Equivariant linear isometries operads over Abelian groups

$N_\infty$-operads are an equivariant generalization of $E_\infty$-operads introduced by Blumberg and Hill to study structural problems in equivariant stable homotopy theory. In the original paper introducing these objects, Blumberg and Hill raised the question of classifying $N_\infty$-operads that are weakly equivalent to a particularly nice kind of $N_\infty$-operad called a linear isometries operad. For some groups there is a known classification of linear isometries operads up to weak equivalence in terms of certain combinatorially defined objects called saturated transfer systems, but this classification is known to be invalid in general. Various authors have made incremental progress on understanding the domain of validity for this classification, but even among cyclic groups the validity is unknown in general. We determine essentially all the finite Abelian groups for which the classification is valid using techniques from algebra and extremal combinatorics.Comment: 36 pages, 0 figures; comments welcome

Composition closed premodel structures and the Kreweras lattice

We investigate the rich combinatorial structure of premodel structures on finite lattices whose weak equivalences are closed under composition. We prove that there is a natural refinement of the inclusion order of weak factorization systems so that the intervals detect these composition closed premodel structures. In the case that the lattice in question is a finite total order, this natural order retrieves the Kreweras lattice of noncrossing partitions as a refinement of the Tamari lattice, and model structures can be identified with stacked triangulations of a particular shape.Comment: 28 pages, 8 figures; comments welcome

Lifting N∞N_\infty operads from conjugacy data

We isolate a class of groups -- called lossless groups -- for which homotopy classes of $G$-$N_\infty$ operads are in bijection with certain restricted transfer systems on the poset of conjugacy classes $\operatorname{Sub}(G)/G$.Comment: v2: updates in response to referee report, accepted version in Tunisian Journal of Mathematics; 24 pages, 16 figures. v1: 22 pages, 13 figures; comments welcome

Dumbo: Faster Asynchronous BFT Protocols

HoneyBadgerBFT, proposed by Miller et al. [32] as the first practical asynchronous atomic broadcast protocol, demonstrated impressive performance. The core of HoneyBadgerBFT (HB-BFT) is to achieve batching consensus using asynchronous common subset protocol (ACS) of Ben-Or et al., constituted with $n$ reliable broadcast protocol (RBC) to have each node propose its input, followed by $n$ asynchronous binary agreement protocol (ABA) to make a decision for each proposed value ($n$ is the total number of nodes). In this paper, we propose two new atomic broadcast protocols (called Dumbo1, Dumbo2) both of which have asymptotically and practically better efficiency. In particular, the ACS of Dumbo1 only runs a small $k$ (independent of $n$) instances of ABA, while that of Dumbo2 further reduces it to constant! At the core of our techniques are two major observations: (1) reducing the number of ABA instances significantly improves efficiency; and (2) using multi-valued validated Byzantine agreement (MVBA) which was considered sub-optimal for ACS in [32] in a more careful way could actually lead to a much more efficient ACS. We implement both Dumbo1, Dumbo2 and deploy them as well as HB-BFT on 100 Amazon EC2 t2.medium instances uniformly distributed throughout 10 different regions across the globe, and run extensive experiments in the same environments. The experimental results show that our protocols achieve multi-fold improvements over HoneyBadgerBFT on both latency and throughput, especially when the system scale becomes moderately large