Ask HN: Mathematicians, what textbooks are best for learning these math topics?
I&#x27;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:<p>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 &quot;just okay&quot; but who can e.g. review and critique your work is better than a textbook.<p>Learning math is very unlike learning programming. It&#x27;s a counterintuitive idea, but the information density of math textbooks (whether they&#x27;re well or poorly written) is generally so high that you can&#x27;t absorb the material unless you read only a few pages per day. Not only that, but it&#x27;s usually not the case that a single textbook has the ideal level of exposition for your needs - for example, you don&#x27;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&#x2F;computational level. Which suits your needs more? Have you studied it at all?<p>If you&#x27;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.<p>If you go at a rate which will actually allow you to absorb the material, doing this &quot;properly&quot; will take you years. With dedication and not much talent I&#x27;d expect it to take as long as an undergraduate degree. With dedication and a <i>lot</i> of talent I could see this being accomplished in two, maybe three years. Once again, I strongly, strongly suggest finding a mentor or instructor.<p>In any case, here is a list of the textbooks <i>most</i> mathematicians will consider to be very good:<p>1. Calculus<p><i>Calculus</i>, by Spivak<p>This gives you a rigorous treatment of calculus, which hopefully you have some familiarity with. After this you can move on to real analysis.<p>2. Real Analysis<p><i>Principles of Mathematical Analysis</i>, by Rudin<p>You might be ready for this after Spivak&#x27;s <i>Calculus</i>, but it can be rough. If you can&#x27;t reproduce a proof of irrationality after reading through the first few pages, work through Tao&#x27;s <i>Analysis I</i> first.<p>3. Topology<p><i>Topology</i>, by Munkres is the absolute gold standard. You should be comfortable with calculus (and hopefully analysis) before tackling this.<p>4. Linear Algebra<p><i>Linear Algebra Done Right</i>, by Axler<p>This is a thorough introduction to the subject at a theoretical level, with a focus on finite-dimensional vector spaces over fields <i>R</i> and <i>C</i>.<p>You should also work through either <i>Linear Algebra</i> by Friedberg, Insel, Spence or <i>Linear Algebra</i> by Hoffman &amp; Kunze for the treatment of more advanced&#x2F;specialized material and, in particular, determinants (which are notably de-emphasized by Axler).<p>Noam Elkies uses Axler for Harvard&#x27;s Math 55 and has written up notes and remarks for his students; be sure to read them: <a href="http:&#x2F;&#x2F;www.math.harvard.edu&#x2F;~elkies&#x2F;M55a.16&#x2F;index.html" rel="nofollow">http:&#x2F;&#x2F;www.math.harvard.edu&#x2F;~elkies&#x2F;M55a.16&#x2F;index.html</a><p>5. Abstract Algebra (Groups, Rings, etc)<p><i>Abstract Algebra</i> by Dummit &amp; Foote is the usual reference text for a first course. It&#x27;s pretty good. If it&#x27;s too advanced for you, try Pinter&#x27;s <i>A Book of Abstract Algebra</i>. For a very challenging (but comprehensive) approach to the subject, try Lange&#x27;s <i>Algebra</i>.<p>6. Category Theory<p>Once you have abstract algebra under your belt, a good introduction to category theory is given by Aluffi&#x27;s <i>Algebra: Chapter 0</i>. I would suggest not trying to dive into this prior to at least encountering fields, groups and rings because it&#x27;s good to have both the traditional and modern (read: categorical) contexts.<p>Also try <i>Category Theory in Context</i>, by Riehl.<p>7. Complex Analysis<p><i>Complex Analysis</i>, by Ahlfors. This is an excellent and concise text. You can theoretically approach this before real analysis, but I wouldn&#x27;t recommend that. Also try <i>Complex Variables</i>, by Churchill &amp; Brown.<p>8. Differential Geometry<p><i>Calculus on Manifolds</i> by Spivak. You will want to have a thorough understanding of analysis and linear algebra before approaching this material.<p>9. Measure Theory<p>This is very advanced material in an analysis sequence; don&#x27;t jump to this unless you&#x27;ve thoroughly worked through analysis first.<p>I would recommend Stein &amp; Shakarchi&#x27;s <i>Real Analysis: Measure Theory, Integration and Hilbert Spaces</i>.<p>10. Probability Theory<p>A really rigorous treatment of probability is measure theoretic, but even if you haven&#x27;t worked with measures before you&#x27;ll need (real) analysis and linear algebra. Tackle those first.<p>Feller&#x27;s <i>Introduction to Probability Theory</i> is usually a good first course. If you don&#x27;t like that, try Ross. For truly advanced probability theory, work through Shiryaev or Kallenberg.<p>The other things you&#x27;ve asked for are a little under-specified or outside my wheelhouse (in particular, I don&#x27;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.<p>1. Numerical Analysis<p><i>Numerical Linear Algebra</i>, by Trefethen &amp; Bau. This is the best all-around introduction. Once you&#x27;ve worked through this, try moving on to <i>Matrix Computations</i> by Golub &amp; van Loan. The latter is much more of a reference text.<p>2. Cryptography<p>You haven&#x27;t specified what you&#x27;re looking for here, but given the mathematical bent of your question I&#x27;d recommend Goldreich&#x27;s <i>Foundations of Cryptography</i> (two volumes). Be forewarned: cryptography is a subfield of complexity theory. You should have a strong understanding of complexity theory before embarking on Goldreich&#x27;s <i>Foundations.</i><p>If you really want to challenge yourself theoretically, work through Galbraith&#x27;s <i>Mathematics of Public Key Cryptography</i>. The most up to date version is available for free: <a href="https:&#x2F;&#x2F;www.math.auckland.ac.nz&#x2F;~sgal018&#x2F;crypto-book&#x2F;crypto-book.html" rel="nofollow">https:&#x2F;&#x2F;www.math.auckland.ac.nz&#x2F;~sgal018&#x2F;crypto-book&#x2F;crypto-...</a><p>On the other hand, if you&#x27;re looking for a more implementation-focused text on cryptography, try Menezes&#x27; <i>Handbook of Applied Cryptography.</i><p>3. Optimization<p>This is extremely broad. There&#x27;s linear programming, mixed integer programming, nonlinear optimization, stochastic optimization...I can&#x27;t recommend textbooks targeted at everything here.<p>For a good start to the subject of optimization and constraints in general, work through Boyd &amp; Vanderberghe&#x27;s <i>Convex Optimization.</i> There are additional exercises available from the authors here: <a href="https:&#x2F;&#x2F;web.stanford.edu&#x2F;~boyd&#x2F;cvxbook&#x2F;bv_cvxbook_extra_exercises.pdf" rel="nofollow">https:&#x2F;&#x2F;web.stanford.edu&#x2F;~boyd&#x2F;cvxbook&#x2F;bv_cvxbook_extra_ex...</a>