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Describing Privacy Enhancing Technologies (PETs) to the general public is challenging but essential to convey the privacy protections they provide. Existing research has explored the explanation of differential privacy in health contexts. Our study adapts well-performing textual descriptions of local differential privacy from prior work to a new context and broadens the investigation to the descriptions of additional PETs. Specifically, we develop user-centric textual descriptions for popular PETs in ad tracking and analytics, including local differential privacy, federated learning with and without local differential privacy, and Google's Topics. We examine the applicability of previous findings to these expanded contexts, and evaluate the PET descriptions with quantitative and qualitative survey data (n=306). We find that adapting a process- and implications-focused approach to the ad tracking and analytics context achieved similar effects in facilitating user understanding compared to health contexts, and that our descriptions developed with this process+implications approach for the additional, understudied PETs help users understand PETs' processes. We also find that incorporating an implications statement into PET descriptions did not hurt user comprehension but also did not achieve a significant positive effect, which contrasts prior findings in health contexts. We note that the use of technical terms as well as the machine learning aspect of PETs, even without delving into specifics, led to confusion for some respondents. Based on our findings, we offer recommendations and insights for crafting effective user-centric descriptions of privacy-enhancing technologies. 

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 [19], crocker plots [55], and multiparameter rank functions [37]. We then introduce a new tool to summarize time-varying metric spaces: a crocker stack. 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 [57]. Altogether, we aim to bring the broader applied mathematics community up-to-date on topological summaries of time-varying metric spaces.