I'm an Industrial Engineering graduate looking to make a switch into Pure Math at the Masters Level/PhD. I have undergrad experience in Calculus (upto Differential Equations) and Probability and Statistics.

I have an MSc/PhD entrance exam exactly a year from now and I shall need to self study topics in:

1) Real Analysis: 7 chapters of Baby Rudin, upto Sequences and Series of Functions

2) Complex Analysis: Introductory Chapters into AV Ahlfors' Complex Analysis

3) Linear Algebra: Artin's Algebra

4) Abstract Algebra: Artin's Algebra, D&F topics like Sylow's Theorem, Finite Field, Maximal and Prime Ideal

5) Differential Equations: Undergrad Level, Tom Apostol Calc II

  • I'm diligent, possess strong work ethic
  • Consistently place 10-12hr work days
  • Large repository of grit and patience
  • Appreciate Mathematical theory and intuition very dearly


My Question: In what order of topics must I cover the above topics, and which courses can/must be taken parallely so that I don't miss any foundation intuition and iteratively cover all topics prior to April-May 2019?

I'd especially appreciate advice from those Math mavericks who'd decided to veer off their traditional path and into Math

@s-stein @jack-bauer @user204305 @ericam @quasar @louis (Since I'd seen ya'll have similar experiences/questions as mine)


closed as primarily opinion-based by Matthew Towers, mrtaurho, Rhys Steele, воитель, The Count Jul 18 at 1:45

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ @jack-bauer (Since I'd seen you have similar experiences/questions as mine) $\endgroup$ – Sumanth Lazarus May 8 '18 at 11:31
  • $\begingroup$ @quasar (Since I'd seen you have a similar experiences/questions as mine) $\endgroup$ – Sumanth Lazarus May 8 '18 at 11:32
  • $\begingroup$ First, way to go. Second, I think it is going very, very tough and long as the ammount of mathematicis needed for a PhD is rather very big taking into account what you did in engineering, and even that IE (because this is perhaps the engineering which requires the less mathematics of them all...). It looks like you'd need to go over the whole, usual undergraduate curriculum. Now, (1) and (3) look to me the most basic. You'd also need some set theory and discrete mathematics. Then abstract algebra, perhaps some Topology and Measure theory and Diff. Eq's, finally complex analysis. \ $\endgroup$ – DonAntonio May 8 '18 at 11:35
  • $\begingroup$ My intention is to do a PhD down the line; may have to get through an MSc program or enroll in an Integrated PhD program. So I don't mind the next 7 years of learning and working in Math. Thanks for your comment anyways :) $\endgroup$ – Sumanth Lazarus May 8 '18 at 11:46
  • 1
    $\begingroup$ In my opinion, your 1 thru 5 will take you 1 and half year, plus topology (you seem to forget), the total will be two years, not one year. Not to discourage you, but you only have 24 hours a day, you need to sleep (too less sleep will make your study in-efficient) and also you need to relax from time to time. So, give yourself two years, not one year. $\endgroup$ – scaaahu May 8 '18 at 14:30

I will answer as if you posted this question today. I was not notified with your "tag". Tagging works when someone has already commented on the original post or an answer or answered, and the tag has to be on the original post, answer or answer, respectively.

About the references, I agree with most.

  • Artin Algebra has Linear Algebra for only Chapters 1 and 3 and you'll be learning mostly Abstract Algebra anyway. How about something simpler for Linear Algebra?

  • Artin Algebra basics are Chapters 1,2,half of 3, half of 11

  • Ahlfor should be up to Chapter 5.1.3 Laurent series, but

  • Ahlfor is hardcore for an engineering. Why not Brown Churchill?

  • Rudin: You can skip the part of Chapter 3 "The Number e" until "Rearrangements", all of Chapter 5 differentiation, the last two sections in Chapter 6: "Integration of Vector-valued functions" and "rectifiable curves" and the last three sections in Chapter 7

  • You're missing topology. Try Munkres Chapters 2-3. Rudin covers a little bit of this but for $\mathbb R$ only I think. Munkres generalizes these.


  1. Real Analysis (The "Elementary" Real Analysis you have suggested)

    • A few topics in complex analysis will be repeats of real analysis like sequences, limit points and uniform convergence, so let's do real analysis first.
  2. Linear Algebra

  3. Complex Analysis Basics (Chapter 1 Ahlfors)

    • Abstract algebra or topology concepts have some examples using roots of unity. It's also easier to view some functions as $e^{2 \pi i x}$ instead of $(\cos(2 \pi x), \sin(2 \pi x))$
  4. Abstract Algebra

  5. Topology

  6. The Rest of Complex Analysis (Chapters 2-5 Ahlfors)

Notes on Order:

After you have finished, you can take up additional topics to see what interests you:

  • Artin and Dummit Foote have a lot for abstract algebra.

  • Ahlfors has more for complex analysis.

  • For real analysis, I think Rudin gives a good introduction of higher real analysis can be learned more in depth in Royden and Fitzpatrick Part One.

  • Functional analysis in Kreyszig, Papa Rudin or Royden and Fitzpatrick Part Two. Kreyszig doesn't require algebra or topology, but I think things will go faster for you if you know those.

  • Algebraic topology in Munkres Part Two. (But if you plan to continue in Munkres, you must know Munkres Chapter 4 as well. You do not need Chapters 5-8 to begin Part Two.)

  • Munkres Part One also has additional topics (Chapters 5-8). (But if you plan to continue in Munkres, you must know Munkres Chapter 4 as well because Chapters 2-4 are core topology.)

  • Algebraic Geometry: Geometry textbooks for university students

  • Differential Geometry: Geometry textbooks for university students


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