Abstract:
In 1960 Puk\'anszky introduced an invariant associating to every masa in a separable $\mathrm{II}_1$ factor a non-empty subset of $\mathbb N\cup\{\infty\}$. This invariant examines the multiplicity structure of the von Neumann algebra generated by the left-right action of the masa. In this paper it is shown that every non-empty subset of $\mathbb N\cup\{\infty\}$ arises as the Puk\'anszky invariant of some masa in a separable McDuff $\mathrm{II}_1$ factor which contains a masa with Puk\'anszky invariant $\{1\}$. In particular the hyperfinite $\mathrm{II}_1$ factor and all separable McDuff $\mathrm{II}_1$ factors with a Cartan masa satisfy this hypothesis. In a general separable McDuff factor we show that every subset of $\mathbb N\cup\{\infty\}$ containing $\infty$ is obtained as a Puk\'anskzy invariant of some masa.

Abstract:
This paper reviews the latest information and perspectives on global phosphorus scarcity. Phosphorus is essential for food production and modern agriculture currently sources phosphorus fertilizers from finite phosphate rock. The 2008 food and phosphate fertilizer price spikes triggered increased concerns regarding the depletion timeline of phosphate rock reserves. While estimates range from 30 to 300 years and are shrouded by lack of publicly available data and substantial uncertainty, there is a general consensus that the quality and accessibility of remaining reserves are decreasing and costs will increase. This paper clarifies common sources of misunderstandings about phosphorus scarcity and identifies areas of consensus. It then asks, despite some persistent uncertainty, what would it take to achieve global phosphorus security? What would a ‘hard-landing’ response look like and how could preferred ‘soft-landing’ responses be achieved?

Abstract:
Phosphorus underpins the world’s food systems by ensuring soil fertility, maximising crop yields, supporting farmer livelihoods and ultimately food security. Yet increasing concerns around long-term availability and accessibility of the world’s main source of phosphorus—phosphate rock, means there is a need to investigate sustainable measures to buffer the world’s food systems against the long and short-term impacts of global phosphorus scarcity. While the timeline of phosphorus scarcity is contested, there is consensus that more efficient use and recycling of phosphorus is required. While the agricultural sector will be crucial in achieving this, sustainable phosphorus measures in sectors upstream and downstream of agriculture from mine to fork will also need to be addressed. This paper presents a comprehensive classification of all potential phosphorus supply- and demand-side measures to meet long-term phosphorus needs for food production. Examples range from increasing efficiency in the agricultural and mining sector, to technologies for recovering phosphorus from urine and food waste. Such measures are often undertaken in isolation from one another rather than linked in an integrated strategy. This integrated approach will enable scientists and policy-makers to take a systematic approach when identifying potential sustainable phosphorus measures. If a systematic approach is not taken, there is a risk of inappropriate investment in research and implementation of technologies and that will not ultimately ensure sufficient access to phosphorus to produce food in the future. The paper concludes by introducing a framework to assess and compare sustainable phosphorus measures and to determine the least cost options in a given context.

Abstract:
Using methods of R.J.Tauer we exhibit an uncountable family of singular masas in the hyperfinite $\textrm{II}_1$ factor $\R$ all with Puk\'anszky invariant $\{1\}$, no pair of which are conjugate by an automorphism of $R$. This is done by introducing an invariant $\Gamma(A)$ for a masa $A$ in a \IIi factor $N$ as the maximal size of a projection $e\in A$ for which $A e$ contains non-trivial centralising sequences for $eN e$. The masas produced give rise to a continuous map from the interval $[0,1]$ into the singular masas in $\R$ equiped with the $d_{\infty,2}$-metric. A result is also given showing that the Puk\'anszky invariant is $d_{\infty,2}$-upper semi-continuous. As a consequence, the sets of masas with Puk\'anszky invariant $\{n\}$ are all closed.

Abstract:
In 1965 Tauer produced a countably infinite family of semi-regular masas in the hyperfinite $\mathrm{II}_1$ factor, no pair of which are conjugate by an automorphism. This was achieved by iterating the process of passing to the algebra generated by the normalisers and, for each $n\in\mathbb N$, finding masas for which this procedure terminates at the $n$-th stage. Such masas are said to have length $n$. In this paper we consider a transfinite version of this idea, giving rise to a notion of ordinal valued length. We show that all countable ordinals arise as lengths of semi-regular masas in the hyperfinite $\mathrm{II}_1$ factor. Furthermore, building on work of Jones and Popa, we obtain all possible combinations of regular inclusions of irreducible subfactors in the normalising tower.

Abstract:
A comparison is made of the behaviour of some evolutionary algorithms in time-varying adaptive recursive filter systems. Simulations show that an algorithm including random immigrants outperforms a more conventional algorithm using the breeder genetic algorithm as the mutation operator when the time variation is discontinuous, but neither algorithm performs well when the time variation is rapid but smooth. To meet this deficit, a new hybrid algorithm which uses a hill climber as an additional genetic operator, applied for several steps at each generation, is introduced. A comparison is made of the effect of applying the hill climbing operator a few times to all members of the population or a larger number of times solely to the best individual; it is found that applying to the whole population yields the better results, substantially improved compared with those obtained using earlier methods.

Abstract:
We investigate possible preduals of the measure algebra $M(G)$ of a locally compact group and the Fourier algebra $A(G)$ of a separable compact group. Both of these algebras are canonically dual spaces and the canonical preduals make the multiplication separately weak$^*$-continuous so that these algebras are dual Banach algebras. In this paper we find additional conditions under which the preduals $C_0(G)$ of $M(G)$ and $C^*(G)$ of $A(G)$ are uniquely determined. In both cases we consider a natural coassociative multiplication and show that the canonical predual gives rise to the unique weak$^*$-topology making both the multiplication separately weak$^*$-continuous and the coassociative multiplication weak$^*$-continuous. In particular, dual cohomological properties of these algebras are well defined with this additional structure.

Abstract:
The groupoid normalisers of a unital inclusion $B\subseteq M$ of von Neumann algebras consist of the set $\mathcal{GN}_M(B)$ of partial isometries $v\in M$ with $vBv^*\subseteq B$ and $v^*Bv\subseteq B$. Given two unital inclusions $B_i\subseteq M_i$ of von Neumann algebras, we examine groupoid normalisers for the tensor product inclusion $B_1\ \overline{\otimes}\ B_2\subseteq M_1\ \overline{\otimes}\ M_2$ establishing the formula $$ \mathcal{GN}_{M_1\,\overline{\otimes}\,M_2}(B_1\ \overline{\otimes}\ B_2)''=\mathcal{GN}_{M_1}(B_1)''\ \overline{\otimes}\ \mathcal{GN}_{M_2}(B_2)'' $$ when one inclusion has a discrete relative commutant $B_1'\cap M_1$ equal to the centre of $B_1$ (no assumption is made on the second inclusion). This result also holds when one inclusion is a generator masa in a free group factor. We also examine when a unitary $u\in M_1\ \overline{\otimes}\ M_2$ normalising a tensor product $B_1\ \overline{\otimes}\ B_2$ of irreducible subfactors factorises as $w(v_1\otimes v_2)$ (for some unitary $w\in B_1\ \overline{\otimes}\ B_2$ and normalisers $v_i\in\mathcal{N}_{M_i}(B_i)$). We obtain a positive result when one of the $M_i$ is finite or both of the $B_i$ are infinite. For the remaining case, we characterise the II$_1$ factors $B_1$ for which such factorisations always occur (for all $M_1, B_2$ and $M_2$) as those with a trivial fundamental group.

Abstract:
We show that a simple separable unital nuclear nonelementary $C^*$-algebra whose tracial state space has a compact extreme boundary with finite covering dimension admits uniformly tracially large order zero maps from matrix algebras into its central sequence algebra. As a consequence, strict comparison implies $\Z$-stability for these algebras.

Abstract:
We prove that faithful traces on separable and nuclear C*-algebras in the UCT class are quasidiagonal. This has a number of consequences. Firstly, by results of many hands, the classification of unital, separable, simple and nuclear C*-algebras of finite nuclear dimension which satisfy the UCT is now complete. Secondly, our result links the finite to the general version of the Toms-Winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dimension. Finally, we confirm the Rosenberg conjecture: discrete, amenable groups have quasidiagonal C*-algebras.