You can hide the Categories and Tags sections - the Navigation (sidebar) menu display control theme component gets you part-way by hiding them from everyone but staff users, or you could create a custom theme component that hides them from everyone including staff. But there isn’t a way to populate the left sidebar with a ‘hot topics’ section that would list the x hot topics from today/this week. You could create a custom section and manually update it with links to topics, but that is likely going to be a chore to keep on top of.
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