It would have been easy for a less imaginative historian of mathematics than Herbert Mehrtens to have portrayed the work of Hilbert, Hausdorff, and other modernists as pioneers, and those who did not subscribe to their program as people who failed, were not good enough to make the turn, and were eventually and convincingly left behind. That he did not do so is not only because this would have been a shallow, selective view of the facts: it is incompatible with his Foucauldian approach to the relations between knowledge and power. Instead, he defined what I see as the most intriguing category of actor in his Moderne—Sprache—Mathematik (1990), the Gegenmoderner, or counter-moderns. The three men who characterize this position are Felix Klein, Henri Poincaré, and Luitzen Brouwer, and each merits a section in the book.

Of the three, Poincaré is the hardest to contain within that category. The range of his work, the nature of his influence, and the shifting standards by which mathematical significance has been evaluated by mathematicians, historians of mathematics, and society at large, all contribute to the problem. After thirty years, the methodological presumptions and aspirations of historians of mathematics have also changed, and I shall suggest that one way to appreciate the richness of Mehrtens’ book, to gain insight into what is meant by mathematical modernism, and to acknowledge a generation of work by other historians since 1990, is to re-examine aspects of Poincaré’s life and work and scholarship about him. Prodded by remarks by Leo Corry, Moritz Epple, and David Rowe, I shall suggest that the simple but useful dichotomy modern/counter-modern must be seen as a way into a more complicated situation, one in which different aspects of mathematics, specifically applied mathematics and the relationship of mathematics to contemporary physics, require fresh accounts of the role of modern mathematics in society.

In the literature, predicativism is connected not only with the Vicious Circle Principle but also with the idea that certain totalities are inherently potential. To explain the connection between these two aspects of predicativism, we explore some approaches to predicativity within the modal framework for potentiality developed in Linnebo (2013) and Linnebo and Shapiro (2019). This puts predicativism into a more general framework and helps to sharpen some of its key theses.

Corresponding to known results on Orlicz–Sobolev inequalities which are stronger than the Poincaré inequality, this paper studies the weaker Orlicz–Poincaré inequality. More precisely, for any Young function $\varPhi$ whose growth is slower than quadric, the Orlicz–Poincaré inequality

$$ \|f\|_\varPhi^2\le C\E(f,f),\qquad\mu(f):=\int f\,\mathrm{d}\mu=0 $$

is studied by using the well-developed weak Poincaré inequalities, where $\E$ is a conservative Dirichlet form on $L^2(\mu)$ for some probability measure $\mu$. In particular, criteria and concrete sharp examples of this inequality are presented for $\varPhi(r)=r^p$ $(p\in[1,2))$ and $\varPhi(r)= r^2\log^{-\delta}(\mathrm{e} +r^2)$ $(\delta>0)$. Concentration of measures and analogous results for non-conservative Dirichlet forms are also obtained. As an application, the convergence rate of porous media equations is described.

We compute the (generalized) Poincaré series of the multi-index filtration defined by a finite collection of divisorial valuations on the ring $\mathcal{O}_{\mathbb{C}^2,0}$ of germs of functions of two variables. We use the method initially elaborated by the authors and Campillo for computing the similar Poincaré series for the valuations defined by the irreducible components of a plane curve singularity. The method is essentially based on the notions of the so-called extended semigroup and of the integral with respect to the Euler characteristic over the projectivization of the space of germs of functions of two variables. The last notion is similar to (and inspired by) the notion of the motivic integration.

AMS 2000 Mathematics subject classification: Primary 14B05; 16W70

Under an extra hypothesis satisfied in every known case, we show that the Euler class of an orientable odd-dimensional Poincaré duality group over any ring has order at most two. We construct groups that are of type FL over the complex numbers but are not FL over the rationals. We construct group algebras over fields for which $K_0$ contains torsion, and construct non-free stably free modules for the group algebras of certain virtually free groups.

AMS 2000 Mathematics subject classification: Primary 19A31. Secondary 16S34; 20J05; 55N15; 57P10

The paper deals with the asymptotic behaviour of infinite mean Galton–Watson processes (denoted by {Zn}). We show that these processes can be classified as regular or irregular. The regular ones are characterized by the property that for any sequence of positive constants {Cn}, for which a.s. exists, The irregular ones, which will be shown by examples to exist, have the property that there exists a sequence of constants {Cn} such that In Part 1 we study the properties of {Zn/Cn} and give some characterizations for both regular and irregular processes. Part 2 starts with an a.s. convergence result for {yn(Zn)}, where {yn} is a suitable chosen sequence of functions related to {Zn}. Using this, we then derive necessary and sufficient conditions for the a.s. convergence of {U(Zn)/Cn}, where U is a slowly varying function. The distribution function of the limit is shown to satisfy a Poincaré functional equation. Finally we show that for every process {Zn} it is possible to construct explicitly functions U, such that U(Zn)/en converges a.s. to a non-degenerate proper random variable. If the process is regular, all these functions U are slowly varying. The distribution of the limit depends on U, and we show that by appropriate choice of U we may get a limit distribution which has a positive and continuous density or is continuous but not absolutely continuous or even has no probability mass on certain intervals. This situation contrasts strongly with the finite mean case.