Reddit mentions: The best algebraic geometry books

I hope others will chime in here, but I'll answer as well as I can.

Euclidean and Non-Euclidean Geometry

Euclidean and non-Euclidean geometries are interesting and important for various reasons, so I certainly wouldn't say it's a bad idea to study them in depth.

If you want to study these subjects first because you find them interesting and you have plenty of years to spend, then go for it! However, it's not necessary (more on this below).

Multivariable Calculus and Linear Algebra

Before attempting even an elementary treatment of differential geometry, you'll want to have a working knowledge of calculus (single and multivariable) and linear algebra.

Elementary Differential Geometry

You could potentially skip the elementary treatments of differential geometry, but these might be useful for tackling more advanced treatments. Studying elementary differential geometry first is perhaps similar to taking a calculus class (with an emphasis on computation and hopefully on intuition) before taking a class in real analysis (with an emphasis on abstraction and rigorous proofs).

If you do want to work through an elementary treatment, then you have options. One well reviewed book, and the one I learned from as an undergraduate, is Elementary Differential Geometry by Barrett O'Neill.

Note that O'Neill lists calculus and linear algebra as prerequisites, but not Euclidean and Non-Euclidean geometry. Experience with Euclidean geometry is definitely relevant, but if you understand calculus and linear algebra, then you already know enough geometry to get started.

Abstract Algebra, Real Analysis, and Topology

The next step would probably be to study a semester's worth of abstract algebra, a year's worth of real analysis, and optionally, a semester's worth of point-set topology. These are the prerequisites for the introduction to manifolds listed below.

Manifolds

An Introduction to Manifolds by Loring W. Tu will give you the prerequisites to take on graduate-level differential geometry.

Note: the point-set topology is optional, since Tu doesn't assume it; he expects readers to learn it from his appendix, but a course in topology certainly wouldn't hurt.

Differential Geometry

After working through the book by Tu listed above, you'd be ready to tackle Differential Geometry: Connections, Curvature, and Characteristic Classes, also by Loring W. Tu. There may be more you want to learn, but after this second book by Tu, it should be easier to start picking up other books as needed.

Caveat

I myself have a lot left to learn. In case you want to ask me about other subjects, I've studied all the prerequisites (multivariable calculus, linear algebra, abstract algebra, real analysis, and point-set topology) and I've tutored most of that material. I've completed an elementary differential geometry course using O'Neill, another course using Calculus on Manifolds by Spivak, and I've studied some more advanced differential geometry and related topics. However, I haven't worked through Tu's books yet (not much). The plan I've outlined is basically the plan I've set for myself. I hope it helps you too!

Sorry for a late reply. I've never read Jones' book. The selection of topics looks great, but the Amazon reviews are worrisome (lack or rigor, not self-contained).

From the preface of Rotman

>By the end of the nineteenth century, there were two main streams of group theory: topological groups (especially Lie groups) and finite groups... It is customary, nowadays, to approach our subject by two paths: "pure" group theory (for want of a better name) and representation theory.

From my experience, on the one hand, physicists don't need to know a lot of topics in "pure" group theory like the proof of structure theorem of finitely generated abelian groups or Sylow theorems. On the other hand, sometimes it's not clear when physicists talk about a group if they're actually talking about its Lie algebra (if it's a Lie group) or their representations. You can come out of a class on Lie groups without having learned much about Lie groups at all (like me)!

So if I have to recommend, the middle way would be to focus on representation theory, starting on finite groups first. Mikhail Khovanov listed a bunch of resources for his course http://www.math.columbia.edu/~khovanov/finite/ including Serre and Etingof. The first third of Serre is good. (It was written for chemists.) And my abstract algebra professor recommended Etingof. (It was from a course for bright high school students and another course for undergrads.)

A lot of results for finite groups carry over to compact infinite groups (as Serre explains). For Lie groups, it's more convenient because they have Lie algebras. Physicists also like the Lie algebra approach because they can bypass differential geometry. (Lee's Smooth Manifolds book also has a few chapters on Lie groups.) Two short books that I like are Cahn and Carter et al.

This is the level that I'm at. Beyond this you will have to look around and see which path you want to take. Hall and Fulton & Harris teach through examples but don't offer a uniform view of the subjects. I begin to like Kirillov Jr. and Vinberg & Onishchik. Procesi is inspiring to me because invariant theory like Schur-Weyl duality are actually useful in quantum information.

Bump is a great book, but it's more of "An Introduction to Langlands Program through Automorphic Forms", than it is an introduction to modular and automorphic forms themselves. If this is to be an introductory course about modular forms, then I think Diamond is a pretty good intro. The first five chapters cover pretty much all the basics you need to talk about modular forms.

Hecke operators really are the meat of the topic. They are what give modular forms nice arithmetic properties, including things like Euler products of L-functions. They're nice operators on nice inner product spaces and so they help pick out orthogonal subspaces, and its the basis vectors of these, called eigenforms, that link to other things like representations of Galois groups. This insight can help setup very useful interpretations of reciprocity, which turn out to be at the heart of things like Class Field Theory and Langlands Program. For instance, for every elliptic curve, there is an eigenform of the Hecke operators so that the elliptic curve L-function is the same as the L-function for this eigenform, which is the hard part in proving Fermat's Last Theorem.

I do not know of any category theory texts I'd say would be accessible to undergrads. But if you are really interested, you might try the first chapter of these very excellent notes due to Ravi Vakil. Someone is probably going to jump down my throat for suggesting these to you (they are considered pretty challenging), but they will give you a good idea of the flavor of category theory. If you are really, really interested in category theory and you have a professor/ grad student you can talk to about these notes, you could really learn a lot here. Another book I hear about a lot is this one but I have never used it myself.

Ok, but really, starting with Ravi's notes might be a little too much. I actually suggest this youtube channel as your first foray into trying to learn category theory. The lectures are fairly accessible and you can always pause them, go back, etc.

If you are really, truly, interested in learning more about category theory, my suggestion is to go to your professor and ask about it. You do not have to be enrolled in a class about category theory to learn it, and this could be a great springboard into some sort of honors project for you. Good luck!

Edit: Ozob beat me on the MacLane! It's also worth noting that, depending on your institution, you may be able to view Springer books for free/ buy them for cheap. Ask your department.

Pick up a copy of Algebraic Geometry: A Problem Solving Approach and work through the first chapter.

It shouldn't require much more than high school algebra, with just a smidgen of understanding of partial derivatives.

The first chapter defines algebraic sets of a polynomial, which is a subset of the plane defined by a polynomial: {(x, y) | P(x, y) = 0}.

The degree of the polynomial determines the degree of the curve. Degree 1 polynomials give straight lines, as you might expect. Degree 2 polynomials give the conic sections. You might remember conic sections from your high school algebra II class, but chances are it was mostly an exercise in memorizing equations.

It goes on to classify the conics up to affine change of coordinates. In R^2, there are ellipses (including the circle), hyperbolas, parabolas, and the degenerate conics, a double-line and a pair of crossing lines.

The chapters are fairly short and filled with super easy exercises that get you thinking about the material you're reading.

The chapter builds up some of the basic notions studied in algebraic geometry. While working over R^2 is great, it is harder to study because not every polynomial will have roots. So you upgrade to C^2 instead. In C^2, though, ellipses and hyperbolas become equivalent, thanks to allowing complex numbers in our affine change of coordinates.

Lastly, it builds up to projective geometry in CP^2. Even in C^2, there are cases where two intersecting lines may fail to meet if they are parallel to each other. By moving to CP^2, we force all lines to eventually greet each other (at some point of infinity if at no finite point).

This final upgrade is a bit technical, but it is a key ingredient to world-famous Bezout's Theorem, studied in chapter 3. But one immediately awesome result is that all nondegenerate conics become equivalent: ellipses, hyperbolas, and parabolas are just three ways of looking at the same geometrical object.

Algebraic geometry is an amazing field whose roots go back to at least Desargues in the 17th century. It has intimate ties with complex analysis (Chow's Theorem says that curves in the projective plane are actually compact Riemann surfaces) and number theory (where we work over the rationals, rather than the reals or complex numbers). In the 1930s, the field was put on a rigorous algebraic basis by Hilbert and Noether (this is essentially what Commutative Algebra is). And in the 1960s, Alexandre Grothendieck went totally ham and rephrased the entire subject in terms of categories and schemes.

I have a few books I read at that age that were great. Most of them are quite difficult, and I certainly couldn't read them all to the end but they are mostly written for a non-professional. I'll talk a little more on this for each in turn. I also read these before my university interview, and they were a great help to be able to talk about the subject outside the scope of my education thus far and show my enthusiasm for Maths.

Fearless Symmetry - Ash and Gross. This is generally about Galois theory and Algebraic Number Theory, but it works up from the ground expecting near nothing from the reader. It explains groups, fields, equations and varieties, quadratic reciprocity, Galois theory and more.

Euler's Gem - Richeson This covers some basic topology and geometry. The titular "Gem" is V-E+F = 2 for the platonic solids, but goes on to explain the Euler characteristic and some other interesting topological ideas.

Elliptic Tales - Ash and Gross. This is about eliptic curves, and Algebraic number theory. It also expects a similar level of knowlege, so builds up everything it needs to explain the content, which does get to a very high level. It covers topics like projective geometry, algebraic curves, and gets on to explaining the Birch and Swinnerton-Dyer conjecture.

Abel's proof - Presic. Another about Galois theory, but more focusing on the life and work of Abel, a contemporary of Galois.

Gamma - Havil. About a lesser known constant, the limit of n to infinity of the harmonic series up to n minus the logarithm of n. Crops up in a lot of places.

The Irrationals - Havil. This takes a conversational style in an overview of the irrational numbers both abstractly and in a historical context.

An Imaginary Tale: The Story of i - Nahin. Another conversational styled book but this time about the square root of -1. It explains quite well their construction, and how they are as "real" as the real numbers.

Some of these are difficult, and when I was reading them at 17 I don't think I finished any of them. But I did learn a lot, and it definitely influenced my choice of courses during my degree. (Just today, I was in a two lectures on Algebraic Number Theory and one on Algebraic Curves, and last term I did a lecture course on Galois Theory, and another on Topology and Groups!)

For compsci you need to study tons and tons and tons of discrete math. That means you don't need much of analysis business(too continuous). Instead you want to study combinatorics, graph theory, number theory, abstract algebra and the like.

Intro to math language(several of several million existing books on the topic). You want to study several books because what's overlooked by one author will be covered by another:

Discrete Mathematics with Applications by Susanna Epp

Mathematical Proofs: A Transition to Advanced Mathematics by Gary Chartrand, Albert D. Polimeni, Ping Zhang

Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers

Numbers and Proofs by Allenby

Mathematics: A Discrete Introduction by Edward Scheinerman

How to Prove It: A Structured Approach by Daniel Velleman

Theorems, Corollaries, Lemmas, and Methods of Proof by Richard Rossi

Some special topics(elementary treatment):

Rings, Fields and Groups: An Introduction to Abstract Algebra by R. B. J. T. Allenby

A Friendly Introduction to Number Theory Joseph Silverman

Elements of Number Theory by John Stillwell

A Primer in Combinatorics by Kheyfits

Counting by Khee Meng Koh

Combinatorics: A Guided Tour by David Mazur


Just a nice bunch of related books great to have read:

generatingfunctionology by Herbert Wilf

The Concrete Tetrahedron: Symbolic Sums, Recurrence Equations, Generating Functions, Asymptotic Estimates by by Manuel Kauers, Peter Paule

A = B by Marko Petkovsek, Herbert S Wilf, Doron Zeilberger

If you wanna do graphics stuff, you wanna do some applied Linear Algebra:

Linear Algebra by Allenby

Linear Algebra Through Geometry by Thomas Banchoff, John Wermer

Linear Algebra by Richard Bronson, Gabriel B. Costa, John T. Saccoman

Best of Luck.

What I meant by study algebra is not so much shed light on classical algebraic objects but rather introduce new algebraic objects which are more robust than conventional algebra and where this extra layer of robustness is truly homotopy-theoretic.

Neither algebraic or geometric topology are that analytic though both certainly have connections to analysis. As far as harder, the algebraic side may involve more machinery (especially for something like algebraic topology a la Lurie), but there is no shortage of interesting and challenging problems in either.

If you want something that is a mix of algebra and topology, then do something in between algebraic and geometric topology. There is not really a clear dividing line between the other; rather is is more of a spectrum. You seem like you would be happiest doing something roughly in the middle of the spectrum.

Here is a book you may find interesting that contains an even mix of algebra and topology:

http://www.amazon.com/Novikov-Conjecture-Geometry-Oberwolfach-Seminars/dp/3764371412/ref=sr_1_1?ie=UTF8&qid=1449770845&sr=8-1&keywords=novikov+conjecture+geometry+and+algebra

It gets into surgery and L-theory, and towards the end it gets into assembly maps. The assembly maps are algebraic (and involve spectra) and can be used to compute K-theory and L-theory which are also algebraic. However, the K and L groups contain obstructions related to geometric questions, and the construction of assembly maps is related to studying classifying spaces for certain families of groups - a topic which very much carries the flavor of geometric group theory.

At some point these "Pop" reading books get wholly unsatisfying and you need textbooks, but I think that's a story for a different semester. Theres a good set of books written by Avner Ash and Robert Gross (Boston College) that anyone with calculus 1 can easily get into:
Elliptic Tales:
https://www.amazon.com/Elliptic-Tales-Curves-Counting-Number/dp/0691151199
Fearless Symmetry:
https://www.amazon.com/Fearless-Symmetry-Exposing-Patterns-Numbers/dp/0691138710/ref=pd_sbs_14_t_1?_encoding=UTF8&psc=1&refRID=JG1NQ2F2XS0WJJ5PBKVV

Well worth the read, entertaining, and great introductions to their respective subjects!

>I was reading Frenkel's book "Love and Math" about the Langlands program.

I'm not sure what level you're coming at it from, but for an intro to Langlands I'd recommend Gelbart which is where I first read about it and still serves a good intro, in more recent times there is Bump et al's An Introduction to the Langlands Program also excellent. Frenkel's book may well whet your appetite but it's pretty light on details.

For me, a "good read" in mathematics should be 1) clear, 2) interestingly written, and 3) unique. I dislike recommending books that have, essentially, the same topics in pretty much the same order as 4-5 other books.

I guess I also just disagree with a lot of people about the
"best" way to learn topology. In my opinion, knowing all the point-set stuff isn't really that important when you're just starting out. Having said that, if you want to read one good book on topology, I'd recommend taking a look at Kinsey's excellent text Topology of Surfaces.

If you're interested in a sequence of books...keep reading.

If you are confident with calculus (I'm assuming through multivariable or vector calculus) and linear algebra, then I'd suggest picking up a copy of Edwards' Advanced Calculus: A Differential Forms Approach. Read that at about the same time as Spivak's Calculus on Manifolds. Next up is Milnor Topology from a Differentiable Viewpoint, Kinsey's book, and then Fulton's Algebraic Topology. At this point, you might have to supplement with some point-set topology nonsense, but there are decent Dover books that you can reference for that. You also might be needing some more algebra, maybe pick up a copy of Axler's already-mentioned-and-excellent Linear Algebra Done Right and, maybe, one of those big, dumb algebra books like Dummit and Foote.

Finally, the books I really want to recommend. Spivak's A Comprehensive Introduction to Differential Geometry, Guillemin and Pollack Differential Topology (which is a fucking steal at 30 bucks...the last printing cost at least $80) and Bott & Tu Differential Forms in Algebraic Topology. I like to think of Bott & Tu as "calculus for grown-ups". You will have to supplement these books with others of the cookie-cutter variety in order to really understand them. Oh, and it's going to take years to read and fully understand them, as well :) My advisor once claimed that she learned something new every time she re-read Bott & Tu...and I'm starting to agree with her. It's a deep book. But when you're done reading these three books, you'll have a real education in topology.

Mathematics papers aren't really a good place to get an introduction to branches of mathematics as they tend to cater to the advanced reader. The most accessible you might find would be Mathematics Magazine or similar.

You would fare far better with an undergraduate level textbook. Springer publish a lot of pretty good undergraduate level texts so you might find something like this or this helpful (although I have not personally read either of those specifically so I cannot speak to their quality, but I find Springer books are usually good).

You might get better advice asking in the main sub (/r/math) as people like to give reading there.

EDIT: or maybe something like this would be more suited to what you're looking for?

Depends on your background. Mac Lane is the standard text and he is a phenomenal author in general, but it builds off knowledge of concepts such as modules, tensor products and homotopy (I still don't have a sufficient background in AT to be honest though). For a more modest background, I would recommend the book "Sets for Mathematics" by Lawvere and Rosebrugh. The book is entirely on category theory, the title is because there is a focus on the category of sets. The first chapter or so is deceptively simple, it gets very difficult as it goes on, but still doesn't require much specific background.

I'll also note that I first got into the subject through a whim purchase in a local Borders of a cheap dover book Topoi by Robert Goldblatt when I was very into mathematical logic. It's 500 pages and requires pretty much no background (I'd know what a topological space is, but I can't think of anything else). It gets very challenging though, and I never got more than 250 pages in before getting overwhelmed, but the first hundred pages really sparked my interest in category theory. Functors (and especially adjoint functors) are postponed much later than you will see in many other sources though. You can find a link to an online version free from the author's webpage too.

I unfortunately don't know of a good generally accessible source on this. It's really one specific (and particularly nice looking) example of a more general theory. As such it's the sort of thing you usually learn as an example while you're learning about the more arithmetic side of modular forms, and I don't know of any good self contained sources for it.

What's your current background in math? If you haven't already taken algebraic number theory, you'd definitely want to start there (although you'd need abstract algebra, and especially Galois theory, as a prerequisite). The most important concept to pick up there would be the notion of a "Frobenius element".

If you're already familiar with algebraic number theory, then (as other people have said here), primes of the form x^(2)+ny^(2) by Cox is a good place to start, although if I remember correctly they don't actually do the n = 23 example there. That combined with a book on modular forms, such as Diamond and Shurman should give you a pretty good understanding of this.

To add to what others have said, it will be helpful to be comfortable with polynomial rings. Most algebra books have a chapter devoted to polynomial rings, so it shouldn't be hard to find a source for this.

I would recommend starting with Ravi Vakil's online notes, or at least reading them concurrently to Hartshorne once you reach chapter 2. In my opinion Vakil gives a lot more motivation and insight than Hartshorne. They're available online here:

http://math.stanford.edu/~vakil/216blog/FOAGdec2915public.pdf

Also, since you're interested in number theory, take a look at Qing Liu's book which is the standard recommendation for arithmetic geometry.

http://www.amazon.com/Algebraic-Geometry-Arithmetic-Graduate-Mathematics/dp/0199202494

Interesting. Did you ever think about self-publishing? Lulu.com or amazon self-publish or something?

My PhD advisor asked me and his other students for some advice on how he should try to get his book priced so that graduate students would be able to buy it. We all suggested between like $25 and $35, largely because we all have read this excellent book which is $36 to buy and free to download. I wonder how Hatcher negotiated that deal, although B&W probably drives the cost down a lot.

If you feel like you have the time, I could recommend http://www.amazon.com/Algebraic-Geometry-Problem-Approach-Mathematical/dp/0821893963 which is a very gentle introduction to the subject using classical curves. Only in the last chapter does it introduce sheaves and cohomology. I suspect something like this might be helpful to place everything in a concrete context, and also build up motivation for all the modern machinery that you'll find in Hartshorne.

As far as algebraic topology goes, while Hatcher is available for free (and legally at that), I've found him quite difficult to use for independent study. He tries a bit too hard, I think, to illustrate his geometric intuition, and ends up with extremely verbose, confusing explanations. His proofs are difficult to follow if you're new to the subject as well (it took me hours to understand his proof that [; \pi_1(S^1) = \mathbb{Z} ;]).

An alternative text that I have greatly enjoyed reading is A Basic Course in Algebraic Topology by Massey. His text is much more appropriate for a student's first introduction to the subject because he explains every relevant detail, rather than assuming some indefinite body of prerequisite knowledge. The text is also split up into many short chapters, rather than the four long chapters of Hatcher, and each chapter includes a generous selection of exercises. It's very readable and very rewarding to work through.

If we do decide to use Massey, make sure to get the text labeled v. 127, as opposed to this one labeled v.56. The latter only includes the material on fundamental groups and covering spaces, without any mention of homology or cohomology.

Loring Tu has a new book which discusses the basics of principal bundles and their characteristic classes towards the end. I have already read a significant chunk of it to get the forms perspective of characteristic classes on regular ol' vector bundles, and I can say that the book is excellent. To get a more detailed or advanced perspective though, Kobayashi & Nomizu seems to be the best place.

I really enjoy Reid's book, but for a first introduction I would look into Cox, Little and O'Shea's book Ideals, Varieties, and Algorithms. It has very good extended examples, and ties in computation in a very interesting way.

I think I see what you mean by that, and if so, a change of basis could be considered a special case of what you're talking about, but change-of-basis is a linear transformation, so you'll only get specific cases of what you're looking for by change-of-basis.

(Actually, I just went back and looked at the video you linked, and he's dealing with explicitly invertible linear transformations, so yes he's basically doing change-of-basis.)

I brought up complex analysis because there are lots of ideas that are similar to what you're talking about in that subject. For instance, you might show that complex inversion transforms circles into circles (in this case, lines are considered circles of infinite radius). It might not be exactly what you're looking for, but it is about transformations and what they do to things like circles and lines.

Another option, if you want something a little more general, is you could look at a book like Garrity's Algebraic Geometry Book. Algebraic geometry is, in its full generality, a very difficult subject, but his book gives a very gentle introduction to the subject, and it starts with a description of affine transformations, which are sort of a generalization of these linear transformations. Affine transformations are linear transformations composed with translations. I bring up this subject because it fits the idea of "an equation defining a subset of the domain". Algebraic geometry is, at least in its simpler form, about studying curves and surfaces (and hypersurfaces) defined by polynomial equations.

As an academic, I don't think it is, at least in my field. The authors of textbooks make 0 money on their books. They do it for the professional exposure. There is almost no need for publishers, and yet it's the publishers who are making a lot of money basically acting as a middle-man for digital content.

You don't need expensive middle men for digital content.

You can buy the best Algebraic Topology book for $40, or you can download it for free on the author's webpage. This is not uncommon. Instead of amazon selling that book, you can imagine Hatcher (the author) just selling the rights to some reasonable online distribution company, and Hatcher would probably see more money that way.

Related: the Elsevier Boycott

Yeah, I've just never been shown a problem where this stuff gives deep insight, and until I see one and understand it these are just gonna be arbitrary definitions that slide right out of my brain when I'm done reading them. I'll definitely give the book a look - is it motivated with examples?

The only book I have on category theory is Conceptual Mathematics: A First Introduction to Categories, and I must say, I'm not a fan of it - too intuitive, not detailed enough, not well organized, not formal enough - should have gone for MacLane instead.

There's a lot of category theory, but this only uses the basics. Galois theory is deeper than anything used so far. Category, functor, natural transformation (co)limit, and maybe adjoint should be plenty (it looks like gibbon's tries to explain everything about adjoints that he uses). That's all in the first four or five (short) chapters of MacLane.

Why are you doing the exercises? Is it for a class? Are you self studying?

I've done all the exercises in Hungerford. When we had a section assigned for homework, I would just do everything in that section. Maybe I was doing 1 or 2 sections a week. I can't really remember. After the class ended, I finished the book in the name of completionism (and because I enjoyed the material). It was a really fun project because my professor had done the same thing years before. She still had her hand-made answer key, and we'd compare solutions during office hours.

I've also done all the exercises in an old version of Vakil's notes and all the exercises in the first few chapters of Hartshorne. However, in both of those books the exercises contain key material so you have to do them.

If I could do it again, I'd force myself to type up everything. I did all this before I learned TeX. Now I have pages and pages of exercises wasting away in a box somewhere...

Perhaps we can get the special flair users in /r/math to setup some of this (the ones with the red background in their flair)?

I know nothing about any of these topics but we could use course notes from a school's Open Courseware.

Here are the relevant ones I've found. If a cell says "none" that just means I've left a placeholder for if people find something I can put in that spot. The ones with all nones means I either wasn't sure what to look for, or if what I found was the right thing (Lie Theory = Lie Groups? for example)



Subject | Source1 | Source2 | Source3| Source4
---|---|----|----|----
Algebraic Topology | MIT Seems to have all relevant readings as PDFs | Introductory Algebraic Topology I don not know the source for this one| Algebraic Topology by Hatcher is free | A Basic Course in Algebraic Topology by Massey - Not free
Algebraic Geometry | MIT Fall 2003 Has lecture notes| MIT Spring 2009 Also has lecture notes | Vakil's course notes| Eyal Goren Syllabus and course notes
Functional Analysis | MIT Lecture notes and assignments with solutions | Nottingham 2010 | Nottingham 2008 These ones not only have lecture notes, but audio of the lecture. | none
Lie Theory | MIT - Intro to Lie Groups | MIT - Topics in Lie Theory: Tensor Categories | none | none
General Relativity | Sean Carroll's Lecture Notes | Stanford video lectures on general relativity, Leonard Susskind | Lecture notes from Nobel Laureate Gerard Hooft on GR | Semi-Riemannian geometry with Applications to Relativity - Not free
Dynamical Systems | Very applied (Strogatz style) course notes for dynamical systems | More theoretical (Perko style) course notes for dynamical systems by the same author | none | none
Numerical Analysis | MIT Spring 2012 | MIT Spring 2004 | none | none

This is obviously not an exhaustive list. I thought Stanford and their own open courseware thing but it seems to just be a list of courses they have on Coursera.

an introduction to manifolds by loring tu has very reasonable exercises that give you a good working feel for the material. many of them have hints or solutions at the back of the book. plus, it’s an excellent book. the same goes for his differential geometry book.

there is also analysis and algebra on differentiable maniflds: a workbook for students and teachers. it has lots of fully worked problems.

Elliptic curves originate from calculus and there is an analytic theory for them (the Weierstrass P-functions and all that), but they are also very interesting from an algebro-geometric viewpoint, closer to the abstract algebra that you mention.

Fundamentally, the points on an elliptic curve form a group (an algebraic structure) under the chord-and-tangent method of addition. The (same) group law can also be defined via the Picard group of the curve. Have a look at the table of contents of Silverman's Arithmetic of Elliptic Curves to see what kind of algebra you can encounter in the theory of elliptic curves.

In that case you may also find Liu's book helpful.

Categories for the Working Mathematician. Of course, that's geared more towards people who need some category theory in their own work. Category theory by itself is like a bookshelf. A nice way to organize your stuff, but nothing substantive is actually there until you fill it with books.

You might want to check out Chapter 5, Robotics and Automated Geometric Theorem Proving, of Cox, Little and O'Shea's Ideals, Varieties and Algorithms for applications of algebraic geometry to motion planning in robotics.

Adding on to this, we need like a workbook for working with sheaves. They are difficult for me to get a feel for

On the other hand, I know there are very concrete problems in https://www.amazon.com/Algebraic-Geometry-Problem-Approach-Mathematical/dp/0821893963

Particularly the last chapter when sheaves and cech cohomology are introduced.

However when I think of sheaves, I cannot see the trees in the forest, I just see the forest

Milnor

I haven't read the following books, but they're supposed to be ultra simple (in this case, easy).

Algebraic Geometry for Scientists and Engineers by Abhyankar

Algebraic Geometry: A Problem Solving Approach by Garrity et al

I am not sure there are AG books more elementary than those listed.