Order of Pure Math Topics to Self Study for PhD Admissions Qualifier

Question

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

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• 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

TL:DR

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)

Answer 1

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.

Order:

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:

• 1 and 2 can be interchanged or learned together

• 4,5 and 6 can be interchanged or learned together

• No more differential equations even partial differential equations. I highly suggest prioritizing everything else because differential equations, statistics and probability are more used in applied math.

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