Stratifying quotient stacks and moduli stacks

Recent results in geometric invariant theory (GIT) for non-reductive linear algebraic group actions allow us to stratify quotient stacks of the form [X/H], where X is a projective scheme and H is a linear algebraic group with internally graded unipotent radical acting linearly on X, in such a way that each stratum [S/H] has a geometric quotient S/H. This leads to stratifications of moduli stacks (for example, sheaves over a projective scheme) such that each stratum has a coarse moduli space.Comment: 25 pages, submitted to the Proceedings of the Abel Symposium 201

The Cohen-Macaulay property of separating invariants of finite groups

In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply that the ring of invariants is non Cohen-Macaulay actually imply that no graded separating algebra is Cohen-Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen-Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen-Macaulay graded separating algebra implies the group is generated by bireflections. Furthermore, we show that, for a $p$-group, the existence of a Cohen-Macaulay graded separating algebra implies the group is generated by bireflections. Additionally, we give an example which shows that Cohen-Macaulay separating algebras can occur when the ring of invariants is not Cohen-Macaulay.Comment: We removed the conjecture which appeared in previous versions: we give a counter-example. We fixed the proof of Lemma 2.2 (previously Remark 2.2). 16 page

Inter-domain Communication Mechanisms in an ABC Importer: A Molecular Dynamics Study of the MalFGK2E Complex

ATP-Binding Cassette transporters are ubiquitous membrane proteins that convert the energy from ATP-binding and hydrolysis into conformational changes of the transmembrane region to allow the translocation of substrates against their concentration gradient. Despite the large amount of structural and biochemical data available for this family, it is still not clear how the energy obtained from ATP hydrolysis in the ATPase domains is “transmitted” to the transmembrane domains. In this work, we focus our attention on the consequences of hydrolysis and inorganic phosphate exit in the maltose uptake system (MalFGK2E) from Escherichia coli. The prime goal is to identify and map the structural changes occurring during an ATP-hydrolytic cycle. For that, we use extensive molecular dynamics simulations to study three potential intermediate states (with 10 replicates each): an ATP-bound, an ADP plus inorganic phosphate-bound and an ADP-bound state. Our results show that the residues presenting major rearrangements are located in the A-loop, in the helical sub-domain, and in the “EAA motif” (especially in the “coupling helices” region). Additionally, in one of the simulations with ADP we were able to observe the opening of the NBD dimer accompanied by the dissociation of ADP from the ABC signature motif, but not from its corresponding P-loop motif. This work, together with several other MD studies, suggests a common communication mechanism both for importers and exporters, in which ATP-hydrolysis induces conformational changes in the helical sub-domain region, in turn transferred to the transmembrane domains via the “coupling helices”