Learn Hardcore German: Heute ist das Problem für mich unwichtig.

German main clauses have the rule “conjugated verb in 2nd position”, but anything (subject, time word, object, etc.) can stand in 1st position. In your two versions:

Heute ist das Problem für mich unwichtig.
– 1st element: Heute (today)
– 2nd element: ist (the verb)
Das Problem ist heute für mich unwichtig.
– 1st element: Das Problem
– 2nd element: ist

Both obey the verb‑second rule.
The difference is mainly emphasis:

Heute ist das Problem … → strong focus on today (“Today, of all days, the problem is unimportant to me”).
Das Problem ist heute … → slightly more neutral; first emphasis on the problem, then you add “today”.

Grammatically, both are fine; the choice is stylistic.

Where else can für mich go in the sentence? Are versions like Heute ist für mich das Problem unwichtig or Heute ist das Problem unwichtig für mich correct?

All of these versions are grammatically correct:

Heute ist das Problem für mich unwichtig. (your sentence)
Heute ist für mich das Problem unwichtig.
Heute ist das Problem unwichtig für mich.

They all keep the verb in 2nd position and just rearrange the “middle” of the sentence.

Naturalness / nuance:

1. Heute ist das Problem für mich unwichtig.
Very natural and neutral. für mich sits right before the adjective it semantically belongs to.
1. Heute ist für mich das Problem unwichtig.
Also correct. Here you slightly highlight für mich (“As for me today, the problem is unimportant”).
1. Heute ist das Problem unwichtig für mich.
Also acceptable. This makes unwichtig für mich sound like one block (“unimportant for me”), almost like an afterthought:
“Today, the problem is unimportant – for me.”

For learners, version 1 is the safest, most neutral word order, but the others are valid and used.

Why is it für mich and not für mir?

Because the preposition für always takes the accusative case in German.

The personal pronoun ich has these main forms:

Nominative (subject): ich
Accusative (direct object / after certain prepositions): mich
Dative (indirect object / after other prepositions): mir

für demands the accusative, so you must use mich:

für mich = for me ✓
für mir ✗ (never correct)

Compare with a preposition that takes the dative:

mit mir = with me (here mit requires dative → mir, not mich)

So the pattern is:
für + Akkusativ → für mich, für dich, für ihn, für sie, …

Could I also say Das Problem ist mir heute unwichtig? Is that the same as für mich?

Yes, Das Problem ist mir heute unwichtig is correct and very idiomatic, and its meaning is very close to Das Problem ist heute für mich unwichtig.

The difference is grammatical and slightly stylistic:

Das Problem ist heute für mich unwichtig.
– für mich = for me / from my point of view
– “Today, the problem is unimportant for me.”
Das Problem ist mir heute unwichtig.
– mir = dative pronoun, no preposition
– Structure: etwas ist mir (un)wichtig / egal / peinlich / unangenehm, etc.
– Literally: “The problem is unimportant *to me today.”*

The dative with “sein” (like ist mir wichtig / unwichtig / egal) is a very natural way to express personal concern or relevance. In many contexts:

Das Problem ist mir heute unwichtig.
Das Problem ist heute für mich unwichtig.

mean practically the same thing; mir often sounds a bit more idiomatic and direct.

What you cannot say is für mir – that’s wrong because für needs the accusative (mich), not the dative (mir).

Why is it unwichtig instead of nicht wichtig? Are they interchangeable?

Both are possible, and in this sentence they are almost interchangeable:

Heute ist das Problem für mich unwichtig.
Heute ist das Problem für mich nicht wichtig.

Both mean roughly: “Today, the problem is not important to me.”

Nuances:

unwichtig = unimportant, a single word.
Often a bit more categorical or compact, and slightly more formal in some contexts.
nicht wichtig = not important.
More transparent for learners, sometimes feels a tiny bit softer than unwichtig, depending on emphasis.

Related forms:

nicht so wichtig = not so important (sounds weaker than unwichtig).
nicht unwichtig = not unimportant → implies “actually somewhat important”.

In your sentence, both unwichtig and nicht wichtig are correct; unwichtig is just a concise, standard choice.

Is unwichtig an adjective or an adverb here? And why doesn’t it have an ending?

Here unwichtig is an adjective used predicatively (as the complement of the verb sein):

Das Problem ist unwichtig.
→ unwichtig describes the noun das Problem via the verb ist.

In German:

Adjectives used after verbs like sein, werden, bleiben (predicative position)
→ normally do not take endings:
Das Problem ist unwichtig.
Die Probleme sind unwichtig.
(No change, no added endings.)
Adjectives used before a noun (attributive position)
→ do take endings according to case, gender, and number, for example:
- ein unwichtiges Problem (acc./nom. neuter singular after ein)
- dieses unwichtige Problem
- die unwichtigen Probleme

So:

In your sentence: unwichtig = predicative adjective → no ending.
There is no special “adverb form” like in English (“important” vs “importantly”); in German, adjectives and adverbs usually look the same in form.

Why is it das Problem and not der Problem or die Problem?

Because the noun Problem is neuter in German, so its definite article in the singular is das.

Key points:

Every German noun has a grammatical gender:
der (masculine), die (feminine), das (neuter).
The gender is mostly lexical (you have to learn it with the noun).

For Problem, dictionaries show it as:

das Problem – neuter, singular
die Probleme – plural

So in the nominative singular you must say:

das Problem (not der Problem or die Problem)

Example contrasts:

das Problem ist groß.
die Probleme sind groß.

The verb form also shows singular vs. plural:

Das Problem ist … (singular)
Die Probleme sind … (plural)

What case is das Problem in, and why?

das Problem is in the nominative case, because it is the subject of the sentence.

Structure:

Heute – adverb of time
ist – verb (sein)
das Problem – subject (nominative)
für mich – prepositional phrase (mich = accusative after für)
unwichtig – predicative adjective describing das Problem

With the verb sein (to be), there is no direct object in the accusative. Instead, sein links a subject (nominative) to a complement:

Das Problem (subject, nominative)
ist
unwichtig (subject complement / predicative adjective)

So:

das Problem → nominative (subject)
mich in für mich → accusative (object of the preposition für)

Can I leave out heute? How does that change the meaning?

Yes, you can:

Das Problem ist für mich unwichtig.
(The problem is unimportant to me.)

Without heute, the sentence sounds like a general statement about how this problem relates to you, without any time limitation.

With heute:

Heute ist das Problem für mich unwichtig.

you imply that this is specifically true today. It suggests that on other days (in the past or future) the problem might be important, but today it isn’t.

So:

With heute → time‑limited, “today in particular”.
Without heute → more timeless, general attitude.

What’s the difference between heute and heutzutage, and could I use heutzutage here?

heute = today (this day)
heutzutage = nowadays / these days / in our time

You can replace heute with heutzutage, but the meaning changes:

Heute ist das Problem für mich unwichtig.
→ “Today (on this specific day), the problem is unimportant to me.”
Heutzutage ist das Problem für mich unwichtig.
→ “Nowadays / these days, the problem is unimportant to me.”
(Over the current period of time, in general.)

Use:

heute if you are talking about today as a specific day.
heutzutage if you mean in today’s era / at the present time in general, not tied to one calendar day.