In 2003, Kechris, Pestov and Todorcevic showed that the structure of certain separable metric spaces--called ultrahomogeneous--is closely related to the combinatorial behavior of the class of their finite metric spaces. The purpose of the present paper is to explore different aspects of this connection.

Let $\mathcal{M}$ denote the space of probability measures on $\mathbb{R}^D$ endowed with the Wasserstein metric. A differential calculus for a certain class of absolutely continuous curves in $\mathcal{M}$ was introduced by Ambrosio, Gigli, and Savare. In this paper the authors develop a calculus for the corresponding class of differential forms on $\mathcal{M}$. In particular they prove an analogue of Green's theorem for 1-forms and show that the corresponding first cohomology group, in the sense of de Rham, vanishes. For $D=2d$ the authors then define a symplectic distribution on $\mathcal{M}$ in terms of this calculus, thus obtaining a rigorous framework for the notion of Hamiltonian systems as introduced by Ambrosio and Gangbo. Throughout the paper the authors emphasize the geometric viewpoint and the role played by certain diffeomorphism groups of $\mathbb{R}^D$.

This paper shows that properties of projective modules over a group ring $\mathbf{Z}_p[\Delta]$, where $\Delta$ is a finite Galois group, can be used to study the behavior of certain invariants which occur naturally in Iwasawa theory for an elliptic curve $E$. Modular representation theory for the group $\Delta$ plays a crucial role in this study. It is necessary to make a certain assumption about the vanishing of a $\mu$-invariant. The author then studies $\lambda$-invariants $\lambda_E(\sigma)$, where $\sigma$ varies over the absolutely irreducible representations of $\Delta$. He shows that there are non-trivial relationships between these invariants under certain hypotheses.

The author considers homomorphisms $H \to K$ from an affine group scheme $H$ over a field $k$ of characteristic zero to a proreductive group $K$. Using a general categorical splitting theorem, Andre and Kahn proved that for every $H$ there exists such a homomorphism which is universal up to conjugacy. The author gives a purely group-theoretic proof of this result. The classical Jacobson-Morosov theorem is the particular case where $H$ is the additive group over $k$. As well as universal homomorphisms, the author considers more generally homomorphisms $H \to K$ which are minimal, in the sense that $H \to K$ factors through no proper proreductive subgroup of $K$. For fixed $H$, it is shown that the minimal $H \to K$ with $K$ reductive are parametrised by a scheme locally of finite type over $k$.

"Volume 209, number 985 (fourth of 5 numbers)."

The multiplier ideals of an ideal in a regular local ring form a family of ideals parameterized by non-negative rational numbers. As the rational number increases the corresponding multiplier ideal remains unchanged until at some point it gets strictly smaller. A rational number where this kind of diminishing occurs is called a jumping number of the ideal. In this manuscript the author gives an explicit formula for the jumping numbers of a simple complete ideal in a two-dimensional regular local ring. In particular, he obtains a formula for the jumping numbers of an analytically irreducible plane curve. He then shows that the jumping numbers determine the equisingularity class of the curve.

This book studies generalized Donaldson-Thomas invariants $\bar{DT}{}^\alpha(\tau)$. They are rational numbers which `count' both $\tau$-stable and $\tau$-semistable coherent sheaves with Chern character $\alpha$ on $X$; strictly $\tau$-semistable sheaves must be counted with complicated rational weights. The $\bar{DT}{}^\alpha(\tau)$ are defined for all classes $\alpha$, and are equal to $DT^\alpha(\tau)$ when it is defined. They are unchanged under deformations of $X$, and transform by a wall-crossing formula under change of stability condition $\tau$. To prove all this, the authors study the local structure of the moduli stack $\mathfrak M$ of coherent sheaves on $X$. They show that an atlas for $\mathfrak M$ may be written locally as $\mathrm{Crit}(f)$ for $f:U\to{\mathbb C}$ holomorphic and $U$ smooth, and use this to deduce identities on the Behrend function $\nu_\mathfrak M$. They compute the invariants $\bar{DT}{}^\alpha(\tau)$ in examples, and make a conjecture about their integrality properties. They also extend the theory to abelian categories $\mathrm{mod}$-$\mathbb{C}Q\backslash I$ of representations of a quiver $Q$ with relations $I$ coming from a superpotential $W$ on $Q$.