<p>If you’re that advanced in math to the point where minute differences in top departments becomes meaningful, you’d also differentiate between department sub specialty. Somehow I doubt you’re at that level which means attempting to differentiate between how far you can go at the above institutions is ridiculous. Fwiw, my former boss, who earned his phD in physics from CalTech did his UG at Waterloo. Multiple math phDs from MIT went to Berkeley for UG.</p>