
Ask HN: Mathematicians, what textbooks are best for learning these math topics? - smithmayowa
So I have just decide to self-learn mathematics up to undergraduate level, and after researching I decided that learning these topics will do the trick for learning to the undergraduate level, but sadly I don&#x27;t know the best resources(textbooks) to use for easily self-learning them.<p>Pure Mathematics<p>1).Group Theory(rubics cube,e.t.c)<p>2).Order Theory<p>3).Combinatorics(trees,graphs,e.t.c)<p>4).Fractal Geometry<p>5).Topology(cup = donut)<p>6).Measure Theory<p>7).Differential Geometry<p>8).Vector Calculus<p>9).Dynamical Systems(Fliud flow,ecosysytems,Control Theory)<p>10).Chaos Theory(Butterfly effect)<p>11).Complex Analysis(Functions with complex numbers)<p>Applied Mathematics<p>1). Numerical Analysis<p>2). Game Theory<p>3). Probability<p>4). Statistics<p>5). Optimization<p>6). Cryptography<p>7). Computer Science<p>Foundations<p>1).Mathematical Logic<p>2).Set Theory<p>3).Category Theory<p>4).Godel Incompleteness Theorems<p>P.s: I have a diploma in Marine Engineering and so I am not a total noob to math in general.
======
govg
I'm not really a mathematician, my experience comes from machine learning /
statistics. Note that most math books tend to be known by the name of the
author, as opposed to a fixed title.

Pure math :

1)Group Theory - Milne

5)Topology - Mukres

6)Measure Theory - Terrence Tao has a course, Robert Ash has a book on
probability theory, and it is recommended that you study a bit of real
analysis before you do this.

7) Differential Geometry and Statistics - Murray and Rice

9, 10 ) Non linear dynamics and chaos - Strogatz

11) Complex analysis - There exists a set of 4 books covering real, complex
and functional analysis by Stein and Shakarchi, which should serve your
purpose

Applied Math :

3) Probability - Grinstead and Snell, Durrett both have good books.

4) All of Statistics by Larry Wasserman for a more ML bent to it

5) Optimization by Boyd and Vanderberghe

7) There's no single topic called "Computer Science", but going with the theme
of the topics you are looking at, Algorithms by Cormen, Leiserson et. al,
Theory of Computation by Michael Sipser should be good starting points.

Note that for a lot of these, you can find high quality material online (Both
videos as well as course material). Just do a search for "Topic MIT OCW",
replacing "Topic" with your choice.

~~~
philip-b
Remark: Optimization by Boyd and Vanderberghe is actually Convex optimization.
It doesn't cover non-convex stuff.

~~~
govg
Of course, you're correct. I chose to ignore that aspect since Boyd's book
covers a lot of basic material, and I'm not actually aware of any such text
covering non-convex methods in general ( and considering how fast research is
in this space, I'm not sure of any general work, in contrast to work like [1]
).

[1] - Non Convex Optimization for Machine Learning - Jain and Kar
([https://arxiv.org/abs/1712.07897](https://arxiv.org/abs/1712.07897))

------
philip-b
Textbooks which people on the internet consider very good:

>Group Theory

Carter - Visual Group Theory

>Complex Analysis

Needham - Visual Complex Analysis

Less confident recommendations:

>Probability

[https://ocw.mit.edu/courses/mathematics/18-05-introduction-t...](https://ocw.mit.edu/courses/mathematics/18-05-introduction-
to-probability-and-statistics-spring-2014/) as a very entry level
introduction. After that you might still need some textbook for more depth.

>Mathematical Logic and Godel Incompleteness Theorems

Start with ForallX by Magnus. Then continue with Computability and Logic by
Boolos (don't read all chapters, check out preface to see what you need for
Godel Incompleteness Theorems).

>Set Theory

Hrbacek - Introduction to Set Theory

\---

I am currently developing an online course which teaches logic, set theory,
and computability theory (this includes Godel Incompleteness Theorem). It uses
the textbooks I've mentioned above. You can check it out at
[https://app.grasple.com/#/course/141?access_token=3HCK4oRipe...](https://app.grasple.com/#/course/141?access_token=3HCK4oRipeFY2ghyYMqDJKYX57KUnzNL).
Use "Fundamentals of formalization" and ignore other tracks.

~~~
smithmayowa
Currently going through your course, thanks for taking the time to make
something like that.

------
mancerayder
I skimmed all the replies below - this is years of learning and study - so I
ask:

Is there some dependency order someone could quickly sketch out for some of
these topics? Eg, linear algebra comes before X?

HN is an incredibly useful crowdsourcing resource for the self-motivated!

~~~
nextos
The minimal classical way to bootstrap your math knowledge is algebra (mostly
linear) + analysis. This is the approach used by e.g. Harvard Math 55, which
is a really famous course.

Harvard Math 55 employed Halmos + baby Rudin as main textbooks. Halmos has now
been replaced by Axler, which is an excellent textbook (but the typographic
changes in the last edition are very distracting).

Rudin is probably too synthetic and dry for a beginner. You could easily
replace it by Hubbard & Hubbard (which was the sole main textbook in one
Harvard Math 55 edition), or use an aid text like Gelbaum & Olmsted. You can
also skip Axler if you go the Hubbard way.

A favorite open question of mine is what would a math bootcamp look like if
you went up one level of abstraction and focus more on logic and abstract
algebra.

~~~
dorchadas
I would love a bootcamp like that. Or even one that helped you with proofs.
I've found that's my biggest issue trying to teach myself pure mathematics. I
just can't _start_ the damn proof. Once I get the start, I can usually finish
it, but starting the proof I just feel so clueless how to do it.

~~~
nextos
You can learn to do basic proofs the way old school Tsarist and Soviet Russia
students did, by studying geometry from Kiselev. It should be pretty
straightforward for any adult, and a nice stepping stone to build your
knowledge on.

A more advanced approach would be to go through a logic book like Velleman
that teaches proofs as structured programming.

------
mojomark
Fellow Marine Engineer (KP) here, and just wrapping up my Master's in CS Data
Science. If you're like me, ou got rushed through a number of higher level
match (between bar crawls), passing tests, but not digesting.

So, the feer of a math intensive grad program (and admission had me worried).
My CS program offered some math refresher, but I ended up just jumpting in
without it. If the program is decent, you will be guided along at a digestable
pace. As a responsible adult with a hunger to learn, and you will enjoy and
digest more. In a curriculum, you'll also have peers/teachers to help you when
absolutely stumped - which will happen.

For books - I just have one that I saw referenced here on HN, which is a
little odd, but highly recommend for understanding Fourier transforms [1].

1\. Who Is Fourier? A Mathematical Adventure 2nd Edition... Amazon link:
[https://www.amazon.com/dp/0964350432/ref=cm_sw_r_other_apa_i...](https://www.amazon.com/dp/0964350432/ref=cm_sw_r_other_apa_i_RoqRCb2FRETNP)

------
chongli
I noticed you left out linear algebra. You may struggle in topics such as
topology and functional analysis without a strong background in linear
algebra. You mentioned studying marine engineering, so I'm not sure, but in my
experience engineering courses tend to cover only the basics up to a first
year level. There is so much more than that.

------
ac2019-04-09
Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by
Hubbard is good book, written somewhat from the point of view that these
subjects are better presented interleaved than as a self-contained book.

------
shoo
> Group Theory

i still have a copy of Fraleigh -- a first course in abstract algebra. not
sure if it is the best, but it did the job.

I really enjoyed the two real analysis & functional analysis courses when i
attended university, but alas, the reading material for these courses were
notes produced by each lecturer, they're not available as published books.

> 5).Topology(cup = donut)

i recall covering the material for "cup = donut" style results in an algebraic
topology course, i think in 3rd or 4th year, after first being drilled with
2-3 years of pure math including real analysis, including lots of basic stuff
about topological spaces, continuous and smooth functions, measure theory,
some abstract algebra, etc.

    
    
      3). Probability
      4). Statistics
    

MacKay's "Information Theory, Inference, and Learning Algorithms" is a great
read:
[http://www.inference.org.uk/itila/book.html](http://www.inference.org.uk/itila/book.html)

    
    
      5). Optimization
    

Sign up for this course: [https://www.coursera.org/learn/discrete-
optimization](https://www.coursera.org/learn/discrete-optimization)

edit:

you didn't mention PDE, but i still have a copy of Evans -- partial
differential equations serving as a monitor stand. you probably want to have 3
years of pure math including linear algebra, lots of real analysis & some
differential equations under your belt first.

------
mirceam
Not a mathematician, but have spent some time self-learning math. I have not
gone through the following books entirely, however I've used them when needed
to learn or clarify a concept:

> Group Theory \- I would look for a good abstract algebra book that also
> covers groups, such as A First Course in Abstract Algebra, Fraleigh or Basic
> Algebra, Knapp. Another popular recommendation is Abstract Algebra, Dummit
> and Foote.

> Topology I think you should first look at a real analysis course and
> understand metric space topology for Euclidean spaces. If you have taken
> calculus, you can try and go through Baby Rudin. For topology specifically
> you can look into Munkres or Introduction to Topological Manifolds, J Lee.

> Measure Theory T. Tao has a good book on this topic.

> Differential Geometry Introduction to Smooth Manifolds by J Lee

> Probability Probability Theory, Achim Klenke

> Game Theory Game Theory by Maschler, Solan & Zamir

------
WhiteSage
For complex analysis the book by Churchill -written for engineers- is a very
down to earth introduction with lots of examples and applications to
resolution of integrals and PDEs.

In the case of set theory I cannot recommend enough Halmos' "Naive set
theory". It's completely rigorous yet easy to read, and lays out all the
basics of ZF, constructing both ordinals and cardinals.

For probability Khinchin has some very thin fun books, written for some
classes he gave in a military school. The books do not get into deep topics
but are a good introduction.

For group theory I would try Isaacs' Algebra book. Although the book is one of
those books which contain too many topics, he was really enthusiastic of group
theory and I have been told the corresponding chapters are very good.

------
aboutruby
I will say it: Wikipedia. You do need to have the common knowledge used to
describe the topics but Wikipedia is an amazing source.

(out of your list, I mainly looked at Game Theory, Probability, and Statistics
there).

------
malms
You'll never read one book on each of each subject.

------
sreeramvenkat
There is a good set of recommendations from mathwonk in this physicsforums
thread (pretty detailed) [https://www.physicsforums.com/threads/the-should-i-
become-a-...](https://www.physicsforums.com/threads/the-should-i-become-a-
mathematician-thread.122924/)

------
z0k
You can study Group Theory and Category Theory simultaneously using Chapter 0
by Aluffi:
[https://www.goodreads.com/book/show/6829004-algebra](https://www.goodreads.com/book/show/6829004-algebra)

------
throwawaymath
I've given a few textbook suggestions for almost all of the topics you
requested, in a preferred order for learning them. But before you look at that
list, consider the following:

I would strongly, strongly advise against trying to learn proof-based
mathematics from a textbook (almost all of the math here will be proof-based).
The absolute best way to learn mathematics is to have an experienced and
competent instructor tailor their pedagogy to you. Failing that, an
experienced instructor who is "just okay" but who can e.g. review and critique
your work is better than a textbook.

Learning math is very unlike learning programming. It's a counterintuitive
idea, but the information density of math textbooks (whether they're well or
poorly written) is generally so high that you can't absorb the material unless
you read only a few pages per day. Not only that, but it's usually not the
case that a single textbook has the ideal level of exposition for your needs -
for example, you don't have linear algebra on here despite it being a
prerequisite for basically everything else. Some textbooks treat this subject
in a highly theoretical manner, while others treat it at a very
applied/computational level. Which suits your needs more? Have you studied it
at all?

If you're actually serious about this, you need to proceed at a slow pace (2 -
5 pages per day) and complete as many exercises as possible. If the exercises
are computationally focused you can do fewer, but you should aim to solve as
many of the proof-based problems as possible.

If you go at a rate which will actually allow you to absorb the material,
doing this "properly" will take you years. With dedication and not much talent
I'd expect it to take as long as an undergraduate degree. With dedication and
a _lot_ of talent I could see this being accomplished in two, maybe three
years. Once again, I strongly, strongly suggest finding a mentor or
instructor.

In any case, here is a list of the textbooks _most_ mathematicians will
consider to be very good:

1\. Calculus

 _Calculus_ , by Spivak

This gives you a rigorous treatment of calculus, which hopefully you have some
familiarity with. After this you can move on to real analysis.

2\. Real Analysis

 _Principles of Mathematical Analysis_ , by Rudin

You might be ready for this after Spivak's _Calculus_ , but it can be rough.
If you can't reproduce a proof of irrationality after reading through the
first few pages, work through Tao's _Analysis I_ first.

3\. Topology

 _Topology_ , by Munkres is the absolute gold standard. You should be
comfortable with calculus (and hopefully analysis) before tackling this.

4\. Linear Algebra

 _Linear Algebra Done Right_ , by Axler

This is a thorough introduction to the subject at a theoretical level, with a
focus on finite-dimensional vector spaces over fields _R_ and _C_.

You should also work through either _Linear Algebra_ by Friedberg, Insel,
Spence or _Linear Algebra_ by Hoffman & Kunze for the treatment of more
advanced/specialized material and, in particular, determinants (which are
notably de-emphasized by Axler).

Noam Elkies uses Axler for Harvard's Math 55 and has written up notes and
remarks for his students; be sure to read them:
[http://www.math.harvard.edu/~elkies/M55a.16/index.html](http://www.math.harvard.edu/~elkies/M55a.16/index.html)

5\. Abstract Algebra (Groups, Rings, etc)

 _Abstract Algebra_ by Dummit & Foote is the usual reference text for a first
course. It's pretty good. If it's too advanced for you, try Pinter's _A Book
of Abstract Algebra_. For a very challenging (but comprehensive) approach to
the subject, try Lange's _Algebra_.

6\. Category Theory

Once you have abstract algebra under your belt, a good introduction to
category theory is given by Aluffi's _Algebra: Chapter 0_. I would suggest not
trying to dive into this prior to at least encountering fields, groups and
rings because it's good to have both the traditional and modern (read:
categorical) contexts.

Also try _Category Theory in Context_ , by Riehl.

7\. Complex Analysis

 _Complex Analysis_ , by Ahlfors. This is an excellent and concise text. You
can theoretically approach this before real analysis, but I wouldn't recommend
that. Also try _Complex Variables_ , by Churchill & Brown.

8\. Differential Geometry

 _Calculus on Manifolds_ by Spivak. You will want to have a thorough
understanding of analysis and linear algebra before approaching this material.

9\. Measure Theory

This is very advanced material in an analysis sequence; don't jump to this
unless you've thoroughly worked through analysis first.

I would recommend Stein & Shakarchi's _Real Analysis: Measure Theory,
Integration and Hilbert Spaces_.

10\. Probability Theory

A really rigorous treatment of probability is measure theoretic, but even if
you haven't worked with measures before you'll need (real) analysis and linear
algebra. Tackle those first.

Feller's _Introduction to Probability Theory_ is usually a good first course.
If you don't like that, try Ross. For truly advanced probability theory, work
through Shiryaev or Kallenberg.

The other things you've asked for are a little under-specified or outside my
wheelhouse (in particular, I don't think chaos theory is still emphasized as a
field distinct from dynamical systems). You should probably add ordinary and
partial differential equations to your list before some of these more
specialized topics.

1\. Numerical Analysis

 _Numerical Linear Algebra_ , by Trefethen & Bau. This is the best all-around
introduction. Once you've worked through this, try moving on to _Matrix
Computations_ by Golub & van Loan. The latter is much more of a reference
text.

2\. Cryptography

You haven't specified what you're looking for here, but given the mathematical
bent of your question I'd recommend Goldreich's _Foundations of Cryptography_
(two volumes). Be forewarned: cryptography is a subfield of complexity theory.
You should have a strong understanding of complexity theory before embarking
on Goldreich's _Foundations._

If you really want to challenge yourself theoretically, work through
Galbraith's _Mathematics of Public Key Cryptography_. The most up to date
version is available for free:
[https://www.math.auckland.ac.nz/~sgal018/crypto-
book/crypto-...](https://www.math.auckland.ac.nz/~sgal018/crypto-book/crypto-
book.html)

On the other hand, if you're looking for a more implementation-focused text on
cryptography, try Menezes' _Handbook of Applied Cryptography._

3\. Optimization

This is extremely broad. There's linear programming, mixed integer
programming, nonlinear optimization, stochastic optimization...I can't
recommend textbooks targeted at everything here.

For a good start to the subject of optimization and constraints in general,
work through Boyd & Vanderberghe's _Convex Optimization._ There are additional
exercises available from the authors here:
[https://web.stanford.edu/%7Eboyd/cvxbook/bv_cvxbook_extra_ex...](https://web.stanford.edu/%7Eboyd/cvxbook/bv_cvxbook_extra_exercises.pdf)

------
fnrslvr
> Mathematical Logic

A Mathematical Introduction to Logic, by Herbert Enderton.

~~~
smithmayowa
Thanks man, will definitely get this book.

~~~
fnrslvr
I'll add, that I strongly advise against beelining for Gödel's Incompleteness
theorems.

Learn the formal language aspects, like grammar and parsing and structural
induction. Learn semantics. Learn a few deductive calculi (say, natural
deduction and Hilbert style) and how they interrelate, and actually use them
to prove some (very simple) results, ideally from some important axiom systems
like Peano arithmetic and ZF set theory. Learn model theory, Gödel's
_completeness_ theorem, the compactness theorem, and their more immediate
implications.

You should also learn how logic interrelates with

* Computation, both in the sense of enumerability of deductive proof systems, and in the sense in which expressability in certain logical theories is Turing-complete; and

* Set theory, both to grasp the sense in which a first-order set theory like ZF seems to suffice to supply an ontology for the rest of mathematics, and to understand the role of cardinality in e.g. model theory.

There isn't really a correct order in which to approach these fields. You'll
find that for a proper understanding of any of these topics, you'll have to
move back and forth between them frequently.

I would put off Gödel's incompleteness theorems until you've done most of
this. In particular, learn the completeness theorem, up to a point of
confidently being able to apply the compactness theorem, first. Many of the
least-informed abuses of the incompleteness theorems come from people who
can't distinguish different notions of entailment, and are unfamiliar with the
successes of deductive calculi and the categoricity shortcomings of first-
order logic.

------
auvrw
i'm not totally sure what 'fractal geometry' is, although there is something
called 'geometric measure theory'..

the canonical text here is Federer, although it's supposedly a tome. Krantz's
_the geometry of domains in space_ appears more approachable.

...

also, note that these topics aren't totally separable.

------
rootsudo
You know,

I never knew Computer Science was an applied math. But I always went with
statistics.

------
jwgarber
Hello! I am a fourth year undergraduate in pure mathematics, and have taken
many of the classes in your list (especially in the first and third
categories), so I'll try to give some advice.

First of all, what you're about to do is an very large endeavor - mathematics
is a difficult subject, and learning math will take great persistence and
self-motivation, especially if you are self-learning. However, it is also
extremely rewarding - mathematics is a beautiful subject, and learning math
has easily been one of the most enjoyable things I have ever done.

For the next point, if you want to go deep into math, then you will have learn
how to prove things. The heart of math is not at all computation, but ideas,
and to know that ideas are true, we need proofs. All of pure mathematics is
based on rigorous formal reasoning and proofs, and sadly, most high schools
and even universities never touch this part of math. If you have never seen
proofs before, I would first recommend reading the book _How to Prove It: A
Structured Approach_ by Daniel J. Velleman, which goes through basic set
theory, logic, and various proof techniques. Most importantly, it will give
exercises for you to practice. Let me say this now: it is impossible to learn
math without doing exercises. Again, this will take some work, and the
beginning may be a bit slow, but as I said above, it is extremely rewarding -
there are few things so satisfying as finding a beautiful, clean, or elegant
proof. I hope you will enjoy this as much as I have.

Now then, let's dive into the courses and textbooks. I'm going to model this
after what I did in my degree. Many of these topics require earlier ones as
prerequisites, so I'm going to organize them into several layers. Some of the
textbook recommendations may be a bit difficult, since in many of my classes
the professors taught out of their own notes and left textbooks only as
references, but I'll do my best. In your "first year", so to speak, there are
three main things to learn:

\- Single variable calculus, differential and integral. You likely know
calculus already, but again, we are now taking the proof based road! The
canonical text for this topic is _Calculus_ by Michael Spivak. It's what I
used in my first year, and most importantly, comes with a solution manual :)

\- Linear algebra. As others have noted, linear algebra is absolutely crucial
for many other subjects. I personally learned from _Algebra_ by Michael Artin,
but have heard very good things about _Linear Algebra Done Right_ by Sheldon
Axler, so I'd probably start there.

\- Graph Theory and Combinatorics. These are I think are somewhat more
accessible than the others (perhaps at least more intuitive), so I might
actually recommend trying these first. For the basics, try _A Walk Through
Combinatorics_ by Miklos Bona.

By the way, whenever I need to find a textbook on a subject, I just Google
"best (subject) textbook", and try to find the Math Stack Exchange post where
someone has asked this question. (Eg. here's [0] the one for graph theory,
which is where I got the combinatorics book.)

Now, this post is already getting long enough, so I'll post this for now and
follow up the rest in another comment.

[0] [https://math.stackexchange.com/questions/27480/what-are-
good...](https://math.stackexchange.com/questions/27480/what-are-good-books-
to-learn-graph-theory)

------
jbottoms
Your background is in engineering so that informs your choices. Further,
math/CS are not linearly enumerated fields. E.g., there is no field of CS, it
is a domain of many fields. Think tree, not list. That domain would take you a
lifetime to get through all fields, even for an undergraduate level. There are
now millions of people around the world and some are developing new material
at the cutting edge. These are the reasons why degrees are now given in
Systems. The only reason to try to learn all these areas is if you are
designing a systems tool or the tool is you, yourself. Systems is a little
appreciated area but one of increasing importance.

My degree is in Computer Systems Engineering and it it is very enjoyable
work...if you like massive puzzles and are a comprehensive abstract thinker
about applied areas. I need to make this more concrete so here are a few
touchstones. Look at the curriculum for schools that offer degrees in systems.
Weed out the ones that let you create your own curriculum, those are the ones
to choose for a school, but the are too ill-defined to help you set your
future. You can also take a look at the Nobel winning economists. Their work
touches on both theory and application.

One of my favorites is Schelling's work on the neighborhood problem of
integration. The demographic issues of income, race, neighborhood size and
shape made it an intractable problem until he stepped in. He used a system
approach of using image processing linear algebra to predict who moves in and
out of a neighborhood to predict the changes across generations. He solved a
problem that others could not and for that, among other contributions, he
received the Nobel prize. His work is demonstrated in the free software tool
Netlogo. It also lead to the work at NorthwesternU in complex adaptive systems
analysis by Scott E Page.

We see it also used in the demograpgic analysis work in the campagns of Obama,
Hillery and Trump by Cambridge Analytical. The way I think about this
discussion is that "All the easy problems have been solved, now we have to
work on the difficult ones." And that will take the use of new tools that
collaborate with us, and tell us which tools to use.

------
orbifold
Basically you should know that people devote their whole life to one sliver of
these subject areas unless they are absolute geniuses. You would be surprised
how little working knowledge a let's say expert in statistics has of category
theory (you can pick basically any pair of subjects that you listed and could
say that). Your list also seems highly idiosyncratic, so let me propose a
different list based on what a typical undergraduate degree in mathematics
could look like in Germany.

First semester (half a year):

\- Analysis I (rigorous study of real analysis, based on parts of Rubin (a
graduate book in the US but never mind) and German equivalent books)

\- Linear Algebra I (rigorous study of linear transformations, fields, vector
spaces, simple group theory, eigenvalues, eigenvectors)

\- Numeric Analysis 0 (introduction to matrix algorithms (QR, LL, Eigenvalue
decomposition, Householder), simple optimisation)

2nd Semester

\- Analysis II (multivariate real analysis, derivatives, Lebesgue Integration,
maybe differential forms)

\- Linear Algebra II (Simple Ring theory, R-Modulus, some more group theory,
depending on lecturer simple homological algebra, chain complexes and spectral
sequences)

\- Numeric I (ODE solver methods (Runge-Kutta etc.)), alternatively Statistics
(you need a good foundation of Lebesgue theory for that) alternatively
Optimisation I

3rd Semester

\- Analysis III (simple analysis and integration on manifolds, differential
forms, stokes theorem, some simple functional analysis, many different things
possible here, depends on the lecturer)

\- Algebra I (only if you totally want to go this route), more Ring theory
(commutative Algebra), Field Extensions, Galois Theory (one of my favourite
things in mathematics, super beautiful, but virtually unknown outside
mathematics)

Maybe

\- Numeric Analysis II (Simple PDE solving, finite differences, finite
elements, needs Analysis II / Analysis III)

alternatively

\- Complex Analysis (Holomorphic functions, residuals, etc., depending on the
lecturer this can mesh really well with what you already learned in Analysis I
and II)

Beyond this point there is lots to explore, many people diverge in their
interest pretty rapidly from this point. Basically what I'm trying to say is
that you absolutely _need_ to no Linear Algebra and some Algebra rigorously
for practically any field of mathematics (or physics) and the same goes for
Real Analysis. Everything is build on top of that.

I would ignore more foundational stuff like Mathematical Logic / Set Theory
etc. because anything that you need from there will be introduced as you go
along. The same goes for anything applied, unless you have a foundation in
rigorous real analysis and linear algebra, you won't be able to appreciate
anything applied beyond a superficial level (for example all the
decompositions in numerical analysis have nice geometric interpretations). In
terms of pure mathematics the first two semesters are absolutely necessary, if
you don't invest massive amounts of time in these foundations, there is no
chance to understand anything more advance, even though you might fool
yourself that you do. In order to test whether you only think that you
understood something or actually understand something, you absolutely _have_
to do _lots_ of exercises. Ideally there should be someone more experienced
that checks some of them and gives you feedback.

Once you've mastered these fundamentals I suspect you will have been exposed
to more advanced concepts along the way and maybe have a better judgement of
what you find interesting (beyond what is known in popsci circles).

As books I recommend

Linear Algebra Done Right, Sheldon Axler
[http://linear.axler.net](http://linear.axler.net) and Baby Rubin Principles
of Mathematical Analysis, Walter Rubin

they should not be your only references, but are pretty solid jumping points I
think. Whenever you don't understand a step in a proof try to fill details in
yourself or consult other books.

