Tenure Track Positions
It is very hard.

A few years ago, a professor I had had told me that he advises students to aim to get a job at a school a tier or two below where they went to grad school, as a general rule. The reasoning is simply that there are far too few jobs. How many graduate students are in a cohort at Columbia? Now compare that to the number of new faculty they hire each year.

The math is straight forward: If every university hires one new faculty a year, that means that you need to be (on average) the best at your university to get a job at an equivalent institution. Most universities have more than 10 incoming graduate students. I don’t know how many new faculty top math universities hire on average, but I would be shocked if the answer was greater than 2.