On period spaces for p-divisible groups

In their book Rapoport and Zink constructed rigid analytic period spaces for Fontaine's filtered isocrystals, and period morphisms from moduli spaces of p-divisible groups to some of these period spaces. We determine the image of these period morphisms, thereby contributing to a question of Grothendieck. We give examples showing that only in rare cases the image is all of the Rapoport-Zink period space.Comment: 6 pages, v2: minor changes, v3: new exposition, no new result

On Fox and augmentation quotients of semidirect products

Let $G$ be a group which is the semidirect product of a normal subgroup $N$ and some subgroup $T$. Let $I^n(G)$, $n\ge 1$, denote the powers of the augmentation ideal $I(G)$ of the group ring $\Z(G)$. Using homological methods the groups $Q_n(G,H) = I^{n-1}(G)I(H)/I^{n}(G)I(H)$, $H=G,N,T$, are functorially expressed in terms of enveloping algebras of certain Lie rings associated with $N$ and $T$, in the following cases: for $n\le 4$ and arbitrary $G,N,T$ (except from one direct summand of $Q_4(G,N)$), and for all $n\ge 2$ if certain filtration quotients of $N$ and $T$ are torsionfree.Comment: 39 pages; paper thoroughly revised: notation and presentation improved, many details and new result added (Theorem 1.7

Bacterial Hsp70 resolves misfolded states and accelerates productive folding of a multi-domain protein

The ATP-dependent Hsp70 chaperones (DnaK in E. coli) mediate protein folding in cooperation with J proteins and nucleotide exchange factors (E. coli DnaJ and GrpE, respectively). The Hsp70 system prevents protein aggregation and increases folding yields. Whether it also enhances the rate of folding remains unclear. Here we show that DnaK/DnaJ/GrpE accelerate the folding of the multi-domain protein firefly luciferase (FLuc) 20-fold over the rate of spontaneous folding measured in the absence of aggregation. Analysis by single-pair FRET and hydrogen/deuterium exchange identified inter-domain misfolding as the cause of slow folding. DnaK binding expands the misfolded region and thereby resolves the kinetically-trapped intermediates, with folding occurring upon GrpE-mediated release. In each round of release DnaK commits a fraction of FLuc to fast folding, circumventing misfolding. We suggest that by resolving misfolding and accelerating productive folding, the bacterial Hsp70 system can maintain proteins in their native states under otherwise denaturing stress conditions. The Hsp70 system prevents protein aggregation and increases folding yields, but it is unknown whether it also enhances the rate of folding. Here the authors combine refolding assays, FRET and hydrogen/deuterium exchange-mass spectrometry measurements to study the folding of firefly luciferase and find that the bacterial Hsp70 actively promotes the folding of this multi-domain protein

Dynamics of spinning test particles in Kerr spacetime

We investigate the dynamics of relativistic spinning test particles in the spacetime of a rotating black hole using the Papapetrou equations. We use the method of Lyapunov exponents to determine whether the orbits exhibit sensitive dependence on initial conditions, a signature of chaos. In the case of maximally spinning equal-mass binaries (a limiting case that violates the test-particle approximation) we find unambiguous positive Lyapunov exponents that come in pairs ± lambda, a characteristic of Hamiltonian dynamical systems. We find no evidence for nonvanishing Lyapunov exponents for physically realistic spin parameters, which suggests that chaos may not manifest itself in the gravitational radiation of extreme mass-ratio binary black-hole inspirals (as detectable, for example, by LISA, the Laser Interferometer Space Antenna)

A Dictionary between Fontaine-Theory and its Analogue in Equal Characteristic

In this survey we explain the main ingredients and results of the analogue of Fontaine-Theory in equal positive characteristic which was recently developed by Genestier-Lafforgue and the author.Comment: 23 page

Quadratic maps between modules

We introduce a notion of $R$-quadratic maps between modules over a commutative ring $R$ which generalizes several classical notions arising in linear algebra and group theory. On a given module $M$ such maps are represented by $R$-linear maps on a certain module $P^2_R(M)$. The structure of this module is described in term of the symmetric tensor square $Sym^2_R(M)$, the degree 2 component $\Gamma^2_R(M)$ of the divided power algebra over $M$, and the ideal $I_2$ of $R$ generated by the elements $r^2-r$, $r\in R$. The latter is shown to represent quadratic derivations on $R$ which arise in the theory of modules over square rings. This allows to extend the classical notion of nilpotent $R$-group of class 2 with coefficients in a 2-binomial ring $R$ to any ring $R$. We provide a functorial presentation of $I_2$ and several exact sequences embedding the modules $P^2_R(M)$ and $\Gamma^2_R(M)$.Comment: 22 page