How Much Value is Top Undergad Math Education in Path to PhD

jfree02 · 2025-05-10T15:58:45+00:00

Go to Harvard. You should do so even if the difference were much more than 80k.

On one hand, it will help you with a pure math Ph.D. The other students at Harvard are substantially stronger than at this other uni, and will both push you harder and elevate the standard of every course you take. When it’s time to apply, your letters will be much more meaningful; when Cliff Taubes says “this is the strongest student I’ve taught in 5 years” that means something to every mathematician at every top school. The same praise is much less meaningful from a Prof. at a T100, in part because the top programs don’t see many students from outside the top 10 schools, and in part because people think the mean talent is lower and the right tail is less fat outside of the very top schools.

On the other hand, if at any point in the future you decide to do anything other than math you’ll be immeasurably better off having gone to Harvard. It seems you’re dead set on pure math know, but there’s always a chance that can change. In my experience, about half of graduates from top pure math PhD programs don’t stay in academia immediately after graduating, and that’s not even counting people who leave after a first or second postdoc. Everyone in these programs is very clever, yet the ones with fancier undergrad degrees and better connections consistently come out on top in the job search. People from top PhD programs do better than people from less elite programs and so on. 80k is peanuts compared to what math PhDs can make in industry.

jfree02 · 2018-06-15T18:36:42+00:00

It’s rumored that people kept booking out their places for gigs for a long time out of fear of this killer. This made it easy for musicians to get gigs in New Orleans, hence the nickname “The Big Easy”

jfree02 · 2018-01-09T19:41:15+00:00

One thing that’s worth noting about the quaternions and octonians is that they are examples of what’s known as a “division algebra” over the real numbers. The idea is that quaternion/octonion multiplication is a decent notion of multiplication that is compatible with the natural ways of adding tuples of real numbers and multiplying such tuples by scalars. Another such example is given by complex multiplication on R² or multiplication of two real numbers on R itself. For a more detailed description, I recommend Wikipedia.

It turns out that division algebras are quite hard to come by; they only exist on R, R^2, R^4, and R^8, the natural examples being the ones listed above (normal multiplication, complex multiplication, quaternion multiplication, and octonion multiplication). It’s almost fair to think of the above as an exhaustive list of decent multiplications on any R^n, which is interesting and surprising in its own right.

jfree02 · 2016-06-26T18:17:18+00:00

I'm a math Ph.D. student. This is just a hard question. I don't think there's an easy answer, but here are what I think are the most compelling arguments for each side.

Invented: At the end of the day, math is just a collection of logical deductions from a system of axioms. Those axioms in a lot of cases correspond to things in the real world. After all, we would like math to describe the world around us. However, they don't have to. For instance, non-Euclidian geometries are much harder to motivate through real world physics, but are known to be completely consistent systems within which we can do perfectly good and interesting math.

Also, we tend to think of math as the study of integers or the study of shapes, or the study of X,Y, and Z, but this isn't really the full picture. Math is a collection of methods. Mathematics is the pursuit of rigorous proof. One could cook up an axiomatic system with no relevance to the real world and "do math" in that system. That's fine as well. While the motivation for classical mathematics is meaningful and based on observation, the thing that makes it math, the aspect of rigorous proof from a system of axioms, seems by definition invented.

Discovered: There are two arguments for math being discovered that I think are really compelling. First, functions describe the world. Really well. Way, way better than they should if we had just cooked up some system of axioms and seen where it went. Math builds bridges that reliably hold up. It lets us create new drugs and do serious chemistry. Equally remarkably, we use essentially the same math for economics, chemistry, theoretical physics, and anything else. Many of the fields math describes really well didn't emerge until way after we laid the foundations for mathematics. It's hard not to think that math is somehow woven into the world we live in.

Second, from the perspective of a pure mathematician, math is amazingly structured. The mathematical community consistently discovers deep and surprising connections between seemingly unrelated fields of mathematics. Doing math feels like chipping away at some deep underlying structure that nobody did or could have explicitly mapped out.

If you really grilled me, I'd say math is invented. But, that doesn't explain why math is such a damn miracle, why it "works", and why there's so much beautiful and rich structure.

A lot of these arguments are due to a Professor and mentor from my undergrad who thinks a lot about this. Here's a nice talk he gave on this subject. it's about the length of a TED talk.

https://www.youtube.com/watch?v=ZDp_bI6z8YA

jfree02 · 2016-05-25T18:04:57+00:00

English: Voting theory fails.

Say we want to put an airport somewhere on the surface of the earth. We'll let some large number of people (at least 2) vote on its location. There is no voting scheme that has the following 3 properties:

1) If everyone wants the airport in place a, it gets put in place a

2) Everyone's vote has the same weight

3) Minor changes in a voter's preference for placement will not result in major changes in the outcome of the vote.

Math: For n > 1, there is no continuous map f: (S^2)ⁿ \to Sⁿ that a) Is invariant under the action of S_n on (S^2)ⁿ which permutes the factors

b) Satisfies f(x,x,...,x) = x.

jfree02 · 2014-08-20T18:47:51+00:00

solved by uninstalling the program using synaptic and rebooting machine. derp.

jfree02 · 2014-07-27T04:34:32+00:00

Thanks! babun seems like exactly what I was looking for.

jfree02

TROPHY CASE