Next up, non-linear dynamics (while I finish the algebraic topology book). Then if you want to get the clock, 'Don't be a jerk' shirt, floating globe, or other STEM items like those you see in this video check out stemerch.com :)
Oh my god I can't wait to see your algebraic topology video. I did a graduate algebraic topology course. I consider myself a good teacher and I would still consider teaching algebraic topology to math students to be one of the difficult subjects. I can't imagine a video that will give a general audience an idea of what algebraic topology but I can imagine that you would be up to the task of making one. I can't wait to see how you'll do it because I really have no clue what it would even take. Been a long time since I algebraic topology so I may remember it as being harder than it is.
I was thinking of algebraic geometry the whole time I wrote this comment. Then I heard you describing algebraic topology in the video and it clicked. I'm sure I'll be impressed with your explanation of algebraic topology but I won't be as impressed as if it had been algebraic geometry (I still think you can do it).
I can't believe how well this was timed. Ten minutes ago, I finished my final exam for Algebra covering this exact same textbook. I sat down to take a break and watch some RUclips, and this video happens to pop up in my subscription feed. How serendipitous!
I was wondering how do you feel about kids in 11th and 12th grade. Do you feel like dumb kids can't do 11th maths or you feel if i would have been there now, i would teach students this way or that way
I feel the need to point out that the operations in a ring are not any more specific than that of a group. The addition and multiplication of a ring are just names, they are not specific operations. The "addition" of a ring can be any operation in the same way that the binary operation of a group can be anything. The "multiplication" of the ring interacts with the addition of the ring through the distributive property. In other words, the terms addition and multiplication in a ring only specify their role in the distributive property relative to each other, not what the operations innately are. For example, if we have a nonempty set X, and we look at its power set P(X), the set P(X) with the operation of symmetric difference as "addition" and set intersection as "multiplication" form a ring, simply because intersection distributes over symmetric difference. Those operations aren't addition and multiplication in any innate sense, only relative to each other, and an operation serving as addition in one ring can serve as multiplication in a different ring.
Thank you, this bugged me a lot when it was said and I was hoping someone in the comments had pointed this out (and you did it very clearly and concisely imo). We like keeping things abstract after all, and "addition" and "multiplication" are more or less just mnemonic in nature.
I think what's meant is that the axioms for a ring are more restrictive than for a group. Any ring is an abelian group under the addition operator, but there's the additional constraint given by the multiplication and the distributive law
When I went to college, I failed abstract algebra, made a C in intermediate analysis, made a C in differential equations and dropped complex analysis. I ran out of that as fast as possible and switched my major to computer science. Then the those maths started to click in my mind. I was using some concepts in there to run programs. Was it efficient? Probably not! otherwise they would teach this math in CS. Was it fun? YES!! Truth be told, I don't think I was suppose to start complex analysis in my junior year. Stupid thing is that I bought all the math books I would need for the major. So I grabbed a book in 2020 and started reading it. It was so simple for me to understand the concepts. I feel stupid that all of this didn't click back in the day. I don't know why I didn't understand it. Although at the time I did take a huge loan out for the first time Then I moved to a cheap apartment close to the college that had bedbugs. It kept me up at night and made me like an OCD person looking at everything that moved on the wall so I wouldn't get bit. I would do rituals to clean my clothes and books so I would not bring them with me to the class. I realized I would have no money by my senior year and I didn't ask for help. I should have applied for a full ride scholarship. My grades were pretty great before the decision to move to that other apartment. The problem is my parents never had the guidance to show me. I should have talked to someone at the school. I was young and thought that I could do everything alone. That my problems were my own to solve. I am a different person now. I realize that not everybody is out to see me fail. If I just go out and ask for help there will be people willing to help. My childhood goal was to dual major in physics and mathematics. Then get a PhD in physics and sit the rest of my life in a research facility somewhere. I learned that my life path isn't set in stone it will change. For some reason, starting a plan in double major physics and mathematics, failing and switching to computer science has made me actually pretty good at data science. The stars aligned for me.
at 2:17 15x + 57y = 1 has no integer solutions and it can be shown with basic algebra (no modulo operations needed): 15x + 57y = 1 3*5x + 3*19y = 1 3*(5x + 19y) = 1 5x + 19y = 1/3 Since if x and y are integers, then 5x+19y must be an integer. However we see above that 5x + 19y = 1/3, a rational number but not an integer, giving us a contradiction.
It's not that the operations for a ring HAVE TO be addition/subtraction/multiplication in the usual sense; they can be anything, too (with some requirements on how they interact). It's just that we CALL them those names because it's something we're familiar with, and they behave largely in the same way.
+1 - In the context of rings, no specific binary operation is innately addition or multiplication. Two operations are only called addition and multiplication with respect to each other.
This semester I had real analysis so I thought how about I go over the course myself during winter and boom I got a notification that you made a video on real analysis. This summer I thought of self studying abstract Algebra, and then I got a notification of this video. Considering I have topology next semester I wouldn't be surprised to get a video on that pretty soon
I'm a fan of your Zack Stars Himself channel. So I checked this out. I'd just like to commend you for being so relatable and funny in your other channel while being so mind blowingly academically gifted. It's rare to see someone with such diverse intelligence.
Abstract Algebra was my favorite math class by far. It was difficult for me, way harder than real analysis but the intuition you get from it was amazing. Like you are doing math from the ground up. Axioms and proofs to describe all the numbers. Nothing is off limits. Everything in Abstract Algebra is built off of proofs, definitions and axioms. It was pretty cool
Analysis should be the same way. A good analysis text starts with the axioms for the real numbers or a construction using the rationals (and proves this construction has those properties) and everything else is developed from there. It's not really different than abstract algebra in terms of it building from the ground up. It just starts from different axioms and of course naturally develops different definitions. If you are looking for the absolute highest quality of rigour, try set theory, category theory, or model theory. I'm studying set theory right now and you would not believe the things you can construct assuming the axiom of choice. We had to construct a set in R^2 using transfinite induction such that given any distance, there was exactly on pair of points in the set that are that distance apart. Things like this make me question math because such a set is completely inconceivable. Then you assume something as harmless as a choice function and you get well orderings of uncountable sets, Banach Tarski, and transfinite induction. You can do induction over the real numbers!!! I suppose at the end of the day, all truth in mathematics is conditional, and in order to make any statements about what is actually true you must rely on faith.
I don't know what's different in this video, you seem... more understanding of the topics you're discussing and more in depth which, in turn, feeds the passion with which you're explaining as well. Can't wait for more of this content!
Yeah, physics uses a lot of the rotation groups, which are subgroups of GL(R^3). These can be more concisely modeled with quaternion multiplication. Kinda cool
I went through so much brain torture to get a B+ in that class... still was one of my favorites, though I never used this information ever again in my life and I doubt I ever will
I would say, depending on the textbook you're using, linear algebra may be a more important prerequisite than you've suggested here. Michael Artin's textbook, for instance, emphasizes matrix groups (such as the group of invertible nXn matrices with real entries) as a key class of examples, especially of noncommutative groups (and similar sets of matrices as examples of noncommutative rings). Matrices also provide a particularly nice example of group action. I agree that it's not necessarily a true prereq, but I'd advise that people should be cautious depending on the choice of textbook/the professor.
Yeah I should probably emphasize more but pretty much everything I say in these videos is in reference to the specific textbook I went through, I don't really have knowledge about any others (or how it's taught in university).
As a mathematically inclined physicist, I am extremely wary of abstract textbooks that use too many matrices, as it perpetuates the unfortunate truth that most physicists don't understand the distinction between a group and a representation of it. I really enjoy the Durbin book, it almost specifically tries to stay away from familiar examples, to make sure you don't accidentally impose too much structure onto the abstract concepts.
@@Saztrah Artin's book is fantastic. It covers very little representation theory, and it's motivation for using the general linear group and it's subgroups a lot are because they are very useful for many examples. All of the theory (most at least) of linear algebra is developed within the text itself. Linear algebra (with vector spaces not just Rn) is quite useful in abstract algebra, especially when you study field extensions and Galois theory. A number of proofs about field extensions can be simplified by considering field extensions as vector spaces over their base field. Then you can use powerful tools such as the dimension of the vector space. In fact, fields and vector spaces have major connections because given some field F the way a vector space V over F is defined is equivalent to V being an Abelian Group and there being a ring homomorphism from F to the group of endomorphisms of G. So in a sense when we multiply V on the left by some element in F, we are doing so in a way that preserves the structure of V. Anyways, the motivation is there to study vector spaces if you are interested in abstract algebra. The definition of vector spaces as I wrote in the last paragraph, is a very concise way of defining vector spaces using the language developed in abstract algebra. You develop intuition for objects such as fields and groups and you can now wield that intuition for studying vector spaces, and vice versa. It's use of linear algebra doesn't really relate to representation theory whatsoever, although I will say, it is somewhat difficult to distinguish a group from its representation. A representation is a homomorphism from a group to the general linear group of a vector space (that is, all automorphisms of the vector space, or in other words, the invertible matrices in the vector space, although this is dependent on your basis). What this means (by the first isomorphism theorem for groups) is that the group factored by some normal subgroup (the kernel of the homomorphism) is isomorphic to some subgroup of the general linear group, ie. as far as group structure is concerned, they are the same. I like to think of this like the group in some way captures the symmetries of the vector space. Sorry for the ramble.
This was by far my favorite class in undergrad. I used the same book. Just candy for the brain. I think some ideas resonate with some people and others with other people. For whatever reason, I just really loved this class. It was two semesters long and covered almost the entire book. I wish I had time to do what you just did again...
I found "A Book of Abstract Algebra" by Charles C. Pinter to be a good, concise introduction to abstract algebra. It's short and easy to understand, even for a self-learner like me whose only previous exposure to group theory was in an inorganic chemistry class. I might look into "Contemporary Abstract Algebra" to expand on the topic a bit.
Hey what is your next book on your " Reading Textbooks I never read in School " list. Please do share as most people who aren’t in maths degree would not get a feel for it.
I'm doing algebraic topology by Hatcher and I'm going to give non-linear dynamics by Strogatz a try. I've been craving some material with direct applications to science and engineering (that's actually discussed thoroughly in the book). After real analysis and this book I haven't gotten much of that lately lol. The next book after this one I think would be galois theory but I don't want to do that quite yet.
@@cogitoergosum2846 Yeah Strogatz is a professor at my college. I'm planning on taking a course by him one of these years cause he's just such a great prof.
Zach, your intellectual curiosity is inspiring and helps keep me motivated as I work through my degree. I'm sure many others feel this way too- thanks for the great content!
Technically, number theory follows from abstract algebra. Those “number theory” concepts are concepts usually first learned in abstract algebra and not the other way around. 🙂 Linear algebra is also usually the first mathematics course where you learn and practice most of your proof skills, and this is because abstract algebra often is taken before the more difficult subject of real analysis.
I disagree -- number theory provides motivation for large swaths of abstract algebra, and was investigated earlier historically. That being said, most students today don't get exposure to the kind of elementary number theory commonly seen in competition/olympiad circles, so abstract algebra is often the first class a math major sees some of those topics, but it's an order of presentation that's only made out of necessity.
@@andresmath I concur about the historical development of the subject, but considering the modern approach to a course in Number Theory would leave most people quite helpless without abstract algebra (for algebraic number theory) or Complex Analysis (for analytic number theory), I would not suggest a course in Number Theory to precede abstract algebra. I also concur that some elementary number theory provides motivation for many topics in abstract algebra, which is why you are typically introduced to those topics within the abstract algebra course. However, the amount of elementary number theory required does not justify an entire course in Number Theory, which would surely cover much more and again would require the concepts of abstract algebra or complex analysis. Without the tools of abstract algebra or complex analysis, many of proofs in Number Theory would be inaccessible to most beginners. EDIT: I did do some additional research and found a “course in Number Theory without abstract algebra.” After reviewing the course materials, I can say it depends on one’s intentions here. If you never intend to take abstract algebra or a more advanced number theory course, then this might be the course for you. I should note that the course is not completely without abstract algebra. It just teaches those concepts within the course. So, I suppose it depends on one’s goals. EDIT2: I should also be clear that if one is interested in Number Theory, a course in abstract algebra is merely the start of that journey that provides the tools needed for that journey, much like a course in elementary algebra is necessary for trigonometry and eventually calculus.
@@stokhosursus well yeah but also there's a difference like you said between elementary number theory and Analytic number theory Analytic is the much more advanced one which abstract algebra would help in but elementary you don't need much background besides a good foundation in arithmetic
Operations in rings don't actully have to be defined. We just need two operations connected by the distributive property, but its helpful to think and notate them as addition and multiplication.
Zach, what was your schedule when doing this book? Did you commit to a certain number of hours per day or per week, or a certain number of days per week? Did you sit down at a set time of day to do it? Were you working a job with regular hours? I'm very interested in how you work a project like this or your previous "trips through the textbook", which is very long term but can be highly variable in page-to-page challenge level, into the rest of your life. I recall in your real analysis project you discussed dealing with really intractable problems that could take hours of effort spread over days or weeks. But how do you keep up the discipline to return to the book day after day and not start letting it slip and eventually lose the continuity you need to finish?
@@RiteshArora-s9x Chapter 1 through Chapter 30, and the last chapter. For this particular text a complete solution manual is readily available online, so it’s well-suited for self-learning.
@@0mon0zz The first derivative of displacement w.r.t time is velocity, second derivative is acceleration and third derivative is called "jerk". I kid you not 😂
I loved abstract algebra in college. We used Printers book, which is phenomenal for developing the subject in a inquiry based way. I can’t wait to continue studying algebra in grad school
Abstract algebra is very important to computer science-especially in its relation to category theory. Category theory might be something interesting for you to learn!
Dummit & Foote is much more dense and thorough than this book. You're done with most of what you see here (excluding the Galois theory briefly mentioned via the insolvability of the quintic, which is later) in the first 9 chapters, and the coverage of modules and then vector spaces in chapters 10 and 11 is very abstract. It's a great book, but be prepared!
Yeah I would encourage you to have a more accessible text like Gallian (the one in the video) on standby when Dummit & Foote gets too incomprehensible. I've had countless friends who hated abstract with Dummit & Foote, but loved it with Gallian.
@@JM-us3fr yeah in general i've found sticking to one book is cumbersome (i used 3 books to get through my graph theory course), so your comment is good reaffirmation of that, much appreciated
Thanks so much for igniting the flame for new knowledge! I am nothing into mathematics, but seeing others go through books and learn so much in diverse topics really inspires me!
I just want to add to your separation of Groups Rings/Fields (which I generally agree with): I am working in the field of calculating so called Feynman integrals. They are integrals that appear in Quantum Field Theoretic processes and are basically impossible to evaluate numerically (they love to diverge) or analytically (only possible in easy cases and too time consuming). Their calculation is crucial for more accurate calculations of elementary particle processes. However, they can be calculated using algebraic methods that are based on the Weyl-Algebra (basically polynomials in n variables and there respective derivatives) structure of the family of all those integrals. The relevant thing here is it's ring structure and both it's ideals and it's quotient rings. In the specific approach I am working on certain Fields of rational function also appear as localizations of specific rings. What I find interesting is that the Weyl-Algebra is more or less omnipresent at the Quantum level as it describes the relationship of space and momentum / time and energy; I think the reason that we see few applications of rings in physics is not that there are none. It's that we haven't explored it well enough yet.
Dude you are on fire, I have deep respect that you are self learning all these advance math subjects. Even I want to study algebraic topology and Riemannian geometry. But Unlike you I take longer to Grasp the subject bit slow learner.
As a math student , at first I HATED abstract algebra for how ,well, abstract it was. But step by step I grew to love Group Theory,then Ring theory, then Number theory..I never took the galois class since I focused more on applied math after the 3rd year but hell, if you like brain puzzle exercises I think these classes have some amazing proofs and exercises and will definitely strengthen your math logic
You just gotta get around the twist of if, i.e. of abstraction. Once done, everything falls into place. The saying goes: once you know some, you know them all.
That was my textbook! Gallian is one of the most accessible abstract algebra textbooks, and I always recommend it. Students who use Dummet & Foote or other in-depth textbooks tend to really struggle with the course. The thing I found most interesting about the course was how you could prove really basic facts like the uniqueness of identities or inverses, and how 0*x=0 for all x in any ring. It really just showed how fundamental the whole subject was. One point of nitpick, being closed under the operation is more important for *sub*groups and *sub*rings. Groups and rings will actually be closed by definition, and typically when a text introduces a new group, it is just assumed the operation is closed. When proving the base operation is well-defined (which is only done for cosets in this text), closure is important to prove, but that is not done for the normal addition or multiplication of which we are familiar.
@@EpicMathTime Makes sense. I plan on using it for my thesis because it’s a great source, but I tried teaching myself with it and it was incredibly dense. My experience with Gallian was that I continued reading it even after the class, and I ended up completing almost all the (non-sage) exercises.
Ah now I understand (vaguely) what the general idea of Abstract Algebra is. I still, however, have no clue what Topology is beyond squishing coffee cups...
Part of it is about limits. Really. When you do limits in 1D, you are talking about deltas and epsilons, in other words about intervals. In 2D, circles, 3+D, balls. But, actually, you are talking about subsets within subsets within subsets, etc. You are taking about sets that have a certain structure to them. That is part of what topology is about.
Physics Major here. In my university it was a required course sequence (Algebra 1 was basically the topics in your book. Algebra 2 was linear algebra---revisited. Algebra 3 was MORE linear algebra). Almost all physicists hated at the time. Then they took Quantum Mechanics and were like "OHHHH! So that is what that is!". Gotta say, one does come to appreciate motivational examples
I really love these videos, despite having covered these topics already, it's made me interested in reading the book(s) suggested (and it gives a new view of these topics!) Not that I'd ever really end up using them much outside of personal interest (bar teaching them to people, when I'm lucky enough to), but hey, that's not a bad thing either(!)
This is awesome! I'm an incoming transfer student to UCB and I hope to take a few of these courses. Math is beautiful, but I'm going CS for those hard job skills. It's inspiring to see someone who is taking on Mathematics of their own accord! The learning is never done!
All your impressions about Abstract Algebra are really impressions about Gallian's text. I would say this is one of the easier texts, in part because it underemphasizes vector space examples and overemphasizes number theory examples. I also learned Abstract Algebra from this text since I couldn't attend lecture due to a schedule conflict. It was easy to learn the basic ideas of Abstract Algebra. It is a good choice for a text if you don't have much in the way of prerequisite coursework. Also, the quotes are super fun.
When he said you just need basic proofs and number theory i thought I was set since we do those in highschool here. On the proof side of things id be set but yeah that number theory stuff was a little ahead of what I've done so far in school lol
I had to learn how Galois fields (finite fields) work (and some related things) in about 1-2 weeks to optimize a combined cipher core and as an EE, they kinda broke me. It is interesting and was able to directly see why it is useful but my timeframe was just way too short.
I’m doing a directed study in algebraic topology right now with my professor and he’s using that textbook as well. The lecture series by Pierre Albin pair very well with the study too, so consider checking those out.
This Video was so good, I screenshoted parts of it to look at again. Thank you so much for sharing your experience and not gating yourself behind a paywall. =]
I almost failed algebraic topology. I have 0 geometric intuition and most of the "proofs" were some drawings of some stuff and you were supposed to understand that. My "favourite" one is the proof for the commutativity of the higher order homotopy groups. Idk how rigorous and understandable the Hatcher is, but I had a lot of problems with this course.
Great video! I really find it quite inspiring that you put have put down so much time and effort into learning abstract algebra and analysis despite not being fascinated by pure math or having to take it for a class. I don't really understand how you found the motivation to study without getting distracted. I personally love pure math, and a lot of the time it doesn't take effort for me to sit down and work on it. But even though I hate to admit it I realize that I'll probably never in my life have the energy to sit down and try to learn how a biological cell works (even though that's one of the most fasciniating things in the universe), because my brain would rather watch youtube and I don't have the pressure of an exam to make me study.
Im taking a linear algebra course this summer and I've already taken a discrete math course so this seems like a fun thing to do this summer after I finish my condensed courses
I imagine that a lot of computer scientists take this class because it's the basis for many topics in theoretical computer science, e.g., programming languages. Category theory is a hot topic, alongside type theory, which heavily relies on information in abstract algebra.
Well, I’m a CS student and we had to take some brief introduction to abstract algebra but it was embedded into one module alongside other discrete math topics like some set theory, relations, general proof-writing, some combinatorics But I wouldn’t be so sure that many really need that. You mean theoretical computer scientists need that which makes sense but not other types of computer scientists
Hi Zach, would it be possible if you gave a summary of the topics you have taught yourself up to now and the things you're planning on learning in the future as well as what order all these should be learnt in?
Great vid! I am doing exactly the same thing but using a textbook on Abstract Algebra by John B. Fraleigh (7th edition). Got it really cheap as a used copy from eBay.
I came across this today, and surprisingly interested in it. I took Language, logic and discrete mathematics, and i see it bridges a little towards this. Definitely gonna self-learn it
You are sooooo amazing!!! I wish i could do what you do. I always loved math but never had enough determination to stick through and just learn it. I'm so jealous!
In all fairness, given his graduate background and the fact that he's already taken tons of math courses, teaching oneself abstract algebra with the Gallian text is actually pretty easy. Gallian is one of the most accessible abstract texts out there, and he organizes it really well. I would definitely encourage anyone (with some background in math) to read it!
In my university, only math majors are required to take abstract algebra. Not the physics or CS majors. This course/type of math is hardcore math (imo), even though I'm required to take it, I do like it.
I was a math major and chose that my junior year. 4 semesters of like 4-5 math classes each. It was so tiring, especially Abstract Algebra and Differential Equations.
Abstract algebra is tightly related to type theory and category theory, a field that makes using computer data structures more composable. Its one of the pillars of typed lambda calculus.
I just finished my algebraic structures course, and I gotta say the class equation was kind of deep o.O. Other than that it was pretty manageable as you were saying, i might go back to revisit some of the stuff that i paid less attention to
Just for example, at my undergrad, the course on abstract algebra (which covered groups, rings, and fields) was called "Group Theory and Introduction to Proof". It was a required course for math majors and directly followed the introductory calculus sequence.
I just finished up my abstract algebra course in school- it looks like we only went up to Ch.11 (of your text book) in terms of topics covered. I might snag this book myself to brush up and study the rest of the topics!
Hatcher is proving to be quite tough lol. But I've already completed general topology from Munkres so I'm making it through so far, difficult but I don't feel like I'm missing any pre-req's (yet).
@@aryamanmishra154 I was thinking about self-studying topology by Munkres and then going to Hatcher's book. Do you recommend anything in particular I should study regarding topology to prepare for Hatcher?
@@harisserdarevic4913 You need to know general topology very well before studying algebraic topology. Munkres should be more than enough because it prepares the gen top. for alg top. and functional analysis (Arzela-Ascoli, compact-open topology (which is important for homotopy theory too), etc.) but I strongly recommend learning it thoroughly. It certainly is okay to crack open Hatcher and parse through it to see which concepts are relevant, and you will learn some even in parallel with Munkres. At the end of Munkres is a chapter on the fundamental group and covering spaces. This is the starting point for algebraic topology, and you should try to understand it in as much detail as possible. I'll mention Lee's Introduction to Topological Manifolds which is a beautiful introduction to topology of manifolds, general topology, and some algebraic topology. I highly recommend it. Good luck!
Abstract álgebra is quite important in physics. Quantum mechanics deals with operators, hermitian matrices and that staff. In the Standard model, symmetries play a fundamental role in some particle Properties
i don't know what it's like in the us, but i study CS and all those things except for vector spaces were in my Discrete Mathematics Exam(groups, rings,fields,modular arithmetic,proofs,and graph theory), while i also took another exam which is required for Physics Majors that is called Geometry(vector spaces, matrix, and so on)
To bound off what you said a bit in the intro of the video, I study mathematical physics and this was a required course. This tends to be especially useful when discussing quantum field theory or even just problems in condensed matter theory, like topological materials and the sort.
Many students can do so as the perquisite is Matrix Algebra, which can be self-taught while taking abstract, if the student has not taken Linear Algebra
Who am I to tell you which algebraic topology book to read first? But, honestly, Hatcher's might not be a good idea, specially because of the insane amount of details the author shows. Sometimes details get in the way of getting a clearer understanding of a topic. I would recommend you to take a look at other books before tackling this one with all of your efforts. Maybe Munkres', Bredon's or even Greenberg's (my personal favorite) will help you as a guide to follow while you go for the details and pictures from Hatcher's. Also, I'm so glad to see you enjoying some hardcore math, it really is a pleasure to study this kind of mathematics, I guess that's the only motivation math majors have!
I practically forget lots of stuff I learned on courses so going back on it through books is nice. Or studying something before course is nice. Relearning is great.
Next up, non-linear dynamics (while I finish the algebraic topology book).
Then if you want to get the clock, 'Don't be a jerk' shirt, floating globe, or other STEM items like those you see in this video check out stemerch.com :)
Look I'm here
Oh my god I can't wait to see your algebraic topology video. I did a graduate algebraic topology course. I consider myself a good teacher and I would still consider teaching algebraic topology to math students to be one of the difficult subjects.
I can't imagine a video that will give a general audience an idea of what algebraic topology but I can imagine that you would be up to the task of making one.
I can't wait to see how you'll do it because I really have no clue what it would even take.
Been a long time since I algebraic topology so I may remember it as being harder than it is.
I was thinking of algebraic geometry the whole time I wrote this comment. Then I heard you describing algebraic topology in the video and it clicked.
I'm sure I'll be impressed with your explanation of algebraic topology but I won't be as impressed as if it had been algebraic geometry (I still think you can do it).
Wait... how are you able to post a comment 4 days before this was uploaded?
My only issue with Hatcher is his covering maps section. Why would you ever have a nonsurjective covering?! Other than that it's phenomenal lol
I can't believe how well this was timed. Ten minutes ago, I finished my final exam for Algebra covering this exact same textbook. I sat down to take a break and watch some RUclips, and this video happens to pop up in my subscription feed. How serendipitous!
RUclips spies on you, just as Facebook does.
And I was just going to start this book ...I m just surprised ⊙.☉
Google can read our minds now
Yo googled this texbook before. That's why youtube feed showed this video
@@victorcossio “subscription feed”, it showed up because he was subscribed, not due to him searching the book
YES I NEEDED THIS, I’m currently on this course and this is the exact book we are using. Going to enjoy the video.
Very fortunate happy for you.
Enjoy.
Gallian is one of the better books out there.
Yeah. I’m currently trying to teach myself about groups so it’s very helpful.
I was wondering how do you feel about kids in 11th and 12th grade. Do you feel like dumb kids can't do 11th maths or you feel if i would have been there now, i would teach students this way or that way
I feel the need to point out that the operations in a ring are not any more specific than that of a group. The addition and multiplication of a ring are just names, they are not specific operations. The "addition" of a ring can be any operation in the same way that the binary operation of a group can be anything. The "multiplication" of the ring interacts with the addition of the ring through the distributive property. In other words, the terms addition and multiplication in a ring only specify their role in the distributive property relative to each other, not what the operations innately are.
For example, if we have a nonempty set X, and we look at its power set P(X), the set P(X) with the operation of symmetric difference as "addition" and set intersection as "multiplication" form a ring, simply because intersection distributes over symmetric difference. Those operations aren't addition and multiplication in any innate sense, only relative to each other, and an operation serving as addition in one ring can serve as multiplication in a different ring.
Thank you, this bugged me a lot when it was said and I was hoping someone in the comments had pointed this out (and you did it very clearly and concisely imo).
We like keeping things abstract after all, and "addition" and "multiplication" are more or less just mnemonic in nature.
I think what's meant is that the axioms for a ring are more restrictive than for a group. Any ring is an abelian group under the addition operator, but there's the additional constraint given by the multiplication and the distributive law
When I went to college, I failed abstract algebra, made a C in intermediate analysis, made a C in differential equations and dropped complex analysis. I ran out of that as fast as possible and switched my major to computer science. Then the those maths started to click in my mind. I was using some concepts in there to run programs. Was it efficient? Probably not! otherwise they would teach this math in CS. Was it fun? YES!! Truth be told, I don't think I was suppose to start complex analysis in my junior year.
Stupid thing is that I bought all the math books I would need for the major. So I grabbed a book in 2020 and started reading it. It was so simple for me to understand the concepts. I feel stupid that all of this didn't click back in the day. I don't know why I didn't understand it. Although at the time I did take a huge loan out for the first time Then I moved to a cheap apartment close to the college that had bedbugs. It kept me up at night and made me like an OCD person looking at everything that moved on the wall so I wouldn't get bit. I would do rituals to clean my clothes and books so I would not bring them with me to the class. I realized I would have no money by my senior year and I didn't ask for help.
I should have applied for a full ride scholarship. My grades were pretty great before the decision to move to that other apartment. The problem is my parents never had the guidance to show me. I should have talked to someone at the school. I was young and thought that I could do everything alone. That my problems were my own to solve.
I am a different person now. I realize that not everybody is out to see me fail. If I just go out and ask for help there will be people willing to help. My childhood goal was to dual major in physics and mathematics. Then get a PhD in physics and sit the rest of my life in a research facility somewhere. I learned that my life path isn't set in stone it will change. For some reason, starting a plan in double major physics and mathematics, failing and switching to computer science has made me actually pretty good at data science. The stars aligned for me.
what book did u grab?
Oof that sounds stressful, I'm glad it worked out.
I wish you success
Awesome 👍
Im kind of in the same position right now, with the whole switching from math/physics to cs thing
at 2:17
15x + 57y = 1 has no integer solutions and it can be shown with basic algebra (no modulo operations needed):
15x + 57y = 1
3*5x + 3*19y = 1
3*(5x + 19y) = 1
5x + 19y = 1/3
Since if x and y are integers, then 5x+19y must be an integer. However we see above that 5x + 19y = 1/3, a rational number but not an integer, giving us a contradiction.
These videos of self learning are really original, and gives a rare opportunity to hear about this from a students/ non expert. Well done!
Thanks!
Abstract algebra is awesome dude. Especially the connections with number theory.
Thanks Itachi!👍
I love the connections with topology, I'm studying schemes now.
It's not that the operations for a ring HAVE TO be addition/subtraction/multiplication in the usual sense; they can be anything, too (with some requirements on how they interact). It's just that we CALL them those names because it's something we're familiar with, and they behave largely in the same way.
+1 - In the context of rings, no specific binary operation is innately addition or multiplication. Two operations are only called addition and multiplication with respect to each other.
Well, sometimes you gotta simplify things.
@@ivarangquist9184 It's not a simplification. It's conceptually the same as how operations on groups was described.
I was gonna say, this is not how I learned this. We literally used operation "star" all the time 😂
This semester I had real analysis so I thought how about I go over the course myself during winter and boom I got a notification that you made a video on real analysis. This summer I thought of self studying abstract Algebra, and then I got a notification of this video.
Considering I have topology next semester I wouldn't be surprised to get a video on that pretty soon
I'm a fan of your Zack Stars Himself channel. So I checked this out. I'd just like to commend you for being so relatable and funny in your other channel while being so mind blowingly academically gifted. It's rare to see someone with such diverse intelligence.
@jow3871 Exactly. I couldn't have said it better. This is an awesome guy.
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Abstract Algebra was my favorite math class by far. It was difficult for me, way harder than real analysis but the intuition you get from it was amazing. Like you are doing math from the ground up. Axioms and proofs to describe all the numbers. Nothing is off limits. Everything in Abstract Algebra is built off of proofs, definitions and axioms. It was pretty cool
Analysis should be the same way. A good analysis text starts with the axioms for the real numbers or a construction using the rationals (and proves this construction has those properties) and everything else is developed from there. It's not really different than abstract algebra in terms of it building from the ground up. It just starts from different axioms and of course naturally develops different definitions.
If you are looking for the absolute highest quality of rigour, try set theory, category theory, or model theory. I'm studying set theory right now and you would not believe the things you can construct assuming the axiom of choice. We had to construct a set in R^2 using transfinite induction such that given any distance, there was exactly on pair of points in the set that are that distance apart. Things like this make me question math because such a set is completely inconceivable. Then you assume something as harmless as a choice function and you get well orderings of uncountable sets, Banach Tarski, and transfinite induction. You can do induction over the real numbers!!! I suppose at the end of the day, all truth in mathematics is conditional, and in order to make any statements about what is actually true you must rely on faith.
I don't know what's different in this video, you seem... more understanding of the topics you're discussing and more in depth which, in turn, feeds the passion with which you're explaining as well. Can't wait for more of this content!
My abstract algebra class was my favorite undergrad math class. Love how it has applications in physics as well.
Application of algebra to physics answers a lot of disapproved theories. Love this.
Yeah, physics uses a lot of the rotation groups, which are subgroups of GL(R^3). These can be more concisely modeled with quaternion multiplication. Kinda cool
This book is a beautiful gem. Love the motivational quotes in the book too. Cheers for the JR Tolkien reference preface.
I went through so much brain torture to get a B+ in that class... still was one of my favorites, though I never used this information ever again in my life and I doubt I ever will
It's an awesome subject! The unsolvability of a general quintic is my favorite part.
I still can't figure that one out.
I would say, depending on the textbook you're using, linear algebra may be a more important prerequisite than you've suggested here. Michael Artin's textbook, for instance, emphasizes matrix groups (such as the group of invertible nXn matrices with real entries) as a key class of examples, especially of noncommutative groups (and similar sets of matrices as examples of noncommutative rings). Matrices also provide a particularly nice example of group action. I agree that it's not necessarily a true prereq, but I'd advise that people should be cautious depending on the choice of textbook/the professor.
Yeah I should probably emphasize more but pretty much everything I say in these videos is in reference to the specific textbook I went through, I don't really have knowledge about any others (or how it's taught in university).
As a mathematically inclined physicist, I am extremely wary of abstract textbooks that use too many matrices, as it perpetuates the unfortunate truth that most physicists don't understand the distinction between a group and a representation of it. I really enjoy the Durbin book, it almost specifically tries to stay away from familiar examples, to make sure you don't accidentally impose too much structure onto the abstract concepts.
@@Saztrah Artin's book is fantastic. It covers very little representation theory, and it's motivation for using the general linear group and it's subgroups a lot are because they are very useful for many examples. All of the theory (most at least) of linear algebra is developed within the text itself. Linear algebra (with vector spaces not just Rn) is quite useful in abstract algebra, especially when you study field extensions and Galois theory. A number of proofs about field extensions can be simplified by considering field extensions as vector spaces over their base field. Then you can use powerful tools such as the dimension of the vector space. In fact, fields and vector spaces have major connections because given some field F the way a vector space V over F is defined is equivalent to V being an Abelian Group and there being a ring homomorphism from F to the group of endomorphisms of G. So in a sense when we multiply V on the left by some element in F, we are doing so in a way that preserves the structure of V.
Anyways, the motivation is there to study vector spaces if you are interested in abstract algebra. The definition of vector spaces as I wrote in the last paragraph, is a very concise way of defining vector spaces using the language developed in abstract algebra. You develop intuition for objects such as fields and groups and you can now wield that intuition for studying vector spaces, and vice versa.
It's use of linear algebra doesn't really relate to representation theory whatsoever, although I will say, it is somewhat difficult to distinguish a group from its representation. A representation is a homomorphism from a group to the general linear group of a vector space (that is, all automorphisms of the vector space, or in other words, the invertible matrices in the vector space, although this is dependent on your basis). What this means (by the first isomorphism theorem for groups) is that the group factored by some normal subgroup (the kernel of the homomorphism) is isomorphic to some subgroup of the general linear group, ie. as far as group structure is concerned, they are the same. I like to think of this like the group in some way captures the symmetries of the vector space.
Sorry for the ramble.
This was by far my favorite class in undergrad. I used the same book. Just candy for the brain. I think some ideas resonate with some people and others with other people. For whatever reason, I just really loved this class. It was two semesters long and covered almost the entire book. I wish I had time to do what you just did again...
I found "A Book of Abstract Algebra" by Charles C. Pinter to be a good, concise introduction to abstract algebra. It's short and easy to understand, even for a self-learner like me whose only previous exposure to group theory was in an inorganic chemistry class. I might look into "Contemporary Abstract Algebra" to expand on the topic a bit.
Oh man you have to do Galois theory next!
yes
That's where things get intense, and all the different subjects come together.
Galois theory is so beautiful
Pretty sure it’s include in the book
Hey what is your next book on your " Reading Textbooks I never read in School " list. Please do share as most people who aren’t in maths degree would not get a feel for it.
The /sci/ wiki has a really good list of textbooks for many different fields.
I'm doing algebraic topology by Hatcher and I'm going to give non-linear dynamics by Strogatz a try. I've been craving some material with direct applications to science and engineering (that's actually discussed thoroughly in the book). After real analysis and this book I haven't gotten much of that lately lol. The next book after this one I think would be galois theory but I don't want to do that quite yet.
@@zachstar You are goanna love Strogatz's book.
@@cogitoergosum2846 Yeah Strogatz is a professor at my college. I'm planning on taking a course by him one of these years cause he's just such a great prof.
@@jeanconnery3740 Oh. How lucky you are! Being lectured by one of the best math professors out there!
Zach, your intellectual curiosity is inspiring and helps keep me motivated as I work through my degree. I'm sure many others feel this way too- thanks for the great content!
Technically, number theory follows from abstract algebra. Those “number theory” concepts are concepts usually first learned in abstract algebra and not the other way around. 🙂
Linear algebra is also usually the first mathematics course where you learn and practice most of your proof skills, and this is because abstract algebra often is taken before the more difficult subject of real analysis.
I disagree -- number theory provides motivation for large swaths of abstract algebra, and was investigated earlier historically. That being said, most students today don't get exposure to the kind of elementary number theory commonly seen in competition/olympiad circles, so abstract algebra is often the first class a math major sees some of those topics, but it's an order of presentation that's only made out of necessity.
@@andresmath I concur about the historical development of the subject, but considering the modern approach to a course in Number Theory would leave most people quite helpless without abstract algebra (for algebraic number theory) or Complex Analysis (for analytic number theory), I would not suggest a course in Number Theory to precede abstract algebra. I also concur that some elementary number theory provides motivation for many topics in abstract algebra, which is why you are typically introduced to those topics within the abstract algebra course. However, the amount of elementary number theory required does not justify an entire course in Number Theory, which would surely cover much more and again would require the concepts of abstract algebra or complex analysis. Without the tools of abstract algebra or complex analysis, many of proofs in Number Theory would be inaccessible to most beginners.
EDIT: I did do some additional research and found a “course in Number Theory without abstract algebra.” After reviewing the course materials, I can say it depends on one’s intentions here. If you never intend to take abstract algebra or a more advanced number theory course, then this might be the course for you. I should note that the course is not completely without abstract algebra. It just teaches those concepts within the course. So, I suppose it depends on one’s goals.
EDIT2: I should also be clear that if one is interested in Number Theory, a course in abstract algebra is merely the start of that journey that provides the tools needed for that journey, much like a course in elementary algebra is necessary for trigonometry and eventually calculus.
@@stokhosursus well yeah but also there's a difference like you said between elementary number theory and Analytic number theory Analytic is the much more advanced one which abstract algebra would help in but elementary you don't need much background besides a good foundation in arithmetic
I love the turn this channel has taken lately. You inspire me to study all the things!
Operations in rings don't actully have to be defined. We just need two operations connected by the distributive property, but its helpful to think and notate them as addition and multiplication.
Zach, what was your schedule when doing this book? Did you commit to a certain number of hours per day or per week, or a certain number of days per week? Did you sit down at a set time of day to do it? Were you working a job with regular hours?
I'm very interested in how you work a project like this or your previous "trips through the textbook", which is very long term but can be highly variable in page-to-page challenge level, into the rest of your life.
I recall in your real analysis project you discussed dealing with really intractable problems that could take hours of effort spread over days or weeks. But how do you keep up the discipline to return to the book day after day and not start letting it slip and eventually lose the continuity you need to finish?
I did every single exercise of this book, it’s a fantastic introductory text on the subject.
how long it took ?
@@RiteshArora-s9x Approximately 9 months since I was taking other courses as well.
@@爸爸到底-s9x how many chapters you did ? it takes me 3 or 4 hours one one hard problem
@@RiteshArora-s9x Chapter 1 through Chapter 30, and the last chapter. For this particular text a complete solution manual is readily available online, so it’s well-suited for self-learning.
I like how his shirt says
"Don't be a jerk"
Thank u
It was supposed to be my comment😂
How?
@@0mon0zz The first derivative of displacement w.r.t time is velocity, second derivative is acceleration and third derivative is called "jerk". I kid you not 😂
@@feynstein1004 Whaaaaat? lol
I loved abstract algebra in college. We used Printers book, which is phenomenal for developing the subject in a inquiry based way.
I can’t wait to continue studying algebra in grad school
Abstract algebra is very important to computer science-especially in its relation to category theory. Category theory might be something interesting for you to learn!
Congrats on 800K subscribers!
Thanks!
Lol your timing is impeccable, I just started the task of learning algebra from Dummit & Foote like 2 days ago
Dummit & Foote is much more dense and thorough than this book. You're done with most of what you see here (excluding the Galois theory briefly mentioned via the insolvability of the quintic, which is later) in the first 9 chapters, and the coverage of modules and then vector spaces in chapters 10 and 11 is very abstract. It's a great book, but be prepared!
@@JPK314 wicked, thanks for the advice!
Yeah I would encourage you to have a more accessible text like Gallian (the one in the video) on standby when Dummit & Foote gets too incomprehensible. I've had countless friends who hated abstract with Dummit & Foote, but loved it with Gallian.
@@JM-us3fr yeah in general i've found sticking to one book is cumbersome (i used 3 books to get through my graph theory course), so your comment is good reaffirmation of that, much appreciated
@@9erik1 yea I used many books throughout that course. Try to get as many different problems in
Thanks so much for igniting the flame for new knowledge! I am nothing into mathematics, but seeing others go through books and learn so much in diverse topics really inspires me!
I just want to add to your separation of Groups Rings/Fields (which I generally agree with):
I am working in the field of calculating so called Feynman integrals. They are integrals that appear in Quantum Field Theoretic processes and are basically impossible to evaluate numerically (they love to diverge) or analytically (only possible in easy cases and too time consuming). Their calculation is crucial for more accurate calculations of elementary particle processes.
However, they can be calculated using algebraic methods that are based on the Weyl-Algebra (basically polynomials in n variables and there respective derivatives) structure of the family of all those integrals. The relevant thing here is it's ring structure and both it's ideals and it's quotient rings. In the specific approach I am working on certain Fields of rational function also appear as localizations of specific rings.
What I find interesting is that the Weyl-Algebra is more or less omnipresent at the Quantum level as it describes the relationship of space and momentum / time and energy; I think the reason that we see few applications of rings in physics is not that there are none. It's that we haven't explored it well enough yet.
bro it's so awesome how you teach yourself advanced classes and show how you do so. Keep up the great work!
Dude you are on fire, I have deep respect that you are self learning all these advance math subjects. Even I want to study algebraic topology and Riemannian geometry. But Unlike you I take longer to Grasp the subject bit slow learner.
As a math student , at first I HATED abstract algebra for how ,well, abstract it was. But step by step I grew to love Group Theory,then Ring theory, then Number theory..I never took the galois class since I focused more on applied math after the 3rd year but hell, if you like brain puzzle exercises I think these classes have some amazing proofs and exercises and will definitely strengthen your math logic
You just gotta get around the twist of if, i.e. of abstraction.
Once done, everything falls into place.
The saying goes: once you know some, you know them all.
Galois theory is amazing. Take a book which focusses on the insolvability of the quintic like the theory was developed. It will blow your mind
Man this is cool! I'm getting this textbook now thank you Zach
I cited this book for some math research I just completed. Super awesome to see an overview of the content, you have great summarizations!
Thanks Zack, you've given me motivation to finish my abstract algebra course!
That was my textbook! Gallian is one of the most accessible abstract algebra textbooks, and I always recommend it. Students who use Dummet & Foote or other in-depth textbooks tend to really struggle with the course. The thing I found most interesting about the course was how you could prove really basic facts like the uniqueness of identities or inverses, and how 0*x=0 for all x in any ring. It really just showed how fundamental the whole subject was.
One point of nitpick, being closed under the operation is more important for *sub*groups and *sub*rings. Groups and rings will actually be closed by definition, and typically when a text introduces a new group, it is just assumed the operation is closed. When proving the base operation is well-defined (which is only done for cosets in this text), closure is important to prove, but that is not done for the normal addition or multiplication of which we are familiar.
Yeah, I think D&F is a first year graduate text, not an undergrad text. At least in most situations.
@@EpicMathTime Makes sense. I plan on using it for my thesis because it’s a great source, but I tried teaching myself with it and it was incredibly dense. My experience with Gallian was that I continued reading it even after the class, and I ended up completing almost all the (non-sage) exercises.
@@EpicMathTime what is your opinion on artin? I found it quite digestible in my intro algebra course.
Teach yourself some category theory and be amazed how interconnected mathematics is.
Ah now I understand (vaguely) what the general idea of Abstract Algebra is. I still, however, have no clue what Topology is beyond squishing coffee cups...
Part of it is about limits. Really. When you do limits in 1D, you are talking about deltas and epsilons, in other words about intervals. In 2D, circles, 3+D, balls. But, actually, you are talking about subsets within subsets within subsets, etc. You are taking about sets that have a certain structure to them. That is part of what topology is about.
Physics Major here. In my university it was a required course sequence (Algebra 1 was basically the topics in your book. Algebra 2 was linear algebra---revisited. Algebra 3 was MORE linear algebra). Almost all physicists hated at the time. Then they took Quantum Mechanics and were like "OHHHH! So that is what that is!". Gotta say, one does come to appreciate motivational examples
I really love these videos, despite having covered these topics already, it's made me interested in reading the book(s) suggested (and it gives a new view of these topics!) Not that I'd ever really end up using them much outside of personal interest (bar teaching them to people, when I'm lucky enough to), but hey, that's not a bad thing either(!)
If you make more of such Pure Mathematics reviews, Imma gonna get a Masters in Mathematics
This is awesome! I'm an incoming transfer student to UCB and I hope to take a few of these courses. Math is beautiful, but I'm going CS for those hard job skills. It's inspiring to see someone who is taking on Mathematics of their own accord! The learning is never done!
All your impressions about Abstract Algebra are really impressions about Gallian's text. I would say this is one of the easier texts, in part because it underemphasizes vector space examples and overemphasizes number theory examples. I also learned Abstract Algebra from this text since I couldn't attend lecture due to a schedule conflict. It was easy to learn the basic ideas of Abstract Algebra. It is a good choice for a text if you don't have much in the way of prerequisite coursework. Also, the quotes are super fun.
More than making me want to read and solve this book. This video made me curious on depth of maths. Thank you for this ❤️
As someone who only has a high school level math ability, this video went way over my head 😅
When he said you just need basic proofs and number theory i thought I was set since we do those in highschool here. On the proof side of things id be set but yeah that number theory stuff was a little ahead of what I've done so far in school lol
@@nope110 the prerequisite is high school level, but not the actual subject lol
I had to learn how Galois fields (finite fields) work (and some related things) in about 1-2 weeks to optimize a combined cipher core and as an EE, they kinda broke me.
It is interesting and was able to directly see why it is useful but my timeframe was just way too short.
That sounds brutal lol. Galois fields came up for one chapter at the end of this book as just an introduction, can't imagine just diving into that.
abstract algebra feels like a deep dive in some really random common stuff you never tought about too much and suddenly DAMN, SOMETHING REALLY COOL
You look great man, amazing content, always blows my mind
I am really amazed by your dedication. I do like algebra, geometry and analysis, but I am not as dedicated as you are
Super fascinating video! Thanks for making this!
I wish you made this video a year ago when I took abstract, i struggled hard but made it out alive lol
I’m doing a directed study in algebraic topology right now with my professor and he’s using that textbook as well. The lecture series by Pierre Albin pair very well with the study too, so consider checking those out.
This Video was so good, I screenshoted parts of it to look at again.
Thank you so much for sharing your experience and not gating yourself behind a paywall. =]
4:08 This look a lot like Rubix Cube algorithms
6:20 Oh there we go
I almost failed algebraic topology. I have 0 geometric intuition and most of the "proofs" were some drawings of some stuff and you were supposed to understand that. My "favourite" one is the proof for the commutativity of the higher order homotopy groups. Idk how rigorous and understandable the Hatcher is, but I had a lot of problems with this course.
10:25, constructible numbers, that's a concept I'd been looking for! Very interested. 😊
When I listened that class, I understood the meaning with making images in my mind at first and then I studied the details. It was very fun for me.
Great video! I really find it quite inspiring that you put have put down so much time and effort into learning abstract algebra and analysis despite not being fascinated by pure math or having to take it for a class. I don't really understand how you found the motivation to study without getting distracted.
I personally love pure math, and a lot of the time it doesn't take effort for me to sit down and work on it. But even though I hate to admit it I realize that I'll probably never in my life have the energy to sit down and try to learn how a biological cell works (even though that's one of the most fasciniating things in the universe), because my brain would rather watch youtube and I don't have the pressure of an exam to make me study.
5:44 I'd say they are less strict since every field is a ring but not every ring is a field
You should do a video on the book concrete mathematics by Knuth, Graham, and patashnik.
Good! Thank you for the review ... I own this book and now maybe I'll give it a read
Im taking a linear algebra course this summer and I've already taken a discrete math course so this seems like a fun thing to do this summer after I finish my condensed courses
I imagine that a lot of computer scientists take this class because it's the basis for many topics in theoretical computer science, e.g., programming languages. Category theory is a hot topic, alongside type theory, which heavily relies on information in abstract algebra.
Well, I’m a CS student and we had to take some brief introduction to abstract algebra but it was embedded into one module alongside other discrete math topics like some set theory, relations, general proof-writing, some combinatorics
But I wouldn’t be so sure that many really need that. You mean theoretical computer scientists need that which makes sense but not other types of computer scientists
A very important video for my general math knowledge. Thanks for making this!
Hi Zach, would it be possible if you gave a summary of the topics you have taught yourself up to now and the things you're planning on learning in the future as well as what order all these should be learnt in?
Great vid! I am doing exactly the same thing but using a textbook on Abstract Algebra by John B. Fraleigh (7th edition). Got it really cheap as a used copy from eBay.
I came across this today, and surprisingly interested in it. I took Language, logic and discrete mathematics, and i see it bridges a little towards this. Definitely gonna self-learn it
You are sooooo amazing!!! I wish i could do what you do. I always loved math but never had enough determination to stick through and just learn it. I'm so jealous!
In all fairness, given his graduate background and the fact that he's already taken tons of math courses, teaching oneself abstract algebra with the Gallian text is actually pretty easy. Gallian is one of the most accessible abstract texts out there, and he organizes it really well. I would definitely encourage anyone (with some background in math) to read it!
In my university, only math majors are required to take abstract algebra. Not the physics or CS majors. This course/type of math is hardcore math (imo), even though I'm required to take it, I do like it.
I was a math major and chose that my junior year. 4 semesters of like 4-5 math classes each. It was so tiring, especially Abstract Algebra and Differential Equations.
Thank you for this fantastic video on teaching yourself abstract algebra.
Abstract algebra is tightly related to type theory and category theory, a field that makes using computer data structures more composable. Its one of the pillars of typed lambda calculus.
I just finished my algebraic structures course, and I gotta say the class equation was kind of deep o.O. Other than that it was pretty manageable as you were saying, i might go back to revisit some of the stuff that i paid less attention to
I am going through this exact text book right now! That's so cool.
Just for example, at my undergrad, the course on abstract algebra (which covered groups, rings, and fields) was called "Group Theory and Introduction to Proof". It was a required course for math majors and directly followed the introductory calculus sequence.
I just finished up my abstract algebra course in school- it looks like we only went up to Ch.11 (of your text book) in terms of topics covered. I might snag this book myself to brush up and study the rest of the topics!
The jump to Hatcher might be a bit steep if you're getting at it from an Engineering angle.
Hatcher is proving to be quite tough lol. But I've already completed general topology from Munkres so I'm making it through so far, difficult but I don't feel like I'm missing any pre-req's (yet).
Hatcher is hard but extremely fun
@@aryamanmishra154 I was thinking about self-studying topology by Munkres and then going to Hatcher's book. Do you recommend anything in particular I should study regarding topology to prepare for Hatcher?
@@harisserdarevic4913 You need to know general topology very well before studying algebraic topology. Munkres should be more than enough because it prepares the gen top. for alg top. and functional analysis (Arzela-Ascoli, compact-open topology (which is important for homotopy theory too), etc.) but I strongly recommend learning it thoroughly. It certainly is okay to crack open Hatcher and parse through it to see which concepts are relevant, and you will learn some even in parallel with Munkres. At the end of Munkres is a chapter on the fundamental group and covering spaces. This is the starting point for algebraic topology, and you should try to understand it in as much detail as possible. I'll mention Lee's Introduction to Topological Manifolds which is a beautiful introduction to topology of manifolds, general topology, and some algebraic topology. I highly recommend it. Good luck!
Taking this class in the Fall, this video made me less anxious haha.
Abstract álgebra is quite important in physics. Quantum mechanics deals with operators, hermitian matrices and that staff. In the Standard model, symmetries play a fundamental role in some particle Properties
um, it's an unrelated question but can I know what is that spinning globe thing you have.
Zack star, this is what I like about algebra it's easy and some drains your Brain but it's worth it.
SYBERMATH DO YOU WATCH HIM?
Poor brian...
@@aashsyed1277 yep
@@cleanseroftheworld lol. Am not saying it's very difficult but for some you need a simple trick to do and that is what I normally do
i don't know what it's like in the us, but i study CS and all those things except for vector spaces were in my Discrete Mathematics Exam(groups, rings,fields,modular arithmetic,proofs,and graph theory), while i also took another exam which is required for Physics Majors that is called Geometry(vector spaces, matrix, and so on)
To bound off what you said a bit in the intro of the video, I study mathematical physics and this was a required course. This tends to be especially useful when discussing quantum field theory or even just problems in condensed matter theory, like topological materials and the sort.
About to take my final for abstract in a week. I hate algebra but love analysis and this is the last course I need to graduate. Wish me luck
Many students can do so as the perquisite is Matrix Algebra, which can be self-taught while taking abstract, if the student has not taken Linear Algebra
Who am I to tell you which algebraic topology book to read first? But, honestly, Hatcher's might not be a good idea, specially because of the insane amount of details the author shows. Sometimes details get in the way of getting a clearer understanding of a topic. I would recommend you to take a look at other books before tackling this one with all of your efforts. Maybe Munkres', Bredon's or even Greenberg's (my personal favorite) will help you as a guide to follow while you go for the details and pictures from Hatcher's. Also, I'm so glad to see you enjoying some hardcore math, it really is a pleasure to study this kind of mathematics, I guess that's the only motivation math majors have!
What is the "Real Analysis" and "Abstract Algebra" of Engineering and Physics?
Nathan Jacobson's Basic Algebra I covers groups, rings and modules in its first three chapters in a very nice way
I have no idea what he is talking about but it makes me feel smart. Keep up the good work funny math man!
hes an Engineering Boi
I practically forget lots of stuff I learned on courses so going back on it through books is nice. Or studying something before course is nice. Relearning is great.
This book is lying around in my shelf, I should make time to read it
Omg! 😭😭 This class was torture. I took abstract algebra I and II 🤯 I was close to changing my major.