arXiv subjects
Publications and source records attributed to Adam Jones.
We solve a long-standing open problem, posed by Landweber in 1974: Let $R = K[x_1, x_2, . . . ]$ be the ring of polynomials in countably many variables over a field $K$. Is the formal power series ring $R[[t]]$ a unique factorization domain? We prove that it is. The proof is based on a new general result in commutative algebra: If $R$ is a Krull domain, and $f \in R[[t]]$ is irreducible, then $f$ is irreducible modulo a finite power of $t$.
Artificial intelligence poses many risks, ranging from familiar present-day harms to unprecedented and potentially catastrophic ones. Effective risk management requires prioritization: we must understand which risks are most severe, who is most vulnerable, and who is most responsible for addressing them. We report results from a three-round Delphi study conducted late 2025 with 272 international AI experts. Experts rated 24 AI risks on harm probability and severity, sector and actor vulnerability, actor responsibility, and overall concern. Experts estimated the five most severe harms in the next 5 years were likely to come from dangerous capabilities, competitive dynamics, weapons & cyberattacks (including CBRNE), power centralization, and false information. In a business-as-usual scenario, experts judged 18 of 24 risks as having a more than 10% probability of catastrophic outcomes (e.g., more than 1 million deaths or more than USD 100B in financial loss) in the next 5 years (2025-2030). In a scenario where pragmatic mitigations are implemented, experts still judged five risks as having a more than 10% probability of catastrophic outcomes: dangerous capabilities, weapons & cyberattacks, environmental harm, inequality & unemployment, and power centralization. All 24 risks were judged as being more than 5% likely to cause catastrophic outcomes. AI users and the general public were judged the most vulnerable to these risks, but experts assigned the highest responsibility for addressing them to general-purpose AI developers and governance actors (including governments, regulators, and standards bodies). Across most risks, experts identified information, finance, and national security as the most vulnerable sectors. These findings can guide AI risk prioritization and clarify expert expectations about who should bear responsibility for mitigation.
Let G be a nilpotent p-valuable (compact p-adic Lie) group. There is an ongoing investigation into the prime ideals of its completed group algebra (Iwasawa algebra), and there remains an open conjecture that they can all be proved to have a canonical standard form. We very this conjecture for several new classes of nilpotent groups, including those corresponding to the positive subalgebra of almost all classical and exceptional types, curiously excluding those of type C.
We address a conjecture (referred to as sur in the literature) in the representation theory of a reductive p-adic Lie group G which has important implications for the relationship between mod-p smooth representations and pro-p Iwahori-Hecke modules, and is currently only known for G of rank 1. We prove that sur follows from exactness of the associated oriented chain complex of a coefficient system, when restricted to a local region of the Bruhat-Tits building for G. Our main result gives strong evidence towards this exactness in the case where G=SL_3(K) for K a totally ramified extension of Q_p. We also develop new combinatorial techniques for analysing the geometric realisation of the A_2 Bruhat-Tits building, which are fundamental to the proof of our main result, and which we hope will inspire further investigation in Bruhat-Tits theory.
We investigate the algebraic properties of the bounded skew power series ring $Q^+[[x;\sigma,\delta]]$ over a (complete, simple) \emph{standard} filtered artinian algebra $Q$ of positive characteristic. Here we are assuming that $(\sigma,\delta)$ is a commuting skew derivation of $Q$, where $\delta$ is inner, satisfying the appropriate compatibility conditions. In a previous work of the authors, it was proved that $Q^+[[x;\sigma,\delta]]$ is a simple ring whenever $\sigma$ has infinite inner order. We now extend this result to the case when $\sigma$ has finite inner order, proving that this ring is often simple, and always prime in cases of interest. This solves an important special case of an open question of Letzter, and yields important consequences for the classification of prime ideals in Iwasawa algebras of solvable groups.
We define and explore the bounded skew power series ring $R^+[[x;\sigma,\delta]]$ defined over a complete, filtered, Noetherian prime ring $R$ with a commuting skew derivation $(\sigma,\delta)$. We establish precise criteria for when this ring is well-defined, and for an appropriate completion $Q$ of $Q(R)$, we prove that if $Q$ has characteristic $p$, $\delta$ is an inner $\sigma$-derivation and no positive power of $\sigma$ is inner as an automorphism of $Q$, then $Q^+[[x;\sigma,\delta]]$ is often prime, and even simple under certain mild restrictions on $\delta$. It follows from this result that $R^+[[x;\sigma,\delta]]$ is itself prime.
We examine the power series ring $R[[X]]$ over a valuation ring $R$ of rank 1, with proper, dense value group. We give a counterexample to Hilbert's syzygy theorem for $R[[X]]$, i.e. an $R[[X]]$-module $C$ that is flat over $R$ and has flat dimension at least 2 over $R[[X]]$, contradicting a previously published result. The key ingredient in our construction is an exploration of the valuation theory of $R[[X]]$. We also use this theory to give a new proof that $R[[X]]$ is not a coherent ring, a fact which is essential in our construction of the module $C$.
Given a complete, positively filtered ring $(R,f)$ and a compatible skew derivation $(\sigma,\delta)$, we may construct its skew power series ring $R[[x;\sigma,\delta]]$. Due to topological obstructions, even if $\delta$ is an \emph{inner} $\sigma$-derivation, in general we cannot ``untwist" it, i.e. reparametrise to find a filtered isomorphism $R[[x; \sigma, \delta]] \cong R[[x'; \sigma]]$, as might be expected from the theory of skew polynomial rings; similarly when $\sigma$ is an inner automorphism. We find general conditions under which it is possible to untwist the multiplication data, and use this to analyse the structure of $R[[x;\sigma,\delta]]$ in the simplest case when $R$ is a matrix ring over a (noncommutative) noetherian discrete valuation ring.
In this paper, we investigate the structure of skew power series rings of the form $S = R[[x;\sigma,\delta]]$, where $R$ is a complete filtered ring and $(\sigma,\delta)$ is a skew derivation respecting the filtration. Our main focus is on the case in which $\sigma\delta = \delta\sigma$, and we aim to use techniques in non-commutative valuation theory to address the long-standing open question: if $P$ is an invariant prime ideal of $R$, is $PS$ a prime ideal of $S$? When $R$ has characteristic $p$, our results reduce this to a finite-index problem. We also give preliminary results in the "Iwasawa algebra" case $\delta = \sigma - \mathrm{id}_R$ in arbitrary characteristic. A key step in our argument will be to show that for a large class of Noetherian algebras, the nilradical is "almost" $(\sigma,\delta)$-invariant in a certain sense.
Given a $p$-adic field $K$ and a nilpotent uniform pro-$p$ group $G$, we prove that all primitive ideals in the $K$-rational Iwasawa algebra $KG$ are maximal, and can be reduced to a particular standard form. Setting $\mathcal{L}$ as the associated $\mathbb{Z}_p$-Lie algebra of $G$, our approach is to study the action of $KG$ on a Dixmier module $\widehat{D(\lambda)}$ over the affinoid envelope $\widehat{U(\mathcal{L})}_K$, and to prove that all primitive ideals can be reduced to annihilators of modules of this form.
The affinoid envelope, $\widehat{U(\mathcal{L})}$ of a free, finitely generated $\mathbb{Z}_p$-Lie algebra $\mathcal{L}$ has proven to be useful within the representation theory of compact $p$-adic Lie groups. Our aim is to further understand the algebraic structure of $\widehat{U(\mathcal{L})}$, and to this end, we will define a Dixmier module over $\widehat{U(\mathcal{L})}$, and prove that this object is generally irreducible in case where $\mathcal{L}$ is nilpotent. Ultimately, we will prove that all primitive ideals in the affinoid envelope can be described in terms of the annihilators of Dixmier modules, and using this, we aim towards proving that these algebras satisfy a version of the classical Dixmier-Moeglin equivalence.
Let p be a prime, K a p-adic field, G a nilpotent, uniform pro-p group. We prove that all faithful, primitive ideals in the Iwasawa algebra KG are controlled by the centraliser of the second term in the upper central series for G.
Let p be a prime, let G be a p-valuable, abelian-by-procyclic group, and let k be a field of characteristic p. We will prove that all faithful prime ideals of the completed group algebra kG are controlled by the centre of G, and a complete decomposition for Spec(kG) will follow. The principal technique we employ will be to study the convergence of Mahler expansions for inner automorphisms.