Hey,
I am applying for Pure math PhD for fall 2022. I was just wondering if any of you guys can share the math courses you all have taken. I was a bit concerned about what are the courses that usually people seem to have done to show for PhD applications.
Courses taken
Re: Courses taken
Here are the math (and cs theory) courses I have taken including what I am taking this Fall semester:
calculus13
diff eq
intro to proof writing class
linear algebra 1&2 (one is computational and two is proofbased using Friedberg et al)
analysis 1&2
abstractalgebra 1&2
graphtheory
combinatorics
graduatelevel graph theory 1&2
theory of computation
graduatelevel algorithms class
graduatelevel computational complexity
graduatelevel analysis
also some independent studies
calculus13
diff eq
intro to proof writing class
linear algebra 1&2 (one is computational and two is proofbased using Friedberg et al)
analysis 1&2
abstractalgebra 1&2
graphtheory
combinatorics
graduatelevel graph theory 1&2
theory of computation
graduatelevel algorithms class
graduatelevel computational complexity
graduatelevel analysis
also some independent studies
Re: Courses taken
My undergrad courses were:

Calc. 13
Differential Equations
Multivariable Calc. 12
Linear Algebra 12 (Undergraduate)
Intro. Proofs
Elementary Real Analysis 12 (Undergraduate)
Complex Analysis 12 (Mixed Grad/Undergrad)
Fourier Analysis 12 (Mixed Grad/Undergrad)
Intro. Real Analysis 13 (Mixed Grad/Undergrad)
Intro. Topology 12 (Mixed Grad/Undergrad)
Intro. to Manifolds (Mixed Grad/Undergrad)
Intro. Differential Geometry (Mixed Grad/Undergrad)
Abstract Linear Algebra (Mixed Grad/Undergrad)
Intro. Abstract Algebra 13 (Mixed Grad/Undergrad)
Abstract Algebra 12 (Graduate)

For reference, I got into a couple schools for Fall 2021, one of which was one of my higher choices (and I'm going there.)
A word of advice: if you wish to apply for pure math  take pure math courses! I think I was rejected by some places because I was missing some important elements of coursework at the time of applying (topology, graduate algebra) since I didn't take those courses until this year (my senior year). It's just a fact that applied math courses (undergrad complex analysis, fourier analysis, physics related courses) are NOT valued very much by committees (when applying for pure math.) Of course this is not true everywhere: some places it is a huge plus to have a wide range of courses and applied math and physics and stuff. But in general, I've received feedback letting me know that it really doesn't help much. But if you are interested in them of course you should take those. Essentially, all I'm saying is make sure youre focusing on pure math and taking a lot of it.
Good luck!

Calc. 13
Differential Equations
Multivariable Calc. 12
Linear Algebra 12 (Undergraduate)
Intro. Proofs
Elementary Real Analysis 12 (Undergraduate)
Complex Analysis 12 (Mixed Grad/Undergrad)
Fourier Analysis 12 (Mixed Grad/Undergrad)
Intro. Real Analysis 13 (Mixed Grad/Undergrad)
Intro. Topology 12 (Mixed Grad/Undergrad)
Intro. to Manifolds (Mixed Grad/Undergrad)
Intro. Differential Geometry (Mixed Grad/Undergrad)
Abstract Linear Algebra (Mixed Grad/Undergrad)
Intro. Abstract Algebra 13 (Mixed Grad/Undergrad)
Abstract Algebra 12 (Graduate)

For reference, I got into a couple schools for Fall 2021, one of which was one of my higher choices (and I'm going there.)
A word of advice: if you wish to apply for pure math  take pure math courses! I think I was rejected by some places because I was missing some important elements of coursework at the time of applying (topology, graduate algebra) since I didn't take those courses until this year (my senior year). It's just a fact that applied math courses (undergrad complex analysis, fourier analysis, physics related courses) are NOT valued very much by committees (when applying for pure math.) Of course this is not true everywhere: some places it is a huge plus to have a wide range of courses and applied math and physics and stuff. But in general, I've received feedback letting me know that it really doesn't help much. But if you are interested in them of course you should take those. Essentially, all I'm saying is make sure youre focusing on pure math and taking a lot of it.
Good luck!

 Posts: 9
 Joined: Mon Jan 04, 2021 1:54 pm
Re: Courses taken
Here are the math and stat courses I have taken.
Intro Level:
Calc 13, Discrete Math, Linear Algebra, Differential Equations, Intro Geometry, Technology for Math (aimed at math education majors)
Pure:
Real Analysis 12, Modern Algebra 12 (only 1 reflected at time of applying), Topology, Number Theory, Senior Seminar, Linear Algebra (graduate)
Applied:
Applications of Math, Partial Differential Equations, Complex Analysis (computational and some proofs, no applications), Numerical Analysis (graduate)
Statistics:
Stat 12, Probability, Mathematical Statistics (like Probability 2), Regression, Statistical Learning
Notes:
For reference, I got into 2 of the 5 schools I applied to, one of which was my top choice (in the 30s on US News rankings). My school allows undergrads a max of 2 grad classes. I took all the undergrad math courses except for 2 aimed at math education majors. My independent studies are not specifically mentioned on my transcript.
I agree with PajamaSam that if your goal is pure math, take pure math courses. Some schools have stricter coursework requirements than others, so you may want to look into those. Generally speaking, your coursework is not the only aspect of your application. If you feel you are lacking in some aspect of coursework (e.g. not enough relevant courses), you may want to address this in your statement of purpose or explain how you will nonetheless be prepared for grad courses (perhaps through research or some other means).
Intro Level:
Calc 13, Discrete Math, Linear Algebra, Differential Equations, Intro Geometry, Technology for Math (aimed at math education majors)
Pure:
Real Analysis 12, Modern Algebra 12 (only 1 reflected at time of applying), Topology, Number Theory, Senior Seminar, Linear Algebra (graduate)
Applied:
Applications of Math, Partial Differential Equations, Complex Analysis (computational and some proofs, no applications), Numerical Analysis (graduate)
Statistics:
Stat 12, Probability, Mathematical Statistics (like Probability 2), Regression, Statistical Learning
Notes:
For reference, I got into 2 of the 5 schools I applied to, one of which was my top choice (in the 30s on US News rankings). My school allows undergrads a max of 2 grad classes. I took all the undergrad math courses except for 2 aimed at math education majors. My independent studies are not specifically mentioned on my transcript.
I agree with PajamaSam that if your goal is pure math, take pure math courses. Some schools have stricter coursework requirements than others, so you may want to look into those. Generally speaking, your coursework is not the only aspect of your application. If you feel you are lacking in some aspect of coursework (e.g. not enough relevant courses), you may want to address this in your statement of purpose or explain how you will nonetheless be prepared for grad courses (perhaps through research or some other means).

 Posts: 5
 Joined: Thu Feb 25, 2021 5:30 am
Re: Courses taken
All of the listed courses were at schools on the quarter system. For reference, I have no research experience, my overall GPA was roughly ~3.4, and my major GPA ended up around ~3.2 or so. I applied to eight graduate programs and got into one. Courses from Winter 2018 through Spring 2019 were taken at a community college, and the remaining courses were taken at the University of Washington.
Winter 2018
 Precalculus I
Spring 2018
 Precalculus II
 Calculus I
Autumn 2018
 Calculus II
 Linear Algebra
Winter 2019
 Calculus III
 Ordinary Differential Equations
Spring 2019
 Calculus IV
Summer 2019
 Intro to Mathematical Logic (proofwriting and set theory)
 Linear Analysis (a nonmajors' course geared more towards engineers covering a little more linear algebra and some basic PDE [heat, wave, and Laplace equations])
 History of Mathematics
Autumn 2019
 Abstract Algebra I (ring theory)
 Knot Theory (a somewhat nonrigorous undergrad special topics course)
Winter 2020
 Abstract Algebra II (group theory)
 Real Analysis I (the reals, sequences, series, continuity)
Spring 2020
 Abstract Algebra III (Galois theory)
 Real Analysis II (sequences and series of functions, differentiation, RiemannStjelties integration, uniform convergence)
Summer 2020
 Probability I (semirigorous course in onevariable probability)
Autumn 2020
 Topology
Winter 2021(The quarter in which I actually applied to the grad schools)
 Differential Geomerty I (curves and surfaces in R^3)
Spring 2021
 Real Analysis III (metric spaces, functions between metric spaces, several real variables)
TL;DR: A year each of abstract algebra and real analysis, a quarter of topology, and a few other classes here and there. No complex analysis or abstract (i.e, proofbased) linear algebra.
Winter 2018
 Precalculus I
Spring 2018
 Precalculus II
 Calculus I
Autumn 2018
 Calculus II
 Linear Algebra
Winter 2019
 Calculus III
 Ordinary Differential Equations
Spring 2019
 Calculus IV
Summer 2019
 Intro to Mathematical Logic (proofwriting and set theory)
 Linear Analysis (a nonmajors' course geared more towards engineers covering a little more linear algebra and some basic PDE [heat, wave, and Laplace equations])
 History of Mathematics
Autumn 2019
 Abstract Algebra I (ring theory)
 Knot Theory (a somewhat nonrigorous undergrad special topics course)
Winter 2020
 Abstract Algebra II (group theory)
 Real Analysis I (the reals, sequences, series, continuity)
Spring 2020
 Abstract Algebra III (Galois theory)
 Real Analysis II (sequences and series of functions, differentiation, RiemannStjelties integration, uniform convergence)
Summer 2020
 Probability I (semirigorous course in onevariable probability)
Autumn 2020
 Topology
Winter 2021(The quarter in which I actually applied to the grad schools)
 Differential Geomerty I (curves and surfaces in R^3)
Spring 2021
 Real Analysis III (metric spaces, functions between metric spaces, several real variables)
TL;DR: A year each of abstract algebra and real analysis, a quarter of topology, and a few other classes here and there. No complex analysis or abstract (i.e, proofbased) linear algebra.