Learn Hardcore Greek: Στο πανεπιστήμιο το τμήμα ελληνικών είναι μικρό, αλλά πολύ ζωντανό. - At the university the Greek department is small, but very lively.

στο πανεπιστήμιο literally = “at the university”, but in many contexts it can also mean “at university” in a general sense.
Saying σε πανεπιστήμιο (without article) sounds incomplete or very unusual; it would typically need something more (e.g. σε κάποιο πανεπιστήμιο = “at some university”).

So the definite article το is the normal, natural form here.

What does τμήμα mean here? Is it like “department” in a university?

Yes. το τμήμα in this context means “(academic) department.”

Typical university use:

το τμήμα ελληνικών – the Greek department
το τμήμα φυσικής – the physics department
το τμήμα ιστορίας – the history department

Outside academia, τμήμα can also mean section / unit / division (e.g. in a company or public service).

Why is it το τμήμα ελληνικών instead of something like το ελληνικό τμήμα?

Both exist, but they don’t mean exactly the same:

το τμήμα ελληνικών
- literally “the department of Greek”
- ελληνικών (genitive plural) indicates the subject/field taught.
- This is the standard way to name university departments.
το ελληνικό τμήμα
- literally “the Greek department”
- Here ελληνικό is an adjective describing the department itself.
- It could mean “the Greek (i.e. Greek-speaking, Greek-national, Greek-side) department,” depending on context.
- It does not automatically mean “department of Greek language.”

So for “the Greek (language) department,” το τμήμα ελληνικών is the natural phrase.

Why is ελληνικών in the genitive plural? Why not just ελληνικά?

Ελληνικά as a language-name is neuter plural (literally “Greek things/words”). When you use a language to name a department, Greek puts the language in the genitive plural:

το τμήμα ελληνικών – department of Greek (studies)
το τμήμα αγγλικών – department of English
το τμήμα γαλλικών – department of French

So:

ελληνικά = Greek (language)
ελληνικών = of Greek (as a subject), genitive plural form used after τμήμα.

Why do μικρό and ζωντανό end in -ό? What are they agreeing with?

Both μικρό and ζωντανό are adjectives describing το τμήμα (neuter singular noun):

το τμήμα → neuter singular
Adjectives must agree in gender, number, and case.

So:

masculine: μικρός, ζωντανός
feminine: μικρή, ζωντανή
neuter: μικρό, ζωντανό

Because τμήμα is neuter singular, the adjectives are also neuter singular: μικρό, ζωντανό.

Why is there a comma before αλλά?

In Greek, αλλά (“but”) is usually preceded by a comma when it connects two clauses or longer phrases:

…είναι μικρό, αλλά πολύ ζωντανό.
= “…it is small, but very lively.”

Similar to English:

“It is small, but very lively.”

So the comma marks the contrast introduced by αλλά.

What nuance does πολύ add before ζωντανό? Could we just say ζωντανό?

πολύ intensifies the adjective: it means “very”.

ζωντανό = lively
πολύ ζωντανό = very lively / really lively

You can say είναι ζωντανό (“it is lively”), but πολύ makes the statement stronger, similar to English.

Does ζωντανό mean “alive” or “lively” here?

Literally, ζωντανός/ζωντανή/ζωντανό can mean “alive.”
In this context (describing a department, a place, a group), it means “lively, vibrant, active.”

So:

Το παιδί είναι ζωντανό. – The child is alive. (or very energetic, depending on context)
Η πόλη είναι πολύ ζωντανή. – The city is very lively.
Το τμήμα είναι πολύ ζωντανό. – The department is very lively/active.

Could the word order be Το τμήμα ελληνικών στο πανεπιστήμιο είναι μικρό…? Is that still correct?

Yes, it is grammatically correct, but the emphasis shifts slightly.

Στο πανεπιστήμιο το τμήμα ελληνικών είναι μικρό…
- Starts with “At the university…”; sets the location first.
Το τμήμα ελληνικών στο πανεπιστήμιο είναι μικρό…
- Starts with “The Greek department at the university…”; focuses first on the department.

Both are natural. Greek word order is fairly flexible, but the beginning of the sentence often carries emphasis or sets the scene.

Why do we need το before τμήμα ελληνικών? Could we just say Στο πανεπιστήμιο, τμήμα ελληνικών είναι μικρό…?

You normally need the article το here:

Το τμήμα ελληνικών = the Greek department (as a specific, known unit).

Leaving out the article:

…τμήμα ελληνικών είναι μικρό… sounds incorrect or at least very unnatural.

In Greek, most singular countable nouns in such subject positions take a definite article unless you specifically want to say “a department” (ένα τμήμα ελληνικών). Here we are clearly talking about the department at that university, so το is required.