It should be mandatory the graduate problem sets be worked on together, in person, with only the textbook and a dedicated 3 hours minimum per day. Students should have no access to the Internet, and can’t communicate with anyone outside of class. Then, every student must present a solution by the next lecture. Professors shouldn’t lecture much at all, and all reading should be required before the next session.

Prelims should exist as a placement test and nothing more. Every student should take every core course, two at a time, per year, and take a qualifying exam after each double core course sequence.

If the students do not pass prelims upon arrival, they should be placed in a remedial equivalent of the graduate sequence, that isn’t undergrad, but is maybe a slower pace than the graduate pace (which only seems to work if you’d had exposure to the material before).

There should be no teaching duties for at least the first two years until qualifying examinations are passed.

Do this, force students to communicate and collaborate, and I bet you, you will have more successful, well-rounded, and educated students. It shouldn’t be the case that a person graduates with a PhD in Mathematics not having seen one of these core courses (Algebra, Analysis, Geometry). Specialization is happening too early, which is why people aren’t discovering new things, or rather… aren’t discovering anything interesting.

So, what Math PhD program is like this in the United States? That’s where I’ll go.

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