system of filters, generalizing the standard definitions of both extenders and towers of normal ideals. This provides a framework to develop the theory of extenders and towers in a more general and concise way. In this framework we investigate the topic of definability of generic large cardinals properties.

In this paper we analyze states on ${{\text{C}}^{*}}$-algebras and their relationship to filter-like structures of projections and positive elements in the unit ball. After developing the basic theory we use this to investigate the Kadison–Singer conjecture, proving its equivalence to an apparently quite weak paving conjecture and the existence of unique maximal centred extensions of projections coming from ultrafilters on the natural numbers. We then prove that Reid's positive answer to this for q-points in fact also holds for rapid p-points, and that maximal centred filters are obtained in this case. We then show that consistently, such maximal centred filters do not exist at all meaning that, for every pure state on the Calkin algebra, there exists a pair of projections on which the state is 1, even though the state is bounded strictly below 1 for projections below this pair. Next, we investigate towers, using cardinal invariant equalities to construct towers on the natural numbers that do and do not remain towers when canonically embedded into the Calkin algebra. Finally, we show that consistently, all towers on the natural numbers remain towers under this embedding.

The towers, plazas, and fortified temple of Chankillo are analyzed within the context of Central Andean culture. Throughout the cultural area, staircases were apparently the scenes of ritual procession, perhaps mimicking shamanic-like movement between the three worlds. The double stairways of the thirteen towers of Chankillo may have been designed for ritual movement. The gradual rotation of the successively higher towers suggests shamanic ascent between the terrestrial and solar realms. The major astronomical feature of Chankillo is its solar axis, oriented to December solstice sunrise and June solstice sunset. Along this axis, to the east and west of the towers, there are prominent plazas in which public ceremonies may have been staged, particularly at the time of June solstice sunset. Celebrants who reached the highest tower on sunset of June solstice would have been silhouetted by the setting sun as viewed by spectators in the eastern plaza just below the tower. In the large plaza west of the towers, a similar public ceremony could have been associated with setting of the June solstice sun over the Temple of the Pillars to the west of the towers. The thirteen towers may have been stations of the moon for public ceremonies during the bright half of the lunar cycle. The presence of Spondylus shells suggests lunar ritual. The duality of private/public ritual, evident at Chavín and elsewhere, may have been present at Chankillo where public ceremonies may been observed from the plazas, while more restricted ceremonies may have occurred behind the walls of the fortified temple. If a horizon calendar had been developed using the profiles of the thirteen towers, it appears to have been an unintended consequence of the initial design of the towers. The monumental size of the towers is incommensurate with the small putative observing stations. The June solstice sun misses the lowest tower by 7 solar diameters, which would have been an unacceptable error if the tower had been built originally to mark June solstice. Another unsatisfactory feature would have been the equal spacing of the towers. If a meaningful calendar had been desired that marked divisions of the year perhaps based upon the moon, it would have involve variable spacing of the towers, with the largest spacing around equinox.

The Spelling Theorem of B. B. Newman states that for a one-relator group (a1, … | Wn), any nontrivial word which represents the identity must contain a (cyclic) subword of W±n longer than Wn−1. We provide a new proof of the Spelling Theorem using towers of 2-complexes. We also give a geometric classification of reduced disc diagrams in one-relator groups with torsion. Either the disc diagram has three 2-cells which lie almost entuirly along the bounday, or the disc diagram looks like a ladder. We use this ladder theorem to prove that a large class of one-relator groups with torsion are locally quasiconvex.