Capturing dynamics of time-varying data via topology

<p style='text-indent:20px;'>One approach to understanding complex data is to study its shape through the lens of algebraic topology. While the early development of topological data analysis focused primarily on static data, in recent years, theoretical and applied studies have turned to data that varies in time. A time-varying collection of metric spaces as formed, for example, by a moving school of fish or flock of birds, can contain a vast amount of information. There is often a need to simplify or summarize the dynamic behavior. We provide an introduction to topological summaries of time-varying metric spaces including vineyards [<xref ref-type="bibr" rid="b19">19</xref>], crocker plots [<xref ref-type="bibr" rid="b55">55</xref>], and multiparameter rank functions [<xref ref-type="bibr" rid="b37">37</xref>]. We then introduce a new tool to summarize time-varying metric spaces: a <i>crocker stack</i>. Crocker stacks are convenient for visualization, amenable to machine learning, and satisfy a desirable continuity property which we prove. We demonstrate the utility of crocker stacks for a parameter identification task involving an influential model of biological aggregations [<xref ref-type="bibr" rid="b57">57</xref>]. Altogether, we aim to bring the broader applied mathematics community up-to-date on topological summaries of time-varying metric spaces.</p>

Photogranules Formed by Filamentous Cyanobacteria and Algae of the Genus Spirogyra Link in the Coastal Zone of Lake Baikal

Biological aggregations, in particular photogranules, can be found in different environments as one of the forms of existence of microorganisms, algae and small invertebrates. As far as they are systems, in which multiple ecological functions are performed by different elements simultaneously, they are considered a promising means for use in wastewater treatment. This paper provides description and composition of photogranules for the first time discovered in mass, 2,5 m away from the water edge in Listvennichnyi Bay of Lake Baikal in October 2015. The photogranules were maroon, free-floating, with a diameter of 0,5 to 1,5 cm, and were not hollow. Their main component was radially divergent filaments of oscillatorian cyanobacteria dominated by the members of the genus Symplocastrum. Photogranules with a diameter of less than 1 cm contained a large number of filaments of Spirogyra sp. ster., which in recent years have become widespread in the littoral of Lake Baikal and in the Angara River. In addition, there were fragments of numerous filamentous green algae, as well as diatom cells with Didymosphenia spp. dominance. The identified taxa are typical for the stony littoral zone of the lake at the depths from 0 to 3 m. Photogranulation in Baikal may be the result of overproduction of bottom communities of algae and cyanobacteria under conditions of increased anthropogenic load in the shallow water zone. Probably, photogranules can act as one of the mechanisms that perform the functions of natural self-purification in Baikal.

The impact of environmental noise on animal communication: pattern formation in a class of deterministic and stochastic hyperbolic models for self-organised biological aggregations

The collective movement of animals occurs as a result of communicationbetween the members of the community. However, inter-individual commu-nication can be aected by the stochasticity of the environment, leading tochanges in the perception of neighbours and subsequent changes in individualbehaviour, which then in uence the overall behaviour of the animal aggre-gations. To investigate the eect of noise on the overall behaviour of animalaggregations, we consider a class of nonlocal stochastic and deterministic hy-perbolic models for the collective movement of animals. We show numericallythat strong noise does not seem to in uence the spatio-temporal pattern (i.e.,travelling aggregations) observed when all neighbours are perceived with thesame intensity (i.e., the environment is homogeneous). However, when neigh-bours ahead/behind are perceived dierently by a reference individual, noisecan lead to the destruction of the spatio-temporal pattern. Moreover, weshow that the increase in noise can lead to dierent transitions between dif-ferent spatio-temporal patterns, and these transitions are relatively similarto the transitions between patterns when we perturb deterministically someparameters.