Learn Hardcore Greek: Δεν είμαι πολύ κοινωνικός, αλλά προσπαθώ να μιλάω με τους συμφοιτητές μου. - I am not very sociable, but I try to talk with my fellow students.

είμαι = I am (1st person singular of είμαι)
So Δεν είμαι πολύ κοινωνικός literally is Not am very sociable, but it means I’m not very sociable.
You could add εγώ for emphasis:
Εγώ δεν είμαι πολύ κοινωνικός = I’m not very sociable (me, as opposed to others), but it’s not required.

What does Δεν mean, and why is it used instead of μην?

Both δεν and μην are negative particles, but they’re used in different environments.

Δεν is used with indicative verbs (normal statements):
- Δεν είμαι = I am not
- Δεν πάω = I am not going / I don’t go
Μην is used mainly with:
- Subjunctive (να
  - verb) in negative commands/warnings:
    Μην πας εκεί! = Don’t go there!
- Some fixed expressions.

In Δεν είμαι πολύ κοινωνικός, we have a simple statement in the present indicative, so δεν is correct; μην would be wrong here.

What exactly does πολύ mean here, and why does it come before κοινωνικός?

Πολύ here means very and modifies the adjective κοινωνικός (sociable).

πολύ κοινωνικός = very sociable
In Greek, degree adverbs like πολύ usually come before the adjective they modify:
- πολύ καλός = very good
- πολύ έξυπνη = very smart (fem.)

Note the accent:

πολύ (with stress) as an adverb = very / much
πολλοί / πολλές / πολλά (with double λ) are plural adjective/pronoun forms meaning many.

Why is it κοινωνικός and not some other ending like κοινωνική or κοινωνικό?

Adjectives in Greek agree in gender, number, and case with the noun (or implied noun) they describe.

The implied subject here is (εγώ) άντρας / άνθρωπος = masculine singular.
So we use the masculine singular nominative form κοινωνικός.
If a woman were speaking, she would normally say:
- Δεν είμαι πολύ κοινωνική (I’m not very sociable – female speaker)
Neuter would be κοινωνικό, used with neuter nouns like παιδί (child):
- Το παιδί δεν είναι πολύ κοινωνικό.

What does αλλά mean, and is the comma before it used the same way as in English?

Αλλά means but.

The structure is very similar to English:
Δεν είμαι πολύ κοινωνικός, αλλά προσπαθώ…
= I’m not very sociable, but I try…

The comma usage is also similar: in Greek it’s normal to place a comma before αλλά when it connects two clauses, just like but in English.

How does προσπαθώ να μιλάω work grammatically? What is this να?

Προσπαθώ means I try / I am trying.
In Greek, many verbs that express effort, desire, intention etc. are followed by να + verb.

Pattern: προσπαθώ να
- (present subjunctive):
  - προσπαθώ να μιλάω ≈ I try to talk / I’m trying to talk

Here να μιλάω is the subjunctive form of μιλάω, and it often expresses an ongoing or repeated action:

προσπαθώ να μιλάω = I make an effort in general to talk (habit, ongoing effort)
A different form, προσπαθώ να μιλήσω, would stress more a single act of speaking (e.g. “I’m trying to say something (once)”).

What is the difference between μιλάω and μιλώ?

They are two alternative present forms of the same verb μιλάω / μιλώ = to speak, to talk.

μιλάω is slightly more colloquial and very common in spoken Greek.
μιλώ is a bit shorter and often feels slightly more formal or written.

In this sentence, you can say either:

προσπαθώ να μιλάω με τους συμφοιτητές μου
προσπαθώ να μιλώ με τους συμφοιτητές μου

Both are correct; learners usually start with μιλάω because it’s more frequent in speech.

Why is it με τους συμφοιτητές μου and not something like στους συμφοιτητές μου?

The preposition changes the meaning slightly:

με
- accusative = with
  - μιλάω με τους συμφοιτητές μου = I talk with my classmates (mutual conversation)
σε / στους
- accusative = to
  - μιλάω στους συμφοιτητές μου = I speak to my classmates (emphasis on addressing them)

In your sentence, the idea is more like interacting with them socially, so με fits well: talking with them.

What does συμφοιτητές mean exactly, and how is it different from συμμαθητές?

Both refer to classmates, but at different levels of education:

συμφοιτητής / συμφοιτήτρια:
- Classmate at university / college (same faculty or department).
συμμαθητής / συμμαθήτρια:
- Classmate at school (primary or secondary).

So οι συμφοιτητές μου is best translated as my fellow students / my classmates at university.

Why is it τους συμφοιτητές μου? What are τους and μου doing here?

This is a noun phrase with article + noun + possessive pronoun:

τους = definite article, masculine plural accusative: the
συμφοιτητές = classmates (uni) – masculine plural
μου = enclitic possessive pronoun: my

So με τους συμφοιτητές μου literally is with the classmates my, i.e. with my classmates.

In Greek, possessive pronouns usually come after the noun (and after the article):

ο φίλος μου = my friend
η αδελφή μου = my sister
τα βιβλία μου = my books

Which case is used after με, and how can I recognize it here?

Με always takes the accusative case.

The base form (nominative singular) is ο συμφοιτητής (male classmate).
Masculine plural nominative: οι συμφοιτητές.
Masculine plural accusative: τους συμφοιτητές (the form used after με).

So με + τους συμφοιτητές (μου) = with my classmates (accusative plural).

Is this sentence considered formal or informal? Could I say it to a professor?

The sentence is neutral in register: natural, everyday Greek that is fine in most contexts.

You could say it:
- To friends and classmates (informal)
- To a teacher or professor (polite and normal)

If you wanted a slightly more formal version, you might just choose μιλώ instead of μιλάω:

Δεν είμαι πολύ κοινωνικός, αλλά προσπαθώ να μιλώ με τους συμφοιτητές μου.

How would the whole sentence change if a woman were speaking?

Only the adjective κοινωνικός needs to agree with a female speaker:

Male speaker:
Δεν είμαι πολύ κοινωνικός, αλλά προσπαθώ να μιλάω με τους συμφοιτητές μου.
Female speaker:
Δεν είμαι πολύ κοινωνική, αλλά προσπαθώ να μιλάω με τους συμφοιτητές μου.

Everything else stays the same.