Hilbert 90 for KnM

This chapter formulates a norm-trace relation for the Milnor 𝐾-theory and étale cohomology of a cyclic Galois extension, herein called Hilbert 90 for 𝐾𝑀 𝑛. To begin, the chapter uses condition BL(n) to establish a related exact sequence in Galois cohomology. It then establishes that condition BL(n − 1) implies the particular case of condition H90(n) for 𝓁-special fields 𝑘 such that 𝐾𝑀 𝑛(𝑘) is 𝓁-divisible. This case constitutes the first part of the inductive step in the proof of Theorem A. The remainder of this chapter explains how to reduce the general case to this particular one. The chapter concludes with some background on the Hilbert 90 for 𝐾𝑀 𝑛.

The Intertwining Lemma

This chapter gives a proof of the Intertwining Lemma. Section 20.2 lists out the formulas for all the maps involved. Section 20.3 recalls the definition of Z* and proves Statement 3 of the Intertwining Lemma. Section 20.4 proves statements 1 and 2 of the Intertwining Lemma for a single point. Section 20.5 decomposes Z* into two smaller pieces as a prelude to giving the inductive step in the proof. Section 20.6 proves the following induction step: If the Intertwining Lemma is true for g ɛ GA then it is also true for g + dTA (0, 1). Section 20.7 explains what needs to be done to finish the proof of the Intertwining Lemma. Section 20.8 proves the Intertwining Theorem for points in Π‎A corresponding to the points gn = (n + 1/2)(1 + A, 1 − A) for n = 0, 1, 2, ... which all belong to GA. This result combines with the induction step to finish the proof, as explained in Section 20.7.

Explicit asymptotic expansions for tame supercuspidal characters

We combine the ideas of a Harish-Chandra–Howe local character expansion, which can be centred at an arbitrary semisimple element, and a Kim–Murnaghan asymptotic expansion, which so far has been considered only around the identity. We show that, for most smooth, irreducible representations (those containing a good, minimal K-type), Kim–Murnaghan-type asymptotic expansions are valid on explicitly defined neighbourhoods of nearly arbitrary semisimple elements. We then give an explicit, inductive recipe for computing the coefficients in an asymptotic expansion for a tame supercuspidal representation. The only additional information needed in the inductive step is a fourth root of unity, which we expect to be useful in proving stability and endoscopic-transfer identities.

What the argument from evil should, but cannot, be

Michael Tooley has developed a sophisticated evidential version of the argument from evil that aims to circumvent sceptical theist responses. Evidential arguments from evil depend on the plausibility of inductive inferences from premises about our inability to see morally sufficient reasons for God to permit evils to conclusions about there being no morally sufficient reasons for God to permit evils. Tooley's defence of this inductive step depends on the idea that the existence of unknown rightmaking properties is no more likely, a priori, than the existence of unknown wrongmaking properties. I argue that Tooley's argument begs the question against the theist, and, in doing so, commits an analogue of the base rate fallacy. I conclude with some reflections on what a successful argument from evil would have to establish.

Are Model Organisms Theoretical Models?

This article compares the epistemic roles of theoretical models and model organisms in science, and specifically the role of non-human animal models in biomedicine. Much of the previous literature on this topic shares an assumption that animal models and theoretical models have a broadly similar epistemic role—that of indirect representation of a target through the study of a surrogate system. Recently, Levy and Currie (2015) have argued that model organism research and theoretical modelling differ in the justification of model-to-target inferences, such that a unified account based on the widely accepted idea of modelling as indirect representation does not similarly apply to both. I defend a similar conclusion, but argue that the distinction between animal models and theoretical models does not always track a difference in the justification of model-to-target inferences. Case studies of the use of animal models in biomedicine are presented to illustrate this. However, Levy and Currie’s point can be argued for in a different way. I argue for the following distinction. Model organisms (and other concrete models) function as surrogate sources of evidence, from which results are transferred to their targets by empirical extrapolation. By contrast, theoretical modelling does not involve such an inductive step. Rather, theoretical models are used for drawing conclusions from what is already known or assumed about the target system. Codifying assumptions about the causal structure of the target in external representational media (e.g. equations, graphs) allows one to apply explicit inferential rules to reach conclusions that could not be reached with unaided cognition alone (cf. Kuorikoski and Ylikoski 2015).

Generating the Mapping Class Group

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

The Pila–Wilkie theorem for subanalytic families: a complex analytic approach

We present a complex analytic proof of the Pila–Wilkie theorem for subanalytic sets. In particular, we replace the use of $C^{r}$-smooth parametrizations by a variant of Weierstrass division. As a consequence we are able to apply the Bombieri–Pila determinant method directly to analytic families without limiting the order of smoothness by a $C^{r}$ parametrization. This technique provides the key inductive step for our recent proof (in a closely related preprint) of the Wilkie conjecture for sets definable using restricted elementary functions. As an illustration of our approach we prove that the rational points of height $H$ in a compact piece of a complex-analytic set of dimension $k$ in $\mathbb{C}^{m}$ are contained in $O(1)$ complex-algebraic hypersurfaces of degree $(\log H)^{k/(m-k)}$. This is a complex-analytic analog of a recent result of Cluckers, Pila, and Wilkie for real subanalytic sets.