Learn Finnish: Näin julkkiksen, kuuluisan laulajan, vilkuttavan yleisölle. - I saw the celebrity, a famous singer, waving to the audience.

It is an apposition, meaning it is a noun phrase that directly renames the noun right before it (julkkiksen). In formal Finnish, an apposition must match the grammatical case of its head noun. Because julkkiksen is in the genitive case, kuuluisan laulajan must also be in the genitive.

Is julkkiksen the direct object of näin or the subject of vilkuttavan?

Grammatically, it functions as the subject of the participle vilkuttavan. In this participial construction, the 'doer' of the observed action is put into the genitive case. The entire phrase julkkiksen vilkuttavan (the celebrity waving) serves as the object of the main verb näin.

Why does the participle vilkuttavan end in -n instead of just being vilkuttava?

This sentence uses a participial construction to express seeing someone do something. When the main verb (näin) is active and has a personal subject, the active present participle (vilkuttava) must take the genitive -n ending to act as the object of the main sentence.

Why does yleisö (audience) take the -lle ending here?