Bredon cohomology and robot motion planning

In this paper we study the topological invariant ${\sf {TC}}(X)$ reflecting the complexity of algorithms for autonomous robot motion. Here, $X$ stands for the configuration space of a system and ${\sf {TC}}(X)$ is, roughly, the minimal number of continuous rules which are needed to construct a motion planning algorithm in $X$. We focus on the case when the space $X$ is aspherical; then the number ${\sf TC}(X)$ depends only on the fundamental group $\pi=\pi_1(X)$ and we denote it ${\sf TC}(\pi)$. We prove that ${\sf TC}(\pi)$ can be characterised as the smallest integer $k$ such that the canonical $\pi\times\pi$-equivariant map of classifying spaces $$E(\pi\times\pi) \to E_{\mathcal D}(\pi\times\pi)$$ can be equivariantly deformed into the $k$-dimensional skeleton of $E_{\mathcal D}(\pi\times\pi)$. The symbol $E(\pi\times\pi)$ denotes the classifying space for free actions and $E_{\mathcal D}(\pi\times\pi)$ denotes the classifying space for actions with isotropy in a certain family $\mathcal D$ of subgroups of $\pi\times\pi$. Using this result we show how one can estimate ${\sf TC}(\pi)$ in terms of the equivariant Bredon cohomology theory. We prove that ${\sf TC}(\pi) \le \max\{3, {\rm cd}_{\mathcal D}(\pi\times\pi)\},$ where ${\rm cd}_{\mathcal D}(\pi\times\pi)$ denotes the cohomological dimension of $\pi\times\pi$ with respect to the family of subgroups $\mathcal D$. We also introduce a Bredon cohomology refinement of the canonical class and prove its universality. Finally we show that for a large class of principal groups (which includes all torsion free hyperbolic groups as well as all torsion free nilpotent groups) the essential cohomology classes in the sense of Farber and Mescher are exactly the classes having Bredon cohomology extensions with respect to the family $\mathcal D$.Comment: This revision contains a few additional comments, among them is Corollary 3.5.

Genetic and epigenetic regulation of abdominal aortic aneurysms

Abdominal aortic aneurysms (AAAs) are focal dilations of the aorta that develop from degenerative changes in the media and adventitia of the vessel. Ruptured AAAs have a mortality of up to 85%, thus it is important to identify patients with AAA at increased risk for rupture who would benefit from increased surveillance and/or surgical repair. Although the exact genetic and epigenetic mechanisms regulating AAA formation are not completely understood, Mendelian cases of AAA, which result from pathologic variants in a single gene, have helped provide a basic understanding of AAA pathophysiology. More recently, genome wide associated studies (GWAS) have identified additional variants, termed single nucleotide polymorphisms, in humans that may be associated with AAAs. While some variants may be associated with AAAs and play causal roles in aneurysm pathogenesis, it should be emphasized that the majority of SNPs do not actually cause disease. In addition to GWAS, other studies have uncovered epigenetic causes of disease that regulate expression of genes known to be important in AAA pathogenesis. This review describes many of these genetic and epigenetic contributors of AAAs, which altogether provide a deeper insight into AAA pathogenesis.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/155527/1/cge13705.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/155527/2/cge13705_am.pd

Bredon cohomology and robot motion planning

Acknowledgements Farber was partially supported by the EPSRC, by the IIAS and by the Marie Curie Actions, FP7, in the frame of the EURIAS Fellowship Programme. Lupton and Oprea were partially supported by grants from the Simons Foundation (# 209575 and # 244393). This research was supported through the programme Research in pairs by the Mathematisches Forschungsinstitut Oberwolfach in 2017.Peer reviewedPostprin

An upper bound for topological complexity

This work was partially supported by a grant from the Simons Foundation: (#244393 to John Oprea). The authors would like to thank the Mathematisches Forschungsinstitut Oberwolfach for its generosity in supporting a July 2017 Research in Pairs stay where this work was begun.Peer reviewedPostprin

Bone marrow transplantation alters the tremor phenotype in the murine model of globoid-cell leukodystrophy

Tremor is a prominent phenotype of the twitcher mouse, an authentic genetic model of Globoid-Cell Leukodystrophy (GLD, Krabbe’s disease). In the current study, the tremor was quantified using a force-plate actometer designed to accommodate low-weight mice. The actometer records the force oscillations caused by a mouse’s movements, and the rhythmic structure of the force variations can be revealed. Results showed that twitcher mice had significantly increased power across a broad band of higher frequencies compared to wildtype mice. Bone marrow transplantation (BMT), the only available therapy for GLD, worsened the tremor in the twitcher mice and induced a measureable alteration of movement phenotype in the wildtype mice. These data highlight the damaging effects of conditioning radiation and BMT in the neonatal period. The behavioral methodology used herein provides a quantitative approach for assessing the efficacy of potential therapeutic interventions for Krabbe’s disease