Learn Hardcore Swedish: Min vän vill studera matematik på universitetet. - My friend wants to study mathematics at the university.

In Swedish, certain verbs behave like modal verbs in English and are followed directly by an infinitive without att.
Vill is one of these verbs.

Common verbs that take an infinitive without att:

vill (want to)
kan (can)
ska (shall / going to)
måste (must / have to)
får (may / allowed to)
brukar (usually do)

So you say:

Min vän vill studera matematik.
Jag kan tala svenska.
Vi ska resa i morgon.

But many other verbs do require att before the infinitive:

Jag tycker om att studera matematik. (I like to study math.)
Hon försöker att lära sig svenska. (She tries to learn Swedish.) – often spoken without att, but grammatically allowed.

In this sentence, vill studera is the correct and natural form; vill att studera is incorrect.

What is the difference between studera, plugga, and läsa?

All three can relate to studying, but they have different nuances and typical uses.

studera
- More formal or academic.
- Often used for higher education or subjects in a more serious context.
- Hon studerar matematik på universitetet.
plugga
- Informal/colloquial.
- Often means to cram / to study hard for a test, or just to study casually in everyday speech.
- Jag måste plugga till tentan. (I have to cram for the exam.)
läsa
- Literally to read, but also to study (a subject).
- Often used for both reading texts and studying subjects.
- Jag läser en bok. (I’m reading a book.)
- Jag läser matematik. (I’m studying mathematics.)

In your sentence, studera fits well because it matches the formal context of på universitetet. Colloquially, someone might say:

Min vän vill plugga matte på universitetet.
(matte = informal for matematik)

Why is there no article before matematik? Why not en matematik or den matematik?

In Swedish, as in English, school subjects and many abstract or mass nouns are used without an article when you talk about them in general:

Jag studerar matematik. – I study mathematics.
Hon gillar musik. – She likes music.
Vi lär oss grammatik. – We are learning grammar.

You would only add a determiner if you specify a particular type or instance, which is less common with matematik:

Den matematik vi läser nu är ganska svår.
(The mathematics we are studying now is quite difficult.)

But in the sentence you gave, matematik is a general field of study, so no article is used.

Why is it på universitetet and not i universitetet?

The choice of preposition is mostly idiomatic. For institutions and some public places, Swedish often uses på where English uses at:

på universitetet – at the university
på jobbet – at work
på sjukhuset – at the hospital
på banken – at the bank
på skolan – at school

I is used more for being inside a physical space or country/city:

i huset – in the house
i Sverige – in Sweden
i Stockholm – in Stockholm

So Min vän vill studera matematik på universitetet is the natural way to say at the university as an institution.

Why is universitetet in the definite form? Why not just på universitet?

Swedish often uses the definite form when referring to institutions in a general but specific way – similar to English the university:

på universitetet – at the university
på sjukhuset – at the hospital
på banken – at the bank
på jobbet – at work (literally at the job)

Universitet is an ett-word:

ett universitet – a university (indefinite)
universitetet – the university (definite)

You can say på ett universitet if you mean at a university (one university or another, not specified), but in many contexts Swedish speakers prefer the definite form because they’re thinking of the university as an institution, not just any random one.

So:

Hon vill studera matematik på universitetet.
= She wants to study mathematics at (the) university (the institution-type).
Hon vill studera matematik på ett universitet i Tyskland.
= … at a university in Germany (not a specific one).

How would I say “My friends want to study mathematics at the university”? How do min and vän change in the plural?

For plural, both the possessive and the noun change:

vän (singular) → vänner (plural)
min (my, singular) → mina (my, plural)

So the full sentence becomes:

Mina vänner vill studera matematik på universitetet.

Breakdown:

Mina – my (plural)
vänner – friends
vill – want to
studera matematik – study mathematics
på universitetet – at the university

Can I move på universitetet earlier in the sentence, like Min vän vill på universitetet studera matematik?

This is grammatically understandable but sounds unnatural and quite odd in normal Swedish. The neutral word order is:

Subject – finite verb – infinitive – object – adverbials
Min vän – vill – studera – matematik – på universitetet.

Typical and natural:

Min vän vill studera matematik på universitetet.

Moving på universitetet before studera usually feels marked or poetic and is not standard in everyday speech:

Min vän vill på universitetet studera matematik. → very unusual.

You can move på universitetet to the front for emphasis or in answers, but then the word order must follow the V2 rule (verb in second position):

På universitetet vill min vän studera matematik.
(At the university, my friend wants to study mathematics.) – acceptable, but emphasizes at the university.

What nuances does vill have? Does it talk about the present or the future?

Vill is the present tense of vilja (to want). It usually expresses a desire or intention, and in context it often has a future meaning, similar to English wants to:

Min vän vill studera matematik på universitetet.
= My friend wants to study mathematics at university (in the future).

So even though the form is present tense, it usually refers to a present wish about something that will happen in the future.

Compare:

Min vän ska studera matematik på universitetet.
– My friend is going to study mathematics at the university (more like a plan / arrangement).
Min vän vill studera matematik, men har inte bestämt sig än.
– My friend wants to study mathematics but hasn’t decided yet.

Is there a more informal way to say this, like how English speakers say “study math at uni”?

Yes. In casual Swedish, you might hear:

Min kompis vill plugga matte på uni.

Informal elements:

kompis – buddy / friend (more casual than vän)
plugga – to study (colloquial, often for students)
matte – math (informal for matematik)
uni – clipped form of universitetet, common in speech

Your original sentence:

Min vän vill studera matematik på universitetet.

is neutral and works very well in both speech and writing. The informal version just sounds younger and more relaxed.