In a sentence like, "I saw him run," what is the rule that explains why "run" is in the present tense when "saw" is in the past tense? A similar construction, "She said he swam," requires past tense for both. I suspect the pronoun is governing verb tense. Is there a term for this?
Hear, see, watch, notice and similar verbs of perception can be followed by object + infinitive without to or object + -ing form.
There is usually a difference of meaning between the two structures.
The infinitive is used after these verbs when we want to say that we hear or see the whole of an action or event. The gerund –ing form is used to suggest that we hear an action or event in progress.
The verbs see, hear, watch, notice etc., can also be followed by an object + past participle.
In this case, the past participle has a passive meaning.
I heard his name repeated several times. (= His name was repeated several times.)