reading books (but they are textbooks on mathematics)

A blog designed to document my experience and progress in math. It is going very well. main blog: regina-mathematicae

[adventures-in-mathematics]

Math Blog 19

I think part of the reason I keep missing a week is because having so few things to do on a weekly basis makes the weeks go by much quicker. Before, I would interact with the world (friends, classes, getting food) basically every day, but now, I'm in somewhat of a bubble; of course, I do still have class on tuesdays and thursdays, but I have no friends on campus, and all the (good) on-campus places are closed, I don't have a meal plan to get free food anyways, and the heat is a large incentive to stay inside. All in all, maybe I'll just stick to the once every two weeks schedule that I have fallen into, until the summer ends.

Anyways, I had originally planned on skipping a lot of the exercises from ii.7, and it might be a good thing that I had that extra week, because I ended up spending a lot more time working through the problems, and I actually solved most of them! Yet another point in favour of only updating every other week. I've been putting off getting started on differentials, though. I did read it through, and I've looked at the exercises (and everything seems relatively doable), but I just haven't actually sat down and done the work.

On the other hand, I have been somewhat better on the algebraic number theory front. I've been bouncing between Fröhlich and Taylor and Milne, because they both (seem to) provide a lot of explicit examples/exercises. However, I think I'm just going to stick with Neukirch, and work out some examples myself. The other two, at least compared to Neukirch, are so dry and difficult to read. As a brief aside, Neukirch is the only place I've read this, so I know I got it from there: in one of his proofs, he writes "[one obtains some result], after Vieta," to mean that one applies Vieta's formulae to the hypotheses to obtain the conclusion. First of all, I realised that I misremembered it, because I always say "après," and I in particular use it quite frequently in applying Nakayama's lemma, so I often write (or say) something like "après Nakayama."

I've been continuing to read Miranda, and I think it was a good idea to read it before I use it for a (hypothetical, still) seminar, because I can just make the executive decision to cut out anything I personally find uninteresting and not worth the time. On a more serious note, it is quite pleasant to see the obvious geometry happening in the proofs; not to say that algebraic geometry à la Hartshorne is nongeometric, but rather the geometry is somewhat nonobvious, and one has to check various technical, algebraic details even in an overall geometric argument. In contrast, Miranda has been pleasingly (though sometimes even jarringly) direct in using topological/geometric arguments. I say jarringly because it's taken me a moment to actually adjust and realise that things do work much more simply in the case of complex curves.

I almost forgot to talk about my research! I think it's going kind of well? It's somewhat amusing, because every time I start to feel like I'm getting the hang of things, I go to our meeting and find out, not only am I wrong, but everything we've already done is also wrong. This most recent time was the worst one, because I made a lot of progress, and we realised that the fundamental assumption I made in literally the very first definition was false. There's a pretty clear way to modify it, and everything else goes through basically unchanged, except for one equality that I don't really know how to make sense of anymore. I think it's just a matter of keeping track of where things go and come from, but I am yet to take the time to actually sit down and work everything out (that's a tomorrow thing).

I also started learning Tamil, which has been pretty easy going. I've finally figured out the differences between all the letters, which seems pretty simple/silly, but that actually might have been my biggest problem, considering I have some familiarity with speaking/hearing Malayalam (which rounds to Tamil, if you really think about it).

I think that's all I have to say for now, so I shall be off, to return in possibly one week, but likely two.

[adventures-in-mathematics]

Math Blog 18

I'm still getting into the habit of posting weekly again, so I unfortunately missed last week. I did all the (unstarred) exercises from II.6, but basically most of them were starred, so I was hoping to finish II.7's problems too; I didn't.

I totally forgot it has been two weeks; I thought I had at least kept up the one a day rate, but alas. In any case, I'm getting really close to Chapter III (I'm planning on skipping the chapter on formal schemes. I might revisit it later, but I want to get back to arithmetic).

I've been focusing more on research (as I ought to, naturally). I fear that it's not going as well as one may hope. I've made basically no process, and my advisor pointed out that a fact I had taken for granted, which would have made my life much easier, was completely false. There should be a very reasonable fix to the situation, which I hope to sit down and actually work out tonight or tomorrow.

I think the problem I've been running into with algebraic number theory is a lack of problems, because I think I'm in the same boat that I was in with algebraic geometry, where I have enough knowledge that more reading won't help, but not enough that I feel comfortable moving on, so I've been reading sources like Milne and Neukirch, which aren't very exercise heavy, and ultimately just been skimming them, because I don't need to learn more theory. Based off of this hypothesis, I'm going to try just working through the problems from Jarvis's Algebraic Number Theory and Gouvêa's p-adic Numbers, An Introduction. If I do one from Jarvis and two from Gouvêa a day, I should finish at the same time as Hartshorne.

On the other hand, it might also be the case that my mathematical reading skills have just atrophied a lot, because it's been so long since I've actually tackled a completely new subject. The real problem is probably a combination of the two, but I think this one is far more detrimental than the first. Therefore, I think I'm going to stop studying Needham (because I've already read through a lot of it years ago), and switch to Miranda for my analytic/"real" geometry book. If that doesn't work out, I'm going to keep moving up the alphabet, so next up will be someone starting with L (Lee or Lang seem like the only two I can think of right now, so let's hope it doesn't come to that).

One thing I want to try tomorrow is finish all of Xu-Gallier's book in a day, because I remember reading about a graduate student who finished Silverman-Tate's Rational Points on Elliptic Curves in a day, and I've always wanted to try something like that, and Xu-Gallier, being half the length of Silverman-Tate, seems like a reasonable goal.

[adventures-in-mathematics]

Math Blog 17

I finished (most of) the chapters to section II.5 of Hartshorne, but I don't think I have much else to update on?

I didn't post last week because I was moving back to Texas for my summer research/class, which went alright. While I'll have roommates over the school year, I have the unit to myself for the summer, which is cool.

I sent my first idea for the research project to my professor around a week ago, and he did not give any feedback (yet). We're meeting on Tuesday to talk about it, and I am on the edge of my seat!

I'm thinking that I don't have much of an excuse to not get on the grind with studying other subjects, which means I'm going to have to finally lock in for real.

I want to pick up a copy of Fröhlich and Taylor's Algebraic Number Theory from the library, and get back on the Needham horse, and while I feel like I should commit to more, I am worried that I won't even be able to hold this much up, so I will not!

I went to my first class this past Thursday; it was actually the third class of the semester, but I was in the wrong state, and my flight was already booked. It was pretty enjoyable. We had a substitute lecturer, and it was material I already knew, but the lecturer was pretty good, and the homework was actually kind of fun too.

Anyways, I shouldn't think that I have anything else to say, and so I shall return next week with updates.

[adventures-in-mathematics]

Math Blog 16

This is the first time in a while that I'm making a post, and there isn't really much for me to say (except that there is).

I've officially finished my first year of college, and it went slightly worse than I would have liked (specifically, the second semester). Unfortunately, I miscalculated my grades near the end of the semester, and on the basis of that miscalculation, did the bare minimum necessary to get an A, which resulted in me only getting a B+. Not too bad, and I still ended with a 3.5 cumulative gpa, but it should have been better.

Now that the boring schoolwork side of things is done, I have a lot more interesting things to talk about.

For one, my independent study on elliptic curves evolved into an elementary study on the Tate curve, and my paper ended up being a retelling of the first part of Roquette's 1970 manuscript. My professor just gave me 100% in the course, and said he'd give me specific review/feedback on my paper at some point in the future; I assumed it would be soon after the year ended, but when I asked, he simply told me that the academic review process tends to take time, and that I should get used to submitting a paper and keeping it out of mind until I eventually receive feedback, which I think means that I will not get feedback for a hundred thousand years, and I've made my peace with that.

On a related note, I'm going to be doing a summer research project with him. I was somewhat worried about my funding situation, because I technically submitted my application after the deadline, and I didn't receive any relevant communication from the Honors College, which would have been what supplied the funding. However, when I reached out, they said that I was already in a different summer research program (that I never applied to), which I guess is good for me, because it means that I can use the Honors College stipend in a different summer (it's a one-time only thing). I was still a little bit antsy until today, when I received a message that the payroll office had a check waiting for me (I wasn't able to set up direct deposit by the deadline, because there was more paperwork on the University's end, I suppose, which only made my anxiety worse). All's well that ends well, I suppose.

As for the actual summer project, my professor wants to try and extend some results he got in a paper for algebraically closed fields to non-algebraically closed fields, and seems confident that we'll actually be able to get a publication out of it some time in the fall! Originally I was going to be working with a different research group he's organising, but that's going to be with 'normal' students, so the background to get them up to speed will likely prevent them from getting as deep (his words, not mine).

I'm also going to be taking a summer course with a different professor I had this past semester, and I'm really excited for that, because, going off of past experiences, he has a habit of going very quickly through the boring bits, and emphasising proofs/concepts over computation/calculation; for example, he seems to center differential forms/the generalised Stokes' theorem, and derives the standard results as a consequence thereof, whereas the standard course only covers differential forms at the end, after everything else (as far as I can tell). Unfortunately, I'll miss the first two classes, because I thought that the course started a week after it actually does, but I should be fine.

I've also been working on organising a seminar in the Spring 2027 semester on the history of mathematics, following the third edition of Stillwell's book. My secret ulterior motive is to get all my friends up to speed and conversational in mathematics, essentially a sort of enculturation into mathematics (which is very different than inculturation, beware). Everyone I would want to join has agreed, and I have two (maybe three) professors who seem happy to supervise.

Finally, as for Hartshorne, I have just finished up to II.4 (minus some problems I chose to skip, but not too many), which I think keeps my head above water on the one problem a day rate. There were a few problems that I really struggled with and used the internet a lot, but I think I did enough to say that I actually did the work, instead of just copying and pasting other work. From here, I have recently reviewed a lot of the topics, and (I think) the exercises get more interesting, so hopefully it goes by more smoothly.

I am still struggling to allot my time in a reasonable way instead of just spending all day grinding out algebraic geometry exercises, and this is especially bad now, because I have to read the paper that I'm supposed to be spending the summer generalising. I don't know if the professor expects me to have the paper fully read by the beginning of the program or not, and I know that if I ask he'll say something like "just do whatever you can, and we can go from there," because that's essentially what he said every time I asked how much work I should do. On one hand, it's a very kind/patient approach, which would hypothetically make me very unstressed out, but on the other hand, it's very stressful because I have no idea if I'm on track or not.

Anyhow, I hope that this hasn't been too long, and my goal is to have some/most of II.5 done by next week, and the introduction and Section 1 of the paper read (those parts he recommended to skim for motivation/intuition, and only really focus on Section 2).

[adventures-in-mathematics]

Math Blog 15

I did not finish Section 2 this week [sad face]. That's all!

On a serious note, it kind of is. I've been looking into algebraic topology, most especially because my friend is a big fan of it, and I don't know enough to answer all of his questions (he really likes the hopf fibration, it seems).

It's been kind of strange proving things that I've been taking for granted for so long--not because I don't know or can't, but because it takes me a moment to parse what actually has to be proved. In addition to that, the proofs are usually (ironically) more tedious, because the majority of the proofs are minor verifications that things work how one would expect, so I either feel bad not including many details, or hate myself for deciding to include every possible detail. I think I've taken to including an aside as to why some step is justified in the specific case one considers.

I decided to work out the snake lemma again, and I forgot how pleasant it is.

I've been working on more non-mathematical things (including to listening to a few albums I've been meaning to get to for a while), so I've nothing to say on that matter.

I have to write a paper on Tate modules and their relation to the Weil conjectures, starting from the basics on profinite groups and such. I think during my last meeting, I really started to grasp the relation between Tate curves and actual elliptic curves.

I don't have much to say, and I'm sure this is enough. At least this time I actually got it in on the weekend(ish; it's technically monday today, but come on). I hope to not only have Section 2 of Hartshorne done by next week, but also get to Act III of Needham (I've but the last chapter of Act II) and one section/chapter of an algebraic topology/number theory book.

[adventures-in-mathematics]

Math Blog 14

I just finished the last problem (read: copied and pasted my answer to the same problem from Görtz-Wedhorn) from Section 1 of Chapter II from Hartshorne, and so I can proclaim victory in my weekly update!

I got a little bit distracted on Saturday and Sunday, which is why I wasn't able to finish it off, and that's the only real excuse I need. Looking at the problems for Section 2, it should be a breeze, so maybe I'll get it done in just one week? I really should, to be honest; the only reason I wouldn't is because I'm not spending as much time as I should on math.

I have been doing a lot less math outside of those problems than I had hoped, but I think that's just because I was pushing myself to finish Hartshorne. Despite what I just said in the last paragraph, I'm going to try and be less single-minded in my approach to math.

I did check out Spivak's book(s), and was shocked to see that he seems to only operate in the context of metric spaces? The thing about writing these out is it forces me to be more measured and rational with my thought processes; quite frequently, I have found myself writing some complaint or frustration about something, with a possible request for help or advice, before realising how completely trivial/nonsensical my issue is. In any case, writing this out makes me see that I was quite possibly being uncharitable towards Spivak, and I should probably give the metric space approach a chance.

Nevertheless, I spent the last week reading Needham, and I think that it's far less rigorous than a traditional book on the subject, it's still actually pretty enjoyable to read, and it does force me to think of things in a different way than I usually do. I'm trying to see the positives, because my school's library doesn't have the book I'd probably be most interested in (Tu's Introduction to Manifolds), but I've officially submitted an interlibrary loan request, so hopefully that goes well.

I feel like I've been moving away from algebraic number theory, which isn't intentional, but a part of me is a touch apprehensive to returning to a place which seemingly so thoroughly lacks geometry. It just feels like algebraic geometry is my current focus, and while differential geometry would help with that, algebraic number theory seems much more parallel. That is to say, I still haven't spent any time with Neukirch, or any other algebraic number theory text.

Actually, I'm sure that the biggest reason for that is that Neukirch is the only number theory book I own and have right now, and he is currently buried away in the bottom shelf in the corner of my room, which makes it a pain to grab as I rush out of my room for something or another. I'll go bring him out for tomorrow morning.

Speaking of tomorrow morning, the professor I'm working with and I have spoken, and I'm probably going to write my paper on Tate modules, Galois representations, the Weil conjectures, and L-functions of elliptic curves. I'm really hoping to work with another professor who is in the more automorphic side of number theory, and I'm hoping the last part of my paper can be a sort of transition. I'm currently in the trenches of the analytic theory of elliptic functions over local fields, and I'm just barely grasping what things are supposed to be.

I did not expect to have so much to write about, and I still have a lot more I want to say! However, I really should get to sleep. Hopefully, I'll actually have an update over the weekend.

[adventures-in-mathematics]

favourite number?

I really struggle with this question, for I love all numbers, but my go-to answer is usually 6:

it's the first product of two distinct primes, which makes it the first real composite number (everyone knows p^n = p, basically) [to make that point more rigorous, Z/6Z is the first nontrivial quotient of Z with more than one point]

2. it is the first perfect number

3. it is a factorial, 3!

4. it is a triangular number 1 + 2 + 3

5. it is the only both factorial and triangular number other than 1

6. S_6 is the only symmetric group with an outer automorphism

I only really separated 3-5 for the sake of getting the list to 6 points, but there it is!

[adventures-in-mathematics]

Hi are you an undergrad? (it says that on your main blog) Cause I am like super impressed with your progress!

yes, I am, and I really appreciate hearing that! My first instinct was to downplay my progress and talk about how much further along I 'should' be, but a moment's reflection recalled to mind that perhaps I am not as far behind as I've been feeling.

Anyways, thank you again for the kind words! I don't really know how else to respond to compliments, so I suppose that's all I'll say

[adventures-in-mathematics]

Math Blog 13

Last week, I decided to try skipping a week to see if I'd prefer the rhythm of semiweekly (I have much distaste for the ambiguity of the word biweekly) updates, and I've decided I don't like it; it feels like it's been a hundred years.

Anyhow, I've been at most kind of busy. I was overwhelmed by a lot of work, but I think I have managed to dig myself out of that hole. I decided to do (at least) one problem a day out of Hartshorne (on average) and some daily reading in algebraic number theory and differential geometry.

I have finished the first nine problems from I.1 of Hartshorne, so by next week, I should have at least 16 problems done, but I really hope to have all of I.1 done. One thing I'm scared of, especially when I get into actual schemes, is not having enough commutative algebra on hand, or not being able to recognise when I need some result. I'm sure I know enough, but I would still like a reference on the side, so I will have to be on the hunt for that. I might switch out the number theory for commutative algebra.

On the differential geometry side, I started with Needham's Visual Differential Geometry and Forms, which occupies a similar space in my heart, soul, and mind as Vakil's book, in the sense that I want so badly to love it, and yet I fear that I simply cannot. It's a really fun book, and I'll probably keep reading it on the side, but I am going to try Spivak's volumes (of course, starting with volume I).

Arithmetically, I haven't actually started anything, because I kind of forgot about it. I really don't know which text to use, and I'm mildly tempted to just put it off. In that case, I would study commutative algebra instead, and I would probably use McLennan's Algebra of Coherent Algebraic Sheaves, which is devoted to an exposition of all the algebra Serre uses in FAC. He states in the introduction that it is meant to be read and understood, instead of having to do lots of work on behalf of the reader, which is really nice and cool for me personally (though I do have my doubts about how useful this approach actually is for learning).

I'm also thinking about writing my own notes about algebraic geometry, not necessarily for publishing, but rather copying how Serge Lang studied topics: write his own book on them. We'll see how that goes.

I think that's all I have to say for now, so I will almost probably be back next week!

[adventures-in-mathematics]

Math Blog 12

It's been pretty much a month since I last made a new blog entry, and I don't really have a reason. I just, for some reason, skipped a week, then another, and I completely lost momentum.

I'm thinking about new ways to get back into the groove of doing math consistently, and I have roughly four ideas: 1. just get back to what I was doing, 2. revise my previous proofs, 3. work on some other algebraic geometry, and 4. take a break from algebraic geometry completely. I'll try and set myself to setting on one of these four paths by next week, as a relatively easy/low pressure thing to work on, mathematically speaking.

I also want to try and incorporate topology and analysis, so I will see how much time that would actually add to my weekly schedule.

I don't have much to say, but I just have to force myself to end this impromptu hiatus into which I have fallen. Anyways, that's all; I will hopefully be back next week!

[adventures-in-mathematics]

Math Blog 11

Due to a combination of factors, I noticeably did not end up posting this on Wednesday. I would like to note, for the record, that I did finish all the exercises from chapter three by Wednesday.

Anyways, I'm getting a little bit tired of Görtz-Wedhorn's style, so I'm thinking about working out the first chapter of Hartshorne to get a feel for more classical geometry and such, and also maybe switch back to Hartshorne fully. I don't think this contradicts my goal of developing patience, because 1. I still intend to work through Hartshorne with the same detail and care (probably more, actually) as I have with Görtz-Wedhorn, and 2. (and to me, more importantly) I feel like I have proven to myself what I wanted to prove; I feel much more confident in my mathematical abilities, and feel far more grounded. My secret third reason is that Hartshorne is much easier and pleasant to carry around then Görtz-Wedhorn, and it's much easier because it's hardcover.

I don't think I have much else to say? I have no idea what kind of schedule to which I should hold myself for Hartshorne; he says in the introduction that he does about one chapter a quarter, but I think I can fairly easily swing that. Maybe I'll get a commutative algebra book and actually learn some of that on the way. I may also take yet another swing at Vakil instead of Hartshorne.

I didn't sleep last night, so my thoughts are relatively disheveled, and it's wholly possible I return next week having kept working on chapter four of Görtz-Wedhorn. I think the whole point of this blog is to keep me accountable though, so I should set down a hard rule. I think I'll work on chapter one of Hartshorne, if only to get a solid understanding of the most classical case down. There are about 90 exercises in the first chapter, and I'm wondering if five weeks is too much time for me. That'd be eighteen a week, on average, which isn't too much. I think by next week, I will set myself the lower bound of the first section of chapter one done, and the goal of the first two sections of chapter one done.

[adventures-in-mathematics]

Math Blog 10

I have decided that I will give myself more time for the next few weeks; that is, I will wait eight days between posting, until I cycle back around to posting on the weekends. This is because I am lazy and don't get much work done anyways.

Speaking of not getting work done, I only finished half of chapter three's exercises so far. I didn't have to do much reading, I just didn't spend much time working out exercises, I guess. Luckily I did say that from here on out, I would allow myself two weeks per chapter, so I don't feel too bad about myself.

I just had my second meeting with the professor, who I think I'll start calling Dr. A, just because "the professor" feels too distant, and I actually think this week went much better than last week. I miserably failed at calculating the canonical sheaf of the projective line, because I was trying to copy what I vaguely remembered the book doing, instead of just trusting myself, but other than that, I was able to answer everything he asked, I think.

I think I'm starting to get a feel for how these meetings are supposed to go, and it makes me feel like a real grad student to be doing the talking, instead of being talked at. I also really appreciate the time to see how a real life mathematician thinks about algebraic geometry in action, instead of poring over the textbooks that cover the foundations.

Overall, I was actually really unproductive last week, relative to what I would have liked to get done, and I don't know why that is. Hopefully, this week, I can get back on my diligent arc.

Anyways, my goal is to be back next Wednesday (!!) with all of chapter three done, and maybe chapter four already started (?). We'll see how that goes.

[adventures-in-mathematics]

Math Blog 9

I have officially finished (most of) chapter two's problems. I was hoping to finish them over the weekend, but at least I managed to finish them in a week. I am simultaneously excited and worried, because we are getting to the point that I might have to start actually reading the chapters before starting the problems, which on the one hand means I am going to start learning new content (yay! woohoo!), but on the other hand, it also means that it will start taking me longer to actually finish each chapter.

As of right now, I am going to stick to the whole one chapter a week schedule, but I am preparing to switch into a two chapters per week schedule, to split my time between reading and doing problems.

Outside of my (personal) algebraic geometry work, I also have to do some algebraic geometry (for school). I have to go through at least the first chapter of Eisenbud and Harris's Practice of Algebraic Curves, but I would certainly like to get through the first two. I have been somewhat (read: extremely) unscrupulous in keeping up with my actual classes, which means I have a little bit of catching up to do but probably not too much, I think. I am not behind on any of my assignments (but I might be after tonight, I have something due at midnight).

I also have to finish my application for an REU. I fully don't intend to get in, and I kind of hope I don't, because in that case, I'll work on elliptic curves over the summer; exciting! Not to say that I wouldn't be extremely glad to get in, but rather that, as of right now, I am in a win-win situation, for which I am incredibly grateful.

Anyways, I think that's all for this week, so I will report back next week, hopefully done with chapter 3.

[adventures-in-mathematics]

Math Blog 8

I have finally finished chapter I of Görtz-Wedhorn, and have written up solutions to all the exercises. Admittedly, a few of them are more on the sketch side than proof side, but in the grand scheme of things, that's probably okay (there is one that hinges on linear algebra that I don't want to do, so I haven't, so I'll probably go back and fix at least that).

I hope to finish the next chapter this coming week, but I actually have a fair bit to do: study the first chapter or two of Eisenbud and Harris's Practice of Algebraic Curves for my reading course, and likewise for the first two chapters of Serre's representation theory book.

Luckily, this will be about half as much work, because the chapter only covers affine schemes. I think I'm going to move forward with the following schedule, for as long as it is feasible: aim for one week per chapter, but have a cap of two weeks. That way, if I theoretically keep the pace for a year, I will be done in a year. More importantly, I think the hard cap is going to give me a motivational push to actually push through things I find boring.

I'm really glad that I managed to articulate that one of the central goals for me with this is actually developing patience, because it provides an extra kick of motivation whenever the "I already know this" wolf inside of me gets a little too bored.

The temptation is so strong to jump back into cool fun stuff right now, but I will hold out because I am also learning patience!

One thing that might be fun is working through chapter 16 in parallel, because chapter 16 is all about examples, specifically citing what in the earlier 15 chapters is being applied at each juncture.

However, it might also be a useful review at the end, to work through all the examples to make sure I actually remember everything from the beginning, before officially picking up volume II for cohomology. As of right now, I'm probably leaning more to this end, if perhaps in part so I can finally be free of "prevarieties" and return to my beloved schemes (even if only the affine ones at first).

[adventures-in-mathematics]

Math Blog 7

I've been somewhat bad; I started Görtz-Wedhorn's book, and instead of being very good and responsible and starting with chapter one, I have sort of been skipping around. I looked at and mentally worked out solutions for the first few chapters (but didn't write anything out formally), and so I started with chapter 6.

I have felt some amount of guilt, because I am only now realising that there is a secondary reason that I want to do this: I want to force myself to slow down and build up patience, instead of constantly needing to rush until the end. I only realised that today, and I think my impatience is probably my biggest mathematical shortcoming (in my opinion).

Anyways, I talked with my professor on Friday, and we decided to switch from étale cohomology to elliptic curves, because he couldn't really see a clear path into étale cohomology, whereas he could with elliptic curves, and also this can tie into a research project for the summer, which étale cohomology can't, really. This redoubles my conviction to study all of Görtz-Wedhorn's duology, because they cover the étale topology in the second book, which I feared would be redundant. It also gives me a lot of time, because that means the next time I have to do a serious algebraic geometry project will be in about two years (this semester is elliptic curves, next two semesters will be representation theory and modular forms, and then hopefully another one on automorphic forms).

I'm hoping to have finished the first chapter + all the exercises by next week, which I'm just now realising I also said last week, which makes me feel like I'm stuck in the time loop, but that was before I realised the whole patience thing was an important factor, and it wasn't just about knowledge and confidence. That is, I now have actual motivation to do the chapter one stuff.

Writing these has been really helpful to isolate and reflect on how I'm feeling; I never would have realised that my real goal is to develop patience without writing this just now, I think.

[adventures-in-mathematics]

Math Blog 6

I'm doing a really good job of posting every weekend and not the monday after. Don't fact-check me on that.

I was posting about Vakil these past few days, and I am going to embark on a long term project to solidify my foundations of algebraic geometry, but not through Vakil, but rather Görtz-Wedhorn's duology on schemes and then their cohomology. I claimed that Vakil was one of the most comprehensive foundational book, but it seems like they are at least in the running, if not ahead.

The nice thing is that it has a more explicitly arithmetic goals (as examples, it talks about Hilbert modular varieties at the end of the first book, and abelian schemes at the end of the second), but more importantly, it is not a book that I know I already dislike. I still hope to one day come to appreciate Vakil's style, like I did with Hartshorne, but I don't think forcing myself to study a book I know I dislike will do me much help right now.

Anyways, I also worked through more of Milne's Abelian Varieties, and while I have a fairly solid understanding up to the Seesaw principle, from there I've been sort of surfing on vague intuition and just believing that my friend Milne wouldn't lie to me. The reason I am so sure that I need to completely revise my foundations in algebraic geometry is I had the sudden and terrifying realisation that if there was some result that he quoted, I not only lacked examples as to why the hypotheses were necessary, but I couldn't even produce mental examples of the phenomena in question.

Since I noticed that, I've sort of been in panic mode, and I think the only way is to start from where I know I'm confident, and push back out, which is why I'm restarting from the ground up again.

I'm even going to try and hold myself to a regular schedule (at least for the earlier chapters) of one chapter a week, and all the exercises.

It's somewhat frustrating to have to keep backpedaling all the time, but I think at the end of the day, it's better for me to overestimate myself now and have to go back than insist that I know what I'm doing and then floundering in two years when I have to actually do things.

I think other than that, I don't have anything specific to report on, but hopefully, next week, I'll have some real results to show off.

[adventures-in-mathematics]

hey do you study algebraic geometry? I relate with a lot of experiences that you wrote about so its fun. Currently I'm studying a algebraic number theory. I think your profile name is really cool. It reminds of Alice in the wonderland. You're awesome!

I first read this as "how do you study algebraic geometry" and I was very worried because I am not qualified to answer that question. However, whether or not I do, while still more in question, is almost one I can definitively answer!

I have bene studying algebraic geometry, with the goal of getting more into arithmetic geometry, and I have been meaning to review algebraic number theory (I have set up a reading group where I teach individuals of a mathematical persuasion about algebraic number theory next semester, for this purpose).

I really appreciate all the kind words! My username in fact comes from the fact that I could not think of a better name, and I was imagining someone exploring a new place/environment and writing their findings/observations in a journal.