Learn Hardcore Norwegian: Jeg tar notater under foredraget slik at jeg husker det viktigste. - I take notes during the talk so that I remember the most important things.

å ta notater = to take notes (most idiomatic)
å skrive notater is understandable, but sounds less natural as a general expression for note-taking. It can be used in specific contexts (e.g. “write notes for someone”), but not as the standard phrase.
You don’t say å gjøre notater; that sounds wrong in this context.

So you should memorize å ta notater as a set phrase.

What does under foredraget mean exactly, and why not i foredraget?

under foredraget here means “during the lecture/talk”.

under
- time period = during that time.
  - under møtet – during the meeting
  - under ferien – during the vacation

i foredraget would more literally be “in the lecture”, and is less natural if you just mean while it is happening. It could be used in sentences like:

Han snakket om miljøet i foredraget. – He talked about the environment in the lecture (i.e. as part of the content).

So to express while it’s happening, use under foredraget.

What is the difference between slik at and sånn at or for at?

All three can translate to so that, but they differ slightly in style and nuance:

slik at
- Quite neutral and common in both speech and writing.
- Can express result or purpose.
- In your sentence: … slik at jeg husker det viktigste = so that I remember the most important points (both purpose and expected result).
sånn at
- More informal/colloquial, common in spoken Norwegian.
- Very often interchangeable with slik at in everyday speech:
  - Jeg tar notater sånn at jeg husker det viktigste.
for at
- Emphasizes purpose/intention a bit more (similar to “in order that”).
- Often used where the focus is on why you do something:
  - Jeg tar notater for at jeg skal huske det viktigste.

In most casual contexts, slik at and sånn at can both be used. for at is more explicitly about intention and is a bit more formal.

Why is it jeg husker and not something like jeg skal huske or jeg kommer til å huske for the future?

Norwegian often uses the present tense to talk about the future when the future meaning is clear from context.

In slik at jeg husker det viktigste, the idea is so that I will remember the most important things. The future meaning comes from the overall situation (taking notes now so you’ll remember later).

You could say:

… slik at jeg skal huske det viktigste.
… slik at jeg kommer til å huske det viktigste.

Both are grammatically fine, but they sound more explicit or heavier. The simple present is more natural here.

Why is it foredraget and not just foredrag?

foredrag = a lecture / talk (indefinite)
foredraget = the lecture / the talk (definite)

In Norwegian, the definite form is usually marked with a suffix:

et foredrag – a lecture
foredraget – the lecture

In your sentence, we are clearly talking about a specific, known lecture, not lectures in general, so Norwegian uses the definite form:

under foredraget – during the lecture (this particular one).

If you were speaking about lectures in general, you might say:

Jeg tar notater under foredrag, slik at jeg husker det viktigste.
(I take notes during lectures, so that I remember the most important points.)

Why is it det viktigste and not something like de viktigste?

viktigste is the superlative of viktig (important). In Norwegian, superlatives can be used as nouns:

det viktigste – the most important (thing/part)
de viktigste – the most important (things/points)

In your sentence, det viktigste is neuter singular definite and functions as a noun:

jeg husker det viktigste
= I remember the most important (part/thing).

If you want to emphasize several points, you can say:

… slik at jeg husker de viktigste punktene.
(…so that I remember the most important points.)

Both are correct; det viktigste is a more compact way of saying “the most important part/thing(s)” without naming a specific noun.

Why is the article det used with viktigste? Is viktigste an adjective or a noun here?

viktigste is originally an adjective in superlative form:

viktig – important
viktigere – more important
viktigst – most important
viktigste – “the most important” (definite form of the superlative)

But in det viktigste, the adjective is used as a noun. Norwegian often turns adjectives into nouns with det or de:

det nye – the new (thing)
det beste – the best (thing)
det viktigste – the most important (thing/part)
de fattige – the poor (people)

So grammatically, viktigste is still an adjective form, but det viktigste works as a noun phrase meaning “the most important (thing).”

Why is the word order slik at jeg husker det viktigste and not slik at husker jeg det viktigste?

slik at introduces a subordinate clause. In Norwegian:

In main clauses, you usually have V2 word order (verb in second position):
- Jeg tar notater. (Subject–Verb)
- I dag tar jeg notater. (Adverb–Verb–Subject)
In subordinate clauses, the verb does not have to be second. The normal pattern is:
[subordinator] + Subject + Verb + …

So:

slik at jeg husker det viktigste
- jeg = subject
- husker = verb

slik at husker jeg det viktigste would incorrectly follow main-clause V2 inside a subordinate clause, which is not allowed in standard Norwegian.

Can I drop the second jeg and say slik at husker det viktigste?

No, you normally cannot drop the subject like that in Norwegian. Each clause needs its own subject:

Jeg tar notater under foredraget slik at jeg husker det viktigste.

You could restructure the sentence, but you still need a subject in the second clause:

Jeg tar notater under foredraget, så jeg husker det viktigste.
(so I remember the most important things)

But slik at husker det viktigste without jeg is ungrammatical.

What is the singular of notater, and when would I use it?

et notat – a note (singular)
notatet – the note
notater – notes (plural)
notatene – the notes

In your sentence you use notater (plural) because you’re talking about taking notes in general during a lecture.

You would use the singular et notat for one specific note:

Jeg skrev et notat til sjefen. – I wrote a note to the boss.
Jeg fant et gammelt notat i boka. – I found an old note in the book.

But the fixed expression is å ta notater (plural).

Does foredrag mean “lecture” or “presentation,” and how formal is it?

foredrag can be translated as “lecture,” “talk,” or “presentation”, depending on context.

University lecture: foredrag (or forelesning, which is more specifically a university lecture)
Conference talk: often foredrag
A public talk at a library or cultural event: foredrag

It sounds a bit formal—you wouldn’t normally call an everyday casual chat a foredrag. For informal “presentation” (e.g. in class), people might still say presentasjon or framføring, depending on the situation.