Post-war reconstruction (c. AD 61–70)

London was rebuilt after the Boudican revolt, chiefly in the period AD 62–4. Military engineers set a new fort over the ashes of the city destroyed by British rebels, rebuilt the harbour with massive new quays, introduced new hydraulic engineering to supply London’s bathhouses, and established new roads and causeways to speed the movement of people and goods. The presence of detachments of auxiliary soldiers used to garrison the city after the revolt is witnessed by exchanges recorded in wooden writing tablets, and by finds of military and cavalry equipment. High status cemeteries included the tomb of the procurator Julius Classicianus, an exceptional group of exotic cinerary urns including one carved from Egyptian porphyry, and the inhumation of a woman dressed in a way that might identify her as a member of the pre-Roman aristocracy. Irregular and fragmented burials are also described, and it is suggested that these may witness practices of corpse abuse and necrophobic ritual. The mutilated corpses of those denied normal burial may have been dispatched to the underworld by disposal in water, and disturbed corpses besides London Bridge may include the victims of Roman retributive violence following the Boudican revolt.

Towards the classification of symplectic linear quotient singularities admitting a symplectic resolution

Over the past 2 decades, there has been much progress on the classification of symplectic linear quotient singularities V/G admitting a symplectic (equivalently, crepant) resolution of singularities. The classification is almost complete but there is an infinite series of groups in dimension 4—the symplectically primitive but complex imprimitive groups—and 10 exceptional groups up to dimension 10, for which it is still open. In this paper, we treat the remaining infinite series and prove that for all but possibly 39 cases there is no symplectic resolution. We thereby reduce the classification problem to finitely many open cases. We furthermore prove non-existence of a symplectic resolution for one exceptional group, leaving $$39+9=48$$ 39 + 9 = 48 open cases in total. We do not expect any of the remaining cases to admit a symplectic resolution.

On the involution fixity of simple groups

Let $G$ be a finite permutation group of degree $n$ and let ${\rm ifix}(G)$ be the involution fixity of $G$ , which is the maximum number of fixed points of an involution. In this paper, we study the involution fixity of almost simple primitive groups whose socle $T$ is an alternating or sporadic group; our main result classifies the groups of this form with ${\rm ifix}(T) \leqslant n^{4/9}$ . This builds on earlier work of Burness and Thomas, who studied the case where $T$ is an exceptional group of Lie type, and it strengthens the bound ${\rm ifix}(T) > n^{1/6}$ (with prescribed exceptions), which was proved by Liebeck and Shalev in 2015. A similar result for classical groups will be established in a sequel.

S.-S. Chern’s study of almost-complex structures on the six-sphere

In April 2003, Chern began a study of almost-complex structures on the six-sphere, with the idea of exploiting the special properties of its well-known almost-complex structure invariant under the exceptional group [Formula: see text]. While he did not solve the (currently still open) problem of determining whether there exists an integrable almost-complex structure on [Formula: see text], he did prove a significant identity that resolves the question for an interesting class of almost-complex structures on [Formula: see text].

Lipid biomarker-based verification of TB infection in mother’s and daughter’s mummified human remains (Vác Mummy Collection, 18th century, CE, Hungary)

The perpetual burden of tuberculosis (TB) keeps drawing the focus of research on this disease. Among other risk factors (e.g., poor living conditions, malnutrition, smoking, HIV infection, etc.), being in close contact with a TB infected person requires special attention. For a better understanding of the disease, paleopathological investigations concerning TB have been carried out with various techniques for a long a time; nevertheless, analysis of incidence among family members is hardly possible in past populations. An exceptional group of naturally mummified individuals, the collection of the Vác mummies (Hungary, 18th century CE), is known about the large TB incidence rate, which has been revealed by aDNA analysis. Besides the high rate of TB infection, another interesting aspect of the collection is that in some cases, the family connections could be reconstructed. In this paper, we present the mycocerosic acid profiles gained by HPLC-HESI-MS measurements of two Vác mummies, who were mother and daughter according to the personal records. Earlier metagenomic analysis already revealed mixed M. tuberculosis infection with the same bacterial strains in both individuals; moreover, the same bacterial strains were recorded in both cases.

Exterior powers of the standard E6-module: An elementary approach

For the standard [Formula: see text]-dimensional representation [Formula: see text] of the exceptional group [Formula: see text] of type [Formula: see text] we prove that [Formula: see text] is a Donkin pair if and only if the characteristic of a ground field is greater than [Formula: see text]. We also develop an elementary approach to describe submodule structure of any exterior power of [Formula: see text].

An exceptional Siegel–Weil formula and poles of the Spin L-function of

We show a Siegel–Weil formula in the setting of exceptional theta correspondence. Using this, together with a new Rankin–Selberg integral for the Spin L-function of $\text{PGSp}_{6}$ discovered by Pollack, we prove that a cuspidal representation of $\text{PGSp}_{6}$ is a (weak) functorial lift from the exceptional group $G_{2}$ if its (partial) Spin L-function has a pole at $s=1$.

Flag-Transitive Non-Symmetric 2-Designs with $(r, \lambda)=1$ and Exceptional Groups of Lie Type

This paper determines all pairs $(\mathcal{D},G)$ where $\mathcal{D}$ is a non-symmetric 2-$(v,k,\lambda)$ design with $(r,\lambda)=1$ and $G$ is the almost simple flag-transitive automorphism group of $\mathcal{D}$ with an exceptional socle of Lie type. We prove that if $T\trianglelefteq G\leq Aut(T)$ where $T$ is an exceptional group of Lie type, then $T$ must be the Ree group or Suzuki group, and there are five classes of designs $\mathcal{D}$.

Albert algebras over rings and related torsors

We study exceptional Jordan algebras and related exceptional group schemes over commutative rings from a geometric point of view, using appropriate torsors to parametrize and explain classical and new constructions, and proving that over rings, they give rise to nonisomorphic structures. We begin by showing that isotopes of Albert algebras are obtained as twists by a certain $\mathrm F_4$ -torsor with total space a group of type $\mathrm E_6$ and, using this, that Albert algebras over rings in general admit nonisomorphic isotopes even in the split case, as opposed to the situation over fields. We then consider certain $\mathrm D_4$ -torsors constructed from reduced Albert algebras, and show how these give rise to a class of generalised reduced Albert algebras constructed from compositions of quadratic forms. Showing that this torsor is nontrivial, we conclude that the Albert algebra does not uniquely determine the underlying composition, even in the split case. In a similar vein, we show that a given reduced Albert algebra can admit two coordinate algebras which are nonisomorphic and have nonisometric quadratic forms, contrary, in a strong sense, to the case over fields, established by Albert and Jacobson.