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Can I really get a good grasp on Algebraic Geometry, Differential Geometry etc. by self studying?

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I self-studied DiffGeo and seem to be doing fine. However, I do have gaps in my understanding (to the point where it’s somewhat affecting my research progress) so I am intending to take a course next year. I think anything is possible if you put your heart, mind, and soul to it.
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Self-studying Algebraic Geometry is much harder, IMO, but maybe it depends on the individual. For DG, you just need to know some analysis, calculus and basic point set topology.

In contrast, for AG, you may also need to pick up some commutative algebra and category theory. Vakil's book walks you through this but the abstraction level is higher and it will take more work building intuitions for the subject.
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That crucially depends on your prior knowledge, your previous experience with self-study and your overall pure mathematics skills. Both topics are very large and go way above what a single person can learn in a life time.

 That being said, if you self study either choose an online lecture and not really self study (maybe Richard  Borcherds yt for advanced undergrad/early graduate level) or follow a book that gives a lot of motivation and examples (maybe miles Reids undergraduate algebraic geometry).

Both topics have very vast formalism but also some down to earth motivating examples and problems in the beginning where geometric intuition and computations in coordinates are as important as formalism. If you are not already Grad level maybe start there.

I am not one for prereqs as in you need to have done commutative algebra for years before starting with geometry, but you should have at least done the equivalent of 2-3 analysis and 2-3 (linear) Algebra courses before this can make any sense to you.
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Self-study is hard, but it’ll help build the independence and discipline required in mathematical research. Slow is perfectly fine, if you’re reading more than 4-5 pages a day you’re probably going too fast. But nothing beats having a dedicated lecture with homework problems, a professor to answer questions, peers to work on problems with, etc. I’d also recommend reading from multiple resources to get a more holistic view of a topic.

By the way, a great book for self-studying manifolds is Janich’s Vector Analysis. It’s designed to emphasize intuition and does a good job at it.
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Differential geometry is nice to self study. With differential geometry you need to sit down and do a few conversions from coordinate free notation to working in charts for every concept you encounter anyways.

I can recommend the books by Tu, the book by jack lee and the book by boothby for differential geometry.


Algebraic geometry is more difficult in my opinion. The problem is that there are a lot of prerequisites for algebraic geometry and it can be hard to stay motivated whilst climbing the mountain alone. I recommend the book by perrin and the books basic algebraic geometry 1&2 by Shafarevich.

I don‘t think one would want to self-study algebraic topology. For algebraic topology you want somebody to guide you through the material and show you how to think about the material. There are also many computational tricks that are hard to find in books in my opinion.
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People have done this, but most don't succeed. With Differential Geometry, the main backstop is a willingness to actually do computations. I tried learning some stuff about manifolds on my own as an undergrad, but the motivation for me to actually sit and compute a transition function or a flow or whatever was at an all time low, especially when I knew how it "should" work. If you end up doing this, don't give in to that temptation. Knowing how it works is the easy part; actually doing the calculation to verify your intuition is the defining barrier between knowing and not knowing differential geometry.

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Algebraic Geometry, on the other hand is not something I'd recommend trying to self study, at least without someone who knows the subject regularly available to answer your questions. There is simply just too much necessary material and necessary machinery, all of which will seem very foreign to you unless you've taken a number of very dense algebra courses. The prerequisite courses required are also quite a barrier to entry. In my opinion, the bare minimum to take algebraic geometry is a graduate course in commutative algebra (at the level of Atiyah Macdonald). To actually get something out of the course, I also feel that it's important to see the concepts used in a simpler, easier context first. For instance, studying curves/surfaces over the complex numbers only makes sense if you've seen some complex analysis. The language of sheaves/schemes is hard to parse unless you've seen some stuff about manifolds (R\^n is to a manifold as an affine scheme is to a scheme, and a manifold's atlas is like a scheme's structure sheaf). Sheaf cohomology is going to make no sense unless you've seen simplicial or DeRham cohomology from either differential or algebraic topology.

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If you've taken (and done well in) all the courses I just mentioned, then \*maybe\* you can self study algebraic geometry. Otherwise, without a professor/expert to help you with the intuition, you will (probably) be dead lost.
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My gut feeling is that most books on the matter are for people who already know it. But Harder’s introduction to algebraic geometry is a bit different. I suggest you try this book. Good luck.
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It is in principal possible to self study an advanced topic in math effectively. On the other hand, it is very hard if you don't get constant feedback from somoone else, either a professor or other students.

The other question I have is how will the graduate school you apply to know that you've learned, say, differential geometry really well by self study? Are you able to do the self study as an independent study or reading course with a professor? If you do that, you will have at least one professor who can write a letter attesting to your understanding of the subject.
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Courses are much more streamlined and closer to where research is headed so the worst you can get from self study is over-saturation, if you’re using a book that is
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Self-study is perfectly fine if you pace yourself correctly.

I learned most of the math I remember through self-study. Most of this was before school when I was a kid so it was during formative years where you remember everything anyway. But even since then, when I was going to take a class in a new subject, I'd generally read the book before class, take some notes, solve some problems, think about it a lot, and then refresh/reorient it during the class itself.

Having to reorient due to having misunderstood the material during self-study but having it corrected during a lecture was rare, but not unheard of.

All-in-all, I'd say self-study is fine if you pace yourself correctly and take it seriously.

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