For example, could a Jewish university student learn mathematics (without a computer) on Shabbat?
The Mishna Shabbat 148b writes that one may not count the number of guests from a list on Shabbat. Abaye in the Gemara 149a explains that this is a rabbinic restriction so as not to come to read a Shtar Hedyot on Shabbat. What is a Shtar Hedyot which one may not read on Shabbat? Rashi 116b s.v. Shtar explains it to be financial calculations or letters. Rambam (Peirush Mishnayot Shabbat 23:2) also explains Shtar Hedyot to mean letters and the reason for this is that on Shabbat one may only read Navi and it’s explanations and not even a book of wisdom and science. Bet Yosef 307:17 quotes the Baal HaMoer (43a s.v. VeHa) who agrees.
However, the Rashba (149a s.v. VeLeInyan) explains Shtar Hedyot as a business document. This is also the definition of Ri quoted in Tosfot 116b s.v "kol sheken" and Rosh (see there where he writes that reading shtarei hedyotot is a problem of mimtzo cheftzecha from Yishayahu 58:13) Shabbat 23:1. Therefore, Sh”t Rashba 7:288 rules that it’s permitted to read books of wisdom and medicine and quotes Ramban who agreed. [Bet Yosef 307:17 learns this out from another Teshuva of the Rashba 1:772 where he says that one may analyze an astrological tool called the Istrolav.]
Shulcha Aruch 307:17 rules like Rambam who says that on Shabbat one should only learn Torah and not books of other disciplines, however, he mentions that some are lenient. Mishna Brurah 307:65 writes that the minhag is like the Rashba, however, a pious person should be strict. Rav Aviner holds like the Mishna Brurah.
However, Yalkut Yosef (Shabbat vol 2, pg 214, 626) rules like Shulchan Aruch that one should only learn Torah on Shabbat and is only lenient to allow a medical student who has a test after Shabbat and is pressured for time to study medicine (except for the study of surgery) on Shabbat.
This is disputed.
Shulchan Arukh, Orach Chayyim 307:17:
אָסוּר לִלְמֹד בְּשַׁבָּת וְיוֹם טוֹב, זוּלַת בְּדִבְרֵי תּוֹרָה; וַאֲפִלּוּ בְּסִפְרֵי חָכְמוֹת אָסוּר; וְיֵשׁ מִי שֶׁמַּתִּיר, וְעַל פִּי סְבָרָתוֹ מֻתָּר לְהַבִּיט באצטרלו''ב בְּשַׁבָּת, וּלְהָפְכָהּ וּלְטַלְטְלָהּ כְּדִלְקַמָּן סי' ש''ח.
My loose translation:
One should not learn anything on SHabbat except Torah related matters. Even studying books of wisdom (I assume this means "adages" or something similar?) is forbidden. But some say that is permitted, and according to them one may use an astrolabe on Shabbat.
On the other hand, it seems that if the sole purpose of studying is for an exam that will occur after Shabbat, then, this may be a prohibition of hachanah - preparing on Shabbat for a weekday. That's also debated. See item #14 in Halachipedia along with its footnote.