Við hefjum fundinn eftir hádegi.

The sentence breaks down like this:

• Við = the subject, we
• hefjum = the finite verb, begin / start
• fundinn = the direct object, the meeting
• eftir hádegi = a time expression, after noon / in the afternoon

So the structure is:

subject + verb + object + time expression

That is a very normal Icelandic main-clause pattern.

Hefjum is the 1st person plural present tense form of að hefja.

So:

• að hefja = to begin, to start
• við hefjum = we begin / we start

A few present-tense forms are:

• ég hef = I begin
• þú hefur = you begin
• við hefjum = we begin

So in this sentence, hefjum matches við.

Because fundinn is the direct object of the verb, and that means it appears in the accusative case.

The dictionary form is:

• fundur = meeting

But in the accusative singular it becomes:

• fund

And with the definite article the, it becomes:

• fundinn = the meeting

So:

• fundur = a meeting, nominative
• fund = a meeting, accusative
• fundinn = the meeting, accusative definite

A useful pattern to notice: many masculine nouns with nominative -ur lose that ending outside the nominative.

In Icelandic, the definite article is very often attached to the end of the noun instead of being a separate word.

So rather than a separate word like English the, Icelandic often uses an ending:

• fundur = meeting
• fundurinn = the meeting (nominative)
• fundinn = the meeting (accusative)

In this sentence, the object is accusative, so you get fundinn, not fundurinn.

Literally, it means after noon or after midday.

In natural English translation, it can mean:

• after noon
• in the afternoon

The exact nuance depends on context. If someone is scheduling something, eftir hádegi often means broadly sometime after midday, not necessarily immediately after 12:00.

Here eftir means after in a time sense, and in that use it takes the accusative.

So hádegi is functioning as the object of eftir.

A tricky point for learners is that hádegi does not visibly change here, so you just see hádegi. Even though the case matters grammatically, the form itself does not make it obvious in this phrase.

In normal Icelandic, yes, you usually keep við.

Even though -um in hefjum already tells you the subject is we, Icelandic generally does not drop subject pronouns the way some other languages do.

So the standard full sentence is:

Við hefjum fundinn eftir hádegi.

Without við, the sentence might still be understandable in some contexts, but it would not be the normal neutral way to say it.

They are close in meaning, but not always identical in tone.

• hefja often sounds a bit more formal or official: begin, commence
• byrja is often more everyday: start, begin

So both of these are possible:

• Við hefjum fundinn eftir hádegi.
• Við byrjum fundinn eftir hádegi.

With fundur specifically, hefja fund is very common in formal or meeting-related language.

Yes. You can also say:

Eftir hádegi hefjum við fundinn.

That is perfectly correct.

This happens because Icelandic main clauses follow a verb-second pattern. If you move the time phrase eftir hádegi to the front, the finite verb hefjum still stays in the second position, and the subject við comes after it.

So:

• Við hefjum fundinn eftir hádegi.
• Eftir hádegi hefjum við fundinn.

Both are natural, but the second version puts more focus on the time.