Hi there!I’m Rithvik Reddy, a freshman math major at UT Austin.
I am mainly interested in homotopy theory, combinatorics(of any kind, with special interest in anything reflection group/Coxeter-related),theoretical computer science and algebraic geometry. I still have a lot to learn and the blog is essentially reflective of my developing mathematical knowledge.
*Though the site is titled ‘Weekly Mathematics’, my posts are rather sporadic.
Every so often, I edit a few Wikipedia articles too.
You can contact me through email.
E-mail ID: rithvikreddy at utexas dot edu
Here is a list of math textbooks, notes etc. which I have used and continue to refer. I haven’t really included papers as that would make the lists too long. For learning knot theory, I used just online notes. I like to jump around from one source to another, change mediums randomly(from textbook to seminal papers to random notes on the internet and if I really want to punish myself, nCatLab).
- Topology, James R. Munkres
- Counterexamples in Topology, Lynn Steen, Arthur Seeback ; Impress your friends by pulling counterexamples out of thin air.
- Algebraic Topology, Allen Hatcher
- Lecture Notes in Algebraic Topology, Paul Kirk, James Davis
- An Introduction to Algebraic Topology, Joseph Rotman; Strangely, I find that this book is not that popular. I wonder why?
- A Concise Course in Algebraic Topology, J.P May
- Algebraic Topology I,II, Sanath Devalapurkar ; I have found these notes compiled by Sanath Devalapurkar, an undergraduate at MIT, to be extremely useful and routinely use it alongside Hatcher.
- A User’s Guide to Spectral Sequences, John McCleary
- Spin Geometry, H. Blaine Lawson, Marie-Louise Michelson
- The Theory of Spinors, Elie Cartan; Spinors are a mind-boggling concoction meant to intentionally deceive and fool you. Let the modern mathematicians and theoretical physicists dance like tricksters all they want. Learn it classically from the master of spinors himself.
- Vector Bundles and K-Theory, Allen Hatcher
- Algebraic K-Theory I, Hymann Bass; A collection of papers
- Topology of fiber bundles, Norman Steenrod
- Lectures on Seiberg-Witten Invariants, John Douglas Moore
- K-Theory, Max Karoubi
- George Whitehead, Elements of Homotopy Theory
- Contemporary Abstract Algebra, Joseph Gallian ; One of the first ‘rigorous’ math textbooks I read alongside Munkres. It is truly an amazing textbook
- Abstract Algebra, Dummit and Foote ; A classic which I regularly use. I even recommend this book to learn commutative algebra and representation theory.
- Algebra, Michael Artin ; I used this for my Algebra I class for the first time. I remember the exercises being fun and the chapters capturing the essential general detail perfectly.
- Algebra, Serge Lang ; I always enjoy Lang
- Linear Algebra, Serge Lang ; I learned Linear Algebra quite a while ago from this book. Never read a page of Axler’s book to date.
- Algebra Chapter 0, Aluffi ; Honestly, I just used this for Galois theory, category theory and homological algebra, not the actual ‘algebra’. I like the amount of detail put into all the explanation
- Introduction to Homological Algebra, Charles Weibel
- Commutative Algebra with a View Towards Algebraic Geometry, David Eisenbud ; It is difficult to approach this book if you try to read linearly. If your aim is AG, I recommend figuring out the ‘broad topics’ from Atiyah,MacDonald or some classic textbook and then reading accordingly. Besides that, it has lots of interesting things.
- Introduction to Commutative Algebra, Atiyah and MacDonald
- Tohoku paper, Grothendieck ; I learned derived nonsense in abelian categories from the original paper and found it be a nice read.
- Stacksproject ; I find this to be a great reference for Commutative Algebra and Homological Algebra. Again, you should know what you are looking for if you plan to use it.
- Representation Theory,A First Course, William Fulton, Joe Harris ; The first two chapters are a little rushed but overall, I think it’s a great book.
- Lie Groups, Lie Algebras and Representations, An Elementary Introduction Brian C. Hall ; I have nothing but praise for this extremely engaging book
- Reflection Groups and Coxeter Groups, James E. Humphreys; I came across this book during my senior year holidays. Really liked the section on Kazdhan-Luztig polynomials and Hecke algebras.
- Representation Theory, A Combinatorial Viewpoint, Amritanshu Prasad ; An excellent book to learn some slick combinatorics. It is also very well organized and accessible.
- Mathematical Analysis, Tom Apostol ; The exposition is clear and detailed. Very suitable for a beginner. It was the first book from which I learned Analysis
- Real Analysis, Elias Stein, Shakarchi ; My favorite so far
- Principles of Mathematical Analysis, Rudin
- Functional Analysis, Elias Stein, Shakarchi
- Multivariate Calculus and Linear Algebra, Tom Apostol ; The first ‘college-level’ I read. Brings back memories of breaking my head trying to understand proofs.
- Calculus of Several Variables, Serge Lang
- Complex Analysis, Serge Lang
- Complex Analysis, Lars V. Ahlfors ; This is very weirdly organized, the complete opposite of Lang yet it is more well regarded. Honestly, I feel like though Lang covers more material, it would be a much better introductory book.
- Fundamentals of Differential Geometry, Serge Lang
- The Rising Sea, Ravi Vakil ; This is GOD, period.
- Algebraic Geometry, Robin Hartshorne ; ‘Do the exercises in the first chapter and switch to Vakil’ is what someone I trust told me. I think I agree.
- Undergraduate Algebraic Geometry, Miles Reid ; I seriously recommend anyone even mildly interested in algebraic geometry to read this gem. It highlights the intuition and motivation behind many seemingly obscure.
- Basic Algebraic Geometry, Shafaravich ; A good place to start after Reid.
- Introduction to Algebraic Geometry, Steven Dale Cutkosky ; I just randomly found this at the UT Austin library. It is not too bad, I think.
- Algebraic Number Theory, Serge Lang
- Lectures on Algebraic Number Theory, Hecke
- Elementary Number Theory, Kenneth Rosen
- Graph Theory, Reinhard Diestel
- Irrational Numbers, Ivan Niven ; This is the first ‘math’ book I read and probably the reason I am studying mathematics right now. I was introduced to lot of seemingly non-intuitive concepts like the cardinality of rationals, Lebesgue measure zero, irrationality measure, transcendentality and proofs of irrationality of hyperbolic, exponential, trigonometric functions etc. I wrote about this on one of my other posts.
- Higher Topos Theory, Jacob Lurie
I do not claim to have read any of these books cover to cover. I just scour through all sources at once looking for the best explanations and examples. If Hatcher is too terse, presents a shoddy proof or if I am in the mood for more category-heavy stuff, I switch to May, Rotman or lecture notes. In fact, it is quite unfortunate that I decided to leave out online notes. I would say that half of my learning is in learning from lecture notes, especially in topology and representation theory. You can also notice that have a weird obsession with Serge Lang. Forgive me, Lang has been there for me from the every beginning step learning calculus, linear algebra and analysis. His precision and detail are admirable. I am not really a fan of number theory so I suppose that is the end of my fascination for him.
I have also left out Wikipedia, ncatlab and Stackexchange from where I clarify many doubts and look for more information.
I think that one of the most important things about learning mathematics is searching for interesting examples and the extent to which these examples use the hypotheses of the result. Often, you’ll find that certain theorems are, on the surface, very general but in application, limited to just a few scenarios. In this regard, problem solving is important in gauging the ‘utility’ or power of a theorem. It is with practice, insight and familiarity that one can hopefully aim to ‘downgrade’ the depth of a theorem to a new ‘tool’.