Deformations of circle-valued Morse functions on surfaces

Let $M$ be a smooth connected orientable compact surface. Denote by $F(M,S^1)$ the space of all Morse functions $f:M\to S^1$ having no critical points on the boundary of $M$ and such that for every boundary component $V$ of $M$ the restriction $f|_{V}:V\to S^1$ is either a constant map or a covering map. Endow $F(M,S^1)$ with the $C^{\infty}$-topology. In this note the connected components of $F(M,S^1)$ are classified. This result extends the results of S. V. Matveev, V. V. Sharko, and the author for the case of Morse functions being locally constant on the boundary of $M$.Comment: 8 pages, 4 figure

Closest horizons of Hsp70 engagement to manage neurodegeneration.

This is the final version. Available on open access from Frontiers Media via the DOI in this record. Our review seeks to elucidate the current state-of-the-art in studies of 70-kilodalton-weighed heat shock proteins (Hsp70) in neurodegenerative diseases (NDs). The family has already been shown to play a crucial role in pathological aggregation for a wide spectrum of brain pathologies. However, a slender boundary between a big body of fundamental data and its implementation has only recently been crossed. Currently, we are witnessing an anticipated advancement in the domain with dozens of studies published every month. In this review, we briefly summarize scattered results regarding the role of Hsp70 in the most common NDs including Alzheimer's disease (AD), Parkinson's disease (PD), and amyotrophic lateral sclerosis (ALS). We also bridge translational studies and clinical trials to portray the output for medical practice. Available options to regulate Hsp70 activity in NDs are outlined, too.Russian Science Foundatio