First semester is over and with it I think it’s time to give a review of books I read over it’s course with few extra books I liked from previous years. This isn’t an amazon review this is just a page that may help a future student start somewhere this my personal opinion about the books and approaches may differ.

Set Theory

I’ll point you the MIRI’s recommended book Naive Set Theory. It contains most set theory you’ll require in subsequent studies and is pretty compact and can also be used as a reference.

Topology

The student’s favorite subject.If you tend to the internet for books you’ll usually get the famous Munkres.But to be honest I don’t recommend it for starting.Topology is difficult in the sense that it’s often described terse and dry there are no particular calculations you need to remember or do but mostly it’s building proofs using defintions and theorems.Which is why I love it, topology will help you build an intuitive imagery of the foundations of Analysis .

Personally I read Bert Mendelson’s Intro to Topology. Bert focuses on building the intuition to attack problems and doesn’t lack rigour, you might find it far more approachable than Munkres. I also used the following set of notes from Prof.Hatcher.

I read Munkres after Bert and the notes and found it more interesting ,the exercices in Munkres are more englobing so you get to practice the usage of all the definitions and theorems you learn along the way.

There’s a book about counter-examples that can be used as a tool box I only skimmed it since I’m not a fan of such ressources.I like to build my own tool box an exercise that helps formulating a better understanding close to creating ex-nihilo .

Differential Calculus

I just went over Spivak’s Calculus On Manifolds this book is amazing in content and language. Falls under what I call perfect textbooks. A perfect textbook is compact in size and use a language that aims for clarity . Personally, I find it more manageable to go trough a 150~200 pages book than 800 pages).

Statistics and Probability

For the subject of probability Papoulis or Sheldon Ross’s book are both recommended.They both cover the subject in depth but I found Sheldon’s great if you’ve never seen P(A) before, Papoulis is also good but Sheldon’s chapters start with intuitive examples that give you a sense of the why and what something Papoulis doesn’t go over. You might like Dartmouth’s intro textbook as well it’s free and covers the same subjects as Sheldon and Papoulis except for Stochastic Processes. For statistics (statistical inference) here again I’ll point to MIRI’s recommended book All Of Statistics it goes from basic probability tools and goes beyond subjects such as p-values, inference ,bootstrap , parametric and non-parametric estimation… It’s a pretty dense book you can read till Chapter 12 like a did,for a more practical view of Part III (Statistical Models) you can use Statistical Models - Theory and Practice by Freedman it’s self contained. I stumbled upon it on a Nassim N.Taleb interview it shows the errors one might make when applying Models to the real world. Emanuel Derman’s Models Behaving Badly is a great read as well it’s not a textbook but more like Antifragile or Fooled by Randomness.

Discrete Mathematics

This wasn’t part of my curriculum but it’s intersting to take a peek at the discrete side of mathematics, Rosen’s is the usual suspect I skimmed it long time ago then found this convincing review about it link . MIT’s own book Mathematics for Computer Science is great I only sampled it,because I found Maurer’s more intersting and colorful. Chapters intersect but I liked Maurer’s more for no particular objective reason. Both are great, If you like watching then MCS’s lecturer Tom Leighton makes the subject fun and intersting,Maurer’s book can serve as supplement given the large nuber of problems he proposes.

Differential Equations

I found this book by accident. And to be honest it was great I didn’t need to look for a second one except maybe use Lamar’s notes as supplement.

Extras

• Functional Analysis by Haase link this was a great book for personal exploration.
• Linear Algebra Done Right by Sheldon Axler
• Vector Calculus (Multivariate Calculus…) by Michael Corral free
• Mathematics It’s Content Methods and Meaning by Aleksandrov et al. WOW this gets a special mention for it’s beauty and the fact that it features around 20 Russian mathematicians work, a true masterpiece that provides references for a more in-depth treatment of each chapter.

Hopefully you’ll find solace in one of these book and manage your difficulties along the way and you’re more than welcome to start a chat with me about books you liked or for remarks about books you didn’t like and found here.