We prove a conjecture of Blumberg and Hill regarding the existence of N 1-operads associated to given sequences F D .F n / n2N of families of subgroups of G † n. For every such sequence, we construct a model structure on the category of Goperads, and we use these model structures to define E F 1-operads, generalizing the notion of an N 1-operad, and to prove the Blumberg-Hill conjecture. We then explore questions of admissibility, rectification, and preservation under left Bousfield localization for these E F 1-operads, obtaining some new results as well for N 1operads.

Chronic non-healing wounds, frequently caused by diabetes, lead to lower quality of life, infection, and amputation. These wounds have limited treatment options. We have previously engineered growth factors to bind to exposed extracellular matrix (ECM) in the wound environment using the heparin-binding domain of placental growth factor-2 (PlGF-2_123–144), which binds promiscuously to ECM proteins. Here, in the type 1 diabetic (T1D) NOD mouse model, engineered growth factors (eGFs) improved both re-epithelialization and granulation tissue formation. eGFs were even more potent in combination, and the “triple therapy” of vascular endothelial growth factor-A (VEGF-PlGF-2_123–144), platelet-derived growth factor-BB (PDGF-BB-PlGF-2_123–144), and heparin-binding epidermal growth factor (HB-EGF-PlGF-2_123–144) both improved wound healing and remained at the site of administration for significantly longer than wild-type growth factors. In addition, we also found that changes in the cellular milieu of a wound, including changing amounts of M1 macrophages, M2 macrophages and effector T cells, are most predictive of wound-healing success in the NOD mouse model. These results suggest that the triple therapy of VEGF-PlGF-2_123–144, PDGF-BB-PlGF-2_123–144, and HB-EGF-PlGF-2_123–144may be an effective therapy for chronic non-healing wounds in that occur as a complication of diabetes.

In an increasingly data-driven world, facility with statistics is more important than ever for our students. At institutions without a statistician, it often falls to the mathematics faculty to teach statistics courses. This paper presents a model that a mathematician asked to teach statistics can follow. This model entails connecting with faculty from numerous departments on campus to develop a list of topics, building a repository of real-world datasets from these faculty, and creating projects where students interface with these datasets to write lab reports aimed at consumers of statistics in other disciplines. The end result is students who are well prepared for interdisciplinary research, who are accustomed to coping with the idiosyncrasies of real data, and who have sharpened their technical writing and speaking skills.

Building upon Hovey's work on Smith ideals for monoids, we develop a homotopy theory of Smith ideals for general operads in a symmetric monoidal category. For a sufficiently nice stable monoidal model category and an operad satisfying a cofibrancy condition, we show that there is a Quillen equivalence between a model structure on Smith ideals and a model structure on algebra morphisms induced by the cokernel and the kernel. For symmetric spectra, this applies to the commutative operad and all †-cofibrant operads. For chain complexes over a field of characteristic zero and the stable module category, this Quillen equivalence holds for all operads. We end with a comparison between the semi-model category approach and the 1-category approach to encoding the homotopy theory of algebras over †-cofibrant operads that are not necessarily admissible.