Revisiting the Afterlife: The Inadequacies of Heaven and Hell

This paper deals with some of the ambiguities that are associated with the intermediate and final states after death. Whereas many in the church have dismissed these concepts as myths of the ancients, this discussion shows how the grounding of such beliefs in the Hebrew mindset was the key to Jesus’ own teachings about the afterlife. The argument begins by developing a biblical anthropology over against the modern naturalistic anthropologies that have largely dominated the philosophical and theological scenes. From here we look at the Old Testament concept of the afterlife, and how the modern view that the Hebrews were ambivalent about such a concept is plainly false. Then it is argued that the New Testament doctrines of heaven and hell, which become very specific at this point, are thoroughly indebted to Jewish underpinnings. Without this foundation there would be no clear divisions within the realms of the dead, but because Jesus and his followers assume the validity of the Old Testament material they are able to flesh out such eschatological questions as where Jesus went after death, and where the saint and reprobate will go today. Far from being a stale theological issue, this study has direct bearing upon how one evangelizes today. For when the specific concepts are grasped, the believer will realize that the lost are not going to hell, at least not yet

Which finite simple groups are unit groups?

We prove that if $G$ is a finite simple group which is the unit group of a ring, then $G$ is isomorphic to either (a) a cyclic group of order 2; (b) a cyclic group of prime order $2^k -1$ for some $k$; or (c) a projective special linear group $PSL_n(\mathbb{F}_2)$ for some $n \geq 3$. Moreover, these groups do (trivially) all occur as unit groups. We deduce this classification from a more general result, which holds for groups $G$ with no non-trivial normal 2-subgroup

Almost purity and overconvergent Witt vectors

In a previous paper, we stated a general almost purity theorem in the style of Faltings: if R is a ring for which the Frobenius maps on finite p-typical Witt vectors over R are surjective, then the integral closure of R in a finite \'etale extension of R[p^{-1}] is "almost" finite \'etale over R. Here, we use almost purity to lift the finite \'etale extension of R[p^{-1}] to a finite \'etale extension of rings of overconvergent Witt vectors. The point is that no hypothesis of p-adic completeness is needed; this result thus points towards potential global analogues of p-adic Hodge theory. As an illustration, we construct (phi, Gamma)-modules associated to Artin Motives over Q. The (phi, Gamma)-modules we construct are defined over a base ring which seems well-suited to generalization to a more global setting; we plan to pursue such generalizations in later work

Satellite operators as group actions on knot concordance

Any knot in a solid torus, called a pattern or satellite operator, acts on knots in the 3-sphere via the satellite construction. We introduce a generalization of satellite operators which form a group (unlike traditional satellite operators), modulo a generalization of concordance. This group has an action on the set of knots in homology spheres, using which we recover the recent result of Cochran and the authors that satellite operators with strong winding number $\pm 1$ give injective functions on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4--dimensional Poincare Conjecture. The notion of generalized satellite operators yields a characterization of surjective satellite operators, as well as a sufficient condition for a satellite operator to have an inverse. As a consequence, we are able to construct infinitely many non-trivial satellite operators P such that there is a satellite operator $\overline{P}$ for which $\overline{P}(P(K))$ is concordant to K (topologically as well as smoothly in a potentially exotic $S^3\times [0,1]$) for all knots K; we show that these satellite operators are distinct from all connected-sum operators, even up to concordance, and that they induce bijective functions on topological concordance classes of knots, as well as smooth concordance classes of knots modulo the smooth 4--dimensional Poincare Conjecture.Comment: 20 pages, 9 figures; in the second version, we have added several new results about surjectivity of satellite operators, and inverses of satellite operators, and the exposition and structure of the paper have been improve

On the Witt vector Frobenius

We study the kernel and cokernel of the Frobenius map on the $p$-typical Witt vectors of a commutative ring, not necessarily of characteristic $p$. We give some equivalent conditions to surjectivity of the Frobenus map on both finite and infinite length Witt vectors; the former condition turns out to be stable under certain integral extensions, a fact which relates closely to a generalization of Faltings's almost purity theorem