Learn Hardcore Korean: jeogi gongwone sarami manhayo. - There are many people in the park over there.

저기 is a locative adverb meaning over there. It sets the general location (the park) as being some distance away from both speaker and listener before you state what’s happening there.

Why is the noun 공원 followed by -에 and not -에서?

-에 marks a static location where something exists, which suits descriptive verbs like 많다. In contrast, -에서 marks the place where an action occurs. Since 사람이 많다 describes how many people are present, you use 공원에.

Why is 사람 followed by the subject marker -이 and not the topic marker -은?

Descriptive verbs like 많다 take a subject, so you use the subject marker -이/가. If you used -은, you’d be making 사람 into the topic (“As for people…”), which changes the nuance.

Why isn’t a plural marker -들 used with 사람?

Korean often omits -들 when plurality is already clear from context or quantifiers. Here, 많다 (“many”) implies more than one, so adding -들 would be redundant.

Is 많아요 a verb or an adjective, and what kind of word is it?

많다 is a descriptive verb (often called an adjective in English grammars). It describes a state—“to be many.” 많아요 is simply its polite present-tense form.

What politeness level is 많아요, and could I instead say 많습니다?

많아요 is informal polite (the “-요” style) used in daily conversation. 많습니다 is formal polite (the “-습니다” style), suited to announcements, presentations, or very polite settings. Both are correct; choose based on context.

Could I replace 저기 with 거기 or 여기 in this sentence?

Yes.

여기 공원에 사람이 많아요 = “There are many people in this park” (near the speaker).
거기 공원에 사람이 많아요 = “There are many people in that park” (near the listener).
Choose 여기/거기/저기 depending on whose relative location you mean.

What difference does adding a topic marker make, as in 저기 공원에는 사람이 많아요?

By using 공원에는, you mark “in that park” as the topic—“As for the park over there…” This adds slight emphasis or contrast on the park itself. Without -는, it’s a neutral location statement.

Between 사람이 많아요 and 사람이 많이 있어요, what’s the difference in nuance?

사람이 많아요 uses the descriptive verb 많다 to state “people are many.”
사람이 많이 있어요 uses the adverb 많이 to modify 있다 (“to exist”), meaning “there are many people.”
Both convey the same idea; 많다 highlights the state of abundance, while 많이 있다 highlights the existence in large numbers.

What is the typical word order in 저기 공원에 사람이 많아요, and can I rearrange it?

The default Korean order is [location/time] + [subject/topic] + [verb/predicate], so 저기 공원에 (location) comes first, 사람이 (subject) next, and 많아요 (predicate) last. You can move elements around for emphasis, but the predicate almost always stays at the end.