He's doing a lecture on modern poetry.
So here what does "doing" mean here?
"Doing" means "giving" here. He's the one who is giving a lecture. He is a speaker, not a student. The verb "do" can often be used to replace another verb. I've found a relevant discussion - please, check this: https://english.stackexchange.com/questions/356416/use-do-instead-of-many-other-verbs.
Hey, John. Do the window. It's cold in here. (Close the window)
Hey, John. Do the window. It's hot in here. (Open the window)
Johanna, would you please do the dishes? The sink is full already. (Wash the dishes)
"Do the lecture" sounds quite similar to me. In my opinion, the verb "do" in our case replaces the verb "give" or "deliver" and makes it sound less formal.
We can also understand that "doing a lecture" means "giving a lecture" by referring to basic meanings of "do". I looked up in the Cambridge Dictionary: https://dictionary.cambridge.org/dictionary/english/do .
do (PERFORM) - to perform, take part in, or achieve something
do (MAKE) - to make, produce, or create something:
(example) I can't go out tonight - I've got to do my history paper.
Hope this helps.