How much math should one have before grad school?

[cfddude123]

I'm looking at getting a PhD in applied math (applied probability, stochastic processes, or PDEs are my main interests) and I feel like people going to grad school have so much more math than I do. Just wanted to know what is normal for someone currently in their sixth semester. I'm a math and statistics major.

 

[chiro]

I'm looking at getting a PhD in applied math (applied probability, stochastic processes, or PDEs are my main interests) and I feel like people going to grad school have so much more math than I do. Just wanted to know what is normal for someone currently in their sixth semester. I'm a math and statistics major.

Hey Nishrito and welcome to the forums.

I would suggest you have an analysis course and cover some measure theory as well.

You mentioned that you have a major in statistics so I'm guessing you have done a fair bit of applied probability and maybe some stochastic analysis of some sort (SDE's, Weiner processes etc).

On top of that anything related to PDE's and DE's that you can take and especially if its applied like Fluid modelling, heat or related systems modelling, insurance or financial product modelling: that kind of thing. Also anything like chaos, weather modelling, oceanography, that kind of thing is also good.

Another suggestion is maybe data mining and any appropriate programming courses or other modeling courses that make extensive use of packages like MATLAB, Maple, R, SAS, etc.

Also I would suggest some kind of topology course as well.

I only say the above because you want to go to graduate school and the advice is based largely on what other people have said here in the past about applied kind of programs.

I am in my last year of a double major in math and I myself haven't taken the rigorous stuff like a proper analysis or topology course, but I have had conversations with a PhD student (probably nearly finished now) who was doing his PhD in financial modelling (not the exact title, but it is a good description) and he spoke about the kinds of stuff he did in his bachelors and honors year subjects before his PhD.

 

[cfddude123]

Hey chiro, thanks for the reply!

You've made me feel a lot better about my chances. My minor is actually in atmospheric science and I've taken classes on atmospheric radiation and two semesters of atmospheric dynamics. I'm also planning on taking measure theory and a numerical fluid dynamics course next semester.

 

[deRham]

I think having mastered the fundamentals to your subject and having a first exposure to the kinds of math used in your field of interest is best. That helps you know what you're getting into.

For instance, if you want to do PDEs, you should take some measure theory, topology, functional analysis, etc as foundations, but you should also try to study an introductory book for graduate students on PDEs (or some suitable equivalent).

I feel like the second step CAN be done without, but it just feels better to recommend that you do it.

 

[cfddude123]

Hey deRham,

That's what's a little worrying to me. I have only one more semester after this one and I will not have taken a functional analysis class nor a topology class. I will have taken two semesters of analysis (the second semester is called "Analysis on Manifolds") and I'm planning on taking measure theory in my next semester.

A lot of the other students in my year are just math majors, so they focus on just the math. So they're already taking measure theory, functional, graduate complex, etc while I'm a few semesters behind in coursework because I was also double majoring in stats. That's why I feel like I'm at a disadvantage.

 

[bpatrick]

Granted, my undergrad was in musicology, grad school in trumpet performance, but I'm going back to do a MD/PhD in mathematical biophysics ... and I kinda feel like a mathematician from time to time, haha. Anyhow, my math prep is almost the same as other students you'd be competing with for grad school spots (in two of the programs I've gotten into I'd actually be earning a M.A. *due to not writing a masters thesis* in applied mathematics along the way to the PhD in biophysics).

My pre-grad school math prep consisted of:

linear algebra
intro ODEs
multivariable / vector calc
1 semester real analysis
1 semester complex analysis
1 semester of PDEs
year long seminar bifurcation theory
1 semester of algebra
advanced linear algebra
1 semester of topology

so 11 courses in math total. I bolded the "upper division" ones I feel were most important for what I'm doing.

if you're into stochastic processes and applied probability, you may not feel those things I bolded were relevant to you. Instead, I'd make sure you were confident with programming at least a few languages and math software so that you can beast your way through that stuff. maybe taking more (or some) numerical analysis if you're getting into applied PDEs / physical stuff.

If you were trying to get into really competitive math programs, you might want to have a lot better math prep than I do. My prep for biophysics is rounded out by the typical pre-med stuff (biochem, microbio, orgo, etc...), neuroscience, and physics courses. So my math prep isn't nearly as good as a true mathematician, that's all I'm getting at.

Your prep sounds like you'll be fine for most applied programs. I'd guess recommendations and gpa in upper division courses will weigh more than the fact you may not have had a course in topology or other "pure" stuff.

 

[cfddude123]

I guess I should have posted my undergraduate background:

Probability/Statistics:
Statistics and Probability 1 and 2
Methods of Applied Statistics
Linear Regression
Time Series Analysis
Analysis of Variance (theoretical)
Markov Chains and Stochastic Processes

Mathematics:
Intro to Proof Writing
Applied/Abstract Linear Algebra
Abstract Algebra 1
Applied Complex Variables
ODEs
PDEs
Honors Analysis I (Level of Baby Rudin)
Honors Analysis II (Analysis on Manifolds-Munkres)
Combinatorics

Going to take Measure Theory, Graduate ODEs or PDEs, and a Numerical Analysis class

Atmospheric Science:
Atmospheric Radiation
Atmospheric Dynamics 1 and 2

 

[deRham]

Don't worry, I hardly meant to scare you! Those are useful things, and if you don't know it all now, first year grad school is a perfect time to catch up. A background in measure theory is a good way to get to the rest of analysis.

Your background in that list looks very fine actually. Just because you did not see some of those subjects at the graduate level does not at all mean you will be in trouble.

In fact, your additional classes like Stochastic stuff will help you appreciate your future coursework. After all, those are the kinds of topics a lot of people interested in analysis go into in more depth. Whereas, things like functional analysis are just foundational, and you may appreciate the need more after seeing all the examples you have. Effectively, whenever you deal with operators or linear functionals on an infinite-dimensional space, you will need functional analysis to get a good feel for what's going on. But effectively, a lot of calculations and techniques probably carry over from your previous coursework (although introducing new beasts is probably part of the fun...).