When you teach the concepts, you have to understand them at a more fundamental level in order to provide metaphors that everyone can understand. This comes naturally through time and practice. Then, once you've learned the concept, you don't ever "unlearn" it so it seems elementary to you. Good professors are those who understand that the things they take for granted were not always easy.
I’m glad myself and many others are finding your channel. This “in the trenches” style of mathematics grad school videos are super informative to many undergrads like myself
In an engineering vector calculus exam, there was a question to solve for the tangent plane. The examiner forgot to specify powers such that the question in fact asked for the tangent plane of a plane rather than a more complex surface like a sphere or cone. I sat there for 10 mins second guessing myself and bit the bullet and just wrote the equation that was on the exam on a blank page and got 15/15 for the question. I love trivial questions!!
it's not too bad, but you do need to take some courses on linear algebra, abstract algebra and possibly field theory, relatively elementary courses should get the job done, the problems shown weren't that difficult. All the problems were pretty standard. it's a skill obviously, nobody is born knowing this stuff
@@geometrividad7716 it's interesting honestly, because put in a sports perspective this makes intuitive sense to anyone. "why can't i kick a 60 yard punt?" well you probably could given the training, practice, and i guess in this case a limb length that would allow that motion. "why does this math look like nonsense to me?" well it won't, given enough time and practice
As a grad student in applied math/astrophysics, it's neat to see what others in doing in other grad programs. Your exams are like bizarro world versions of my exams with different content. The classic grad exam setup seems to be: the questions are quite complicated, quite a few of them were homework set examples from homework sets that took bloody ages to do, and you sort of have to have done the questions beforehand to get them right in the exams. We all get to suffer together.
Hey man! I am applying for an Astrophysics master's course. I do have basic understanding of the varied fields in it. Stellar physics, Observational, Computational, Theory of relativity. And my Maths is decent, that I am quite fine with Differential Equations. What else should I be aware of, before I join the course. Could you help me please?
@@ordenax That's about it, It's also very helpful to be good at latex as well as python/julia/matlab. A good understanding of physical mechanics and electromagnetics is also super valuable. This was the case for my programs.
@@zakaryjaynicholls9867 Yeah Physical Mechanics and Electromagnetism and even a basics of Quantum Mechanics I understand. To help with QFT. Sure, The maths would be advanced, but it is something I hope I can manage
@@ordenax In astrophysics, the math is really not that hard compared to doing graduate courses in actual math. So as long as you're good at single var calc/multi variable calc/basic ODEs/basic PDEs/numerical modelling/statistical modelling as well as numerical computing, you can probably learn what you need as you go along.
Algebra is by far my favorite subject, and this exam looks approachable, except for the field theory part. I really like that this connects to Analysis and Topology I believe we should not separate it, it's really exciting when it all makes sense finally why in school they make us do functions and equations in a certain manner because R is a field, and why we extend it to C to make it algebraically closed having them as integral domains because every field is sol, while this all happening in a topological space of collection of basis as domain and range. it's reassuring that every math discovery came out natural. I wish you a happy career wherever it leads you.
Thank you so much! Finishing calculus 2 this semester and starting Group Theory either 3rd or 4th semester (parents want me to leave off math for just Major courses in the Fall 2023). Truly shocked at how you wrap your brain around this, its like how I worked during High School Honors Geometry ("Pre-AP") but with confident whereas I was the study crammer before tests. Pursuing a Computer Science major but want to continue Math courses for a while since this was just my Freshman University year. Thanks again!
The actuarial exam compared to this is like a Ford focus to Ferrari 😅. I used to learn a lot of financial mathematics subjects mostly in the Stochastic field but Algebra is mind boggling 😅
In reality people with the passing mark on actuarial exam make 200K a year while guys who pass Algebra PhD Qual exam usually barely make any descend income and that is the fact. @@RegularSizeRick
Im using the Sheldon Alxler book for advanced linear right now. Its the first pure math course I've taken other than formal logic and Its super interesting to see how abstracted all the things I learned in my first linear algebra course were and I didnt even realize.
I mean....it's not necessarily "advanced linear algebra" it's undergraduate linear algbera. Advanced lin alg would be something like Advanced Linear Algebra by Steven Roman
Congratulations on your accomplishments with graduate level Math. Not a lot of people are skilled at proof oriented kind of Math, pure Math, etc. I used to be a Math major. I changed my major from Math to accounting. The reason why I dropped out of being a Math major was due to being bad at proof oriented kind of Math. I'm very good at computational kind of Math, but I'm terrible at the proof oriented kind of Math that pure Math majors take. Math is computational, calculation based, etc all the way from Pre-K to College Calculus and College Differential Equations, but once you finish with College Calculus and College Differential Equations, most of the Math classes you take are more proof oriented than computational, calculation based, etc. I struggled with courses such as Discrete Math, Topology, Abstract Algebra, etc. I got A's and B's in Math all throughout elementary school, middle school, and high school, and college in Math, but I did terrible in the proof oriented kinds of Math courses that Math majors take once they finish with Calculus and Differential Equations. Those proof oriented kinds of Math courses are no joke. Those proof oriented Math courses make classes like Calculus, Statistics, Intermediate Accounting, engineering, physics, organic chemistry, anatomy, pre-med, etc seem like mickey mouse classes. Congratulations on your accomplishments in life. I wish you the best.
My interests are applied math but I’m taking the required analysis courses right now… respect to anyone studying this at higher level because it is hard work
shouild i go for that shit? i wanna do computers but back in highschool i tried figuring out how neural networks worked. Then i got hooked into math. Now I feel like doing some math shit too in some combined degree. Hopefully itl help me differentiate between other junior devs and hopefully i wont get stuck churning out fucking webdev soycode. Is it interesting shit? What are some applications of the shit u learn?
@@honkhonk8009 im doing cs and math at uni. i wouldnt say pure math has an obvious application to cs. but its defo very fun and makes you stand out i reckon. itll also be much more difficult than your cs units so be prepared for that haha
@@Castellante Same. It's quite discouraging. They ended up just becoming software engineers at startups (one of them quit shortly thereafter). The pure math career path seems brutal. It's one thing to do it for the love of it but I would hedge my bets in that regard with some more directly applicable studies
@@honkhonk8009 I recently graduated with a CS degree. Take as much math as you can, so long as you enjoy it. It will only serve to make you better at your craft.
Great channel! I'm also an analysis PhD student, working with operator means in C*-algebras. My abstract algebra skills are quite rusty. :D However one problem from the test seems trivial: P2 from the ring theory part. One can use the binomial theorem in commutative rings, and if x^m=0 and y^n=0 then it follows trivially from the binomial theorem that (x+y)^{m+n}=0. For the ideal property, we already proved that sum of nilpotents is nilpotent, and the product property is even easier. If x is nilpotent with x^n=0 and r \in R arbitrary then from commutativity we get that (xr)^n=x^nr^n=0r^n=0. Seems super easy compared to your analysis exam problems :O
Undergrad here! Maybe it’s just easy bc I’m taking galois this semester, but problem 3 from field theory is also super straightforward: A we know a polynomial f has repeated roots iff deg(f)>1 and gcd(f, f’) != 1. Consider an irreducible polynomial with degree >1. Its gcd with its formal derivative must be 1 or itself. f’ is non-zero (characteristic 0) and lesser degree than f, so the gcd can’t be f and must be 1. Therefore f does not have repeated roots. This is true for all irreducible in our char 0 field, so the field is perfect
Thank you so much! Finishing calculus 2 this semester and starting Group Theory (i think if its allowed). Going to perform Group Theory for Undergrad either 3rd or 4th semester (parents want me to leave off math for just Major courses in the Fall 2023). Truly shocked at how you wrap your brain around this, its like how I worked during High School Honors Geometry ("Pre-AP") but with confident whereas I was the study crammer before tests. Pursuing a Computer Science major but want to continue Math courses for a while since this was just my Freshman University year. Thanks again!
I love your video because it can be applied to life in general. That's how scary math is... If you have the capacity to imagine more abstractions, you are scratching the itch of existence.
Thank you for sharing this! Going to be going for a Ph.D. in Mathematics where I will be studying Algebraic Geometry. This is just getting me excited, yet nervous. Thanks again!
Really interesting vids! Math undergraduate here(4th year),it is interesting compared to your Real analysis exam (i could not do much) your algebra exam is something i would be able to solve a few. Linear section problem number 2,Group Theory problem number 2....not looking good so far but in Ring theory can do problems 2,3,4, and Field theory can actually do 1,2,3,4,5 except 4 c. 4 was actually an exam problem in Galois theory (Without C).
i found your channel just yesterday and dude i am obsessed with you ...i soo relate to u (i am an undergrad student studying electronics) ...love your videos ..hope to see more of your content in future ..keep it up ..ps would really like a entire day in a life type video
I like your channel very much. As a side note: I also learn English with you! You said that you didn't pass the exam with "flying colours." In German we say "mit fliegenden Fahnen" (with flying flags). All the best!
I'm surprised tbh, i'd expect it to be harder. Especially the field theory bit given that phd level field theory is so technical. Though i imagine the marking is very strict.
this is very reminiscent of my exams, considering I took group theory and rings and fields in second year its scary how little I can recall immediately when looking back at questions like these I'm sure relearning them wouldn't be crazy difficult and it wouldn't take long to click again but its always funny looking at previous content after a break and realising how baffling it is
It's pretty funny that PhD exercises are similar to my undergraduate Abstract Algebra courses from Universidad de Buenos Aires. I can share anyone who don't believe me, the only inconvenient is that you will have to translate it from Spanish to English with Chat GPT.
Goodness! This is the kind of thing that stops me from more seriously pursuing a pure mathematics degree at any level. I have a BS in Statistics, and I picked up some things along the way, but I never got quite *this* deep into math. Whether I'm trying to read a paper from google results or the /r/math subreddit, everyone seems obsessed with talking about almost literally anything in abstract algebra terms. Groups, fields, rings, sets, magmas, abelian or not, commutative or not...I know what sets are, I know what operations are, and I know what commutativity, associativity, plus, and times are. (thanks to Google while composing this, I know distribution as well, but I never remember to count that with commutativity and associativity as an operational axiom or property.) But to me, this kind of qualifying test just looks more like a vocabulary test than a math test. I feel like if I knew the definitions of any words on that paper, or if the structures were given by operation and element properties instead of by special name, I could probably handle this. But I just cannot master the abstract algebra vocabulary, so I can't just know what identities, inverses, operations, properties, and axioms I have available when I only see the word. It's like if I try to learn Chinese. Even if I read about the radicals and practice drawing each one for a while, how will I ever comfortably remember what it means in the long term?
I'm disappointed not to see Pugh's Real Mathematical Analysis in the stack. I may be a bit biased, but I hear that's a solid work. :) Keep up the good work!
Once I looked up the definition of ker, Im, and a plus inside a circle I figured out I-3 pretty quickly. Enough for tonight. When I took linear algebra 53 (?) years ago it was pretty much all matrices. I remember having an A going into the final on the first two quizzes but then slacking off the rest of the course and ending up with a B. Took grad level Real Analysis and got a D and although I had done enough to fulfill my undergrad major that was that. Eventually did an MBA - so easy, even at a top University. Just put in the time to go through the cases thoroughly and talk in class - and always come to a decision, any decision.
Hi, I'm a fellow grad student at a different institution (who likes algebra). I can tell you how to do 5 on Ring Theory - take any unit u, and the conjugation map r --> uru^-1 is a ring automorphism. By assumption, this is the identity, so r commutes with all units. Since every nilpotent+1 is a unit, nilpotent elements must commute! The rest is easy. You know, I envy you for the fact that at your uni, your quals are not crazy hard (like they've been getting at my uni)! I wish you luck with your future research!
Love your content from Colorado ❤️ never got to pursue a math education formally, living vicariously through you. Would love to see what you’re curious about research wise
Surprised how in my second year of undergrad math and those exact 3 questions of the linear algebra section were on my example sheets, and in fact the same with some of the group and ring theory problems; not the easiest, but still very much doable! Gosh graduate level maths must be so stressful still
Yeah I was surprised how much of this I have seen. All of those questions must have been in the scope of my first three semesters in my bachelors in mathematics. (I probably would have failed the Ring and Field theory parts)
I've extensively applied linear algebra (quantum mechanics), group theory (X-ray structure analysis), field theory (applications on differential geometry), and you're beginning to see ring theory being applied in a lot chemistry/molecular physics especially dealing with aromatic structures. We skim over the algebras but dear all that is unholy, seeing this type of mathematics in the forms of proofs... that's a completely different animal. Thank you for studying such subjects. Physics and chemistry are heavily indebted to the mathematicians.
This is much easier than the algebra qualification we had in the mid 1980’s. Any old timers like myself think’s so? That group theory looks almost first year undergraduate…. No offense btw.
Am I missing something with the 4th group theory question, because it seems too trivial to me: Suppose g is in G_b, so g*b = b. By transitive action, we have b = h*a for some h in G. So g*(h*a) = h*a, then (gh)*a = h*a, then left multiply by h^(-1) and use group action property to get (h^(-1)gh)*a = a. So h^(-1)gh is in G_a = {1}, so h^(-1)gh = 1. Rearrange, or notice that 1 is trivially central, to get g = 1. Any element of G_b is 1, and 1 is trivially in G_b, so G_b = {1}. This feels like the sort of thing you would see as just an example when you first learn about group actions, so like first year (or earlier) group theory - the entire proof is just basic use of the properties of a group action and rearranging group elements, so most of the steps are so trivial that ordinarily I would just leave them out and write the trivial fact: if b = h*a then G_b = G_a^h (where ^h means conjugation by h).
Yes, it really is trivial. Idk, the entire exam seems to be fairly trivial to me, but then again, I've taken much more advanced classes in algebra. The only one that I thought was somewhat nontrivial was #5 on Ring Theory, but I've figured it out!
It’s esoteric, abstract, and thus hard to stay interested in for many people. Not necessarily very complex. The rules for many mathematical structures are often quite simple, and yet it results in so much interesting/useful results and connections. Science such as chemistry is the study of finding and using models, almost always formulated strictly in mathematics, for very specific situations. These specificity can actually make things much more complex than the generality sought out by more abstract math. The objects of interest are more tangible though, so that compensates for it again.
I remember that on college ( I studied economics) we had this kind of exams, there were horrible and intense because each paper that I used it was a must to put my name and enrollment number.
Q3 of linear algebra too, you can just use the Kernel Lemma : if P and Q are relatively primes polynomials and f is a linear map then Ker(PQ(f)) = Ker(P(f)) + Ker(Q(f)) and the sum is a direct sum
@@valentinmassicot1725 I don't understand any of what you wrote but can you just prove that the sum is direct (take v in both, v = Tu = T²u = Tv = 0) and then the sum of the dim of the sum is the sum of the dim which is dimV due to rank-nullity thm?
As an undergraduate student going into his fourth year, doing a math minor I’m happy that I can do all the group theory questions in this video after taking a course on group theory lol
Hey! I am a PhD student in pure math who is preparing for qualifying exams also. Is there any way you could link the website you mention at 2:49 ? Please don't if you would get in trouble with your program for some reason, but this would be immensely helpful. Thanks! Sam
10:24 "I remember that was a homework problem and I didn't study it too much because I didn't like the problem, and then it showed up on the test" so fucking real this happens to me all the time lol
I have a stupid question. So P4 in the Rings section looks easy enough if we can use the fact that if R is an integral domain, then so is R[x]. (The result follows immediately if we can use that fact without proof: For any p(x) € R[x], if deg(p)>0, then it cannot be a unit--e.g., f(x)=x does not have an inverse in R[x] because 1/x is not in R[x], and similarly with higher order polynomials--and accordingly, f(x) is a constant polynomial and thus is in R.) But do you have to write out the proof on such an exam that R being an integral domain implies that R[x] is too, or can you just assume that without proof? Also--thanks for sharing this! Best of luck.
I am pretty sure that you do not need to reprove Noehter's theorem and just assume it. But you dont need it. Look, if f(X)=aX^n+... has degree n>0 and you multiply it by some polynomial h(x)=bX^m+... of degree m - it is enough to look at the coefficient at X^{n+m} which is ab, and which is non-zero unless h(X)=0. Thus fh can not be 1.
Haven't done much abstract algebra tbh (having a masters in probability theory (and mathematical finance)), and have only scratched group theory surface a bit out of curiosity.
I've studied Abstract Algebra in depth several times in my life, but it never remains in my head. Next time I might try to read a book on classical algebraic number theory. I hope my interest in number theory helps me on this alergy.
Is 2B basically that since the action is transitive, the stabilizer of b is conjugate to the stabilizer of a, and the only thing conjugate to the trivial subgroup is the trivial subgroup? A lot of the other ones looked pretty hard but i think this one looked pretty approachable to me
Haven’t seen you other videos but how do you support yourself through the intense studying? Did you have time for a social life? Have some sort of balance is probably a good idea?
I am a GTA, so I am employed by the university. As far as social life is concerned, if they don't work in the math department, then I don't speak to them lol
Just wanted to say your videos are super awesome. I’m pure math PhD student as well in analysis. Im currently working on some ongoing research with my advisor on groupoid cross products over Fell bundles (advisor is an operator algebraist, and im hoping to become one myself lol). Which subfield of analysis are you interested in?
I am still figuring this out, I did research in the past on Cantor sets in the complex plane and measured Hausdorff dimension. So I will hope to do more with this but ultimately it will be whatever my advisor thinks I should do
What always intrigued me is how deep math gets. What also intrigues me is how professors see these deep concepts as elementary.
When you teach the concepts, you have to understand them at a more fundamental level in order to provide metaphors that everyone can understand. This comes naturally through time and practice. Then, once you've learned the concept, you don't ever "unlearn" it so it seems elementary to you. Good professors are those who understand that the things they take for granted were not always easy.
@@brennenhorton2493
Sometimes...
@@brennenhorton2493 with the exception of my teacher
burden of knowledge
Many students learn this stuff just to pass a test. Professors keep re-using and re-using what they learned, so eventually it has to become elementary
I’m glad myself and many others are finding your channel. This “in the trenches” style of mathematics grad school videos are super informative to many undergrads like myself
And to high school students like me! I think getting exposure to more complex mathematics at an earlier age is more beneficial.
its also just entertaining and interesting
In an engineering vector calculus exam, there was a question to solve for the tangent plane. The examiner forgot to specify powers such that the question in fact asked for the tangent plane of a plane rather than a more complex surface like a sphere or cone. I sat there for 10 mins second guessing myself and bit the bullet and just wrote the equation that was on the exam on a blank page and got 15/15 for the question. I love trivial questions!!
After putting my name on the paper, I'd be lost. That's why I'm not now, nor ever will be a PhD candidate with math.
Haha same 😅🤣
it's not too bad, but you do need to take some courses on linear algebra, abstract algebra and possibly field theory, relatively elementary courses should get the job done, the problems shown weren't that difficult. All the problems were pretty standard. it's a skill obviously, nobody is born knowing this stuff
PhD qualifying exam for Microeconomics Theory is also hard and requires advanced Algebra techniques particularly group theory and set theory.
@@geometrividad7716 it's interesting honestly, because put in a sports perspective this makes intuitive sense to anyone. "why can't i kick a 60 yard punt?" well you probably could given the training, practice, and i guess in this case a limb length that would allow that motion.
"why does this math look like nonsense to me?" well it won't, given enough time and practice
And that’s totally OK 👍🏼
As a grad student in applied math/astrophysics, it's neat to see what others in doing in other grad programs. Your exams are like bizarro world versions of my exams with different content. The classic grad exam setup seems to be: the questions are quite complicated, quite a few of them were homework set examples from homework sets that took bloody ages to do, and you sort of have to have done the questions beforehand to get them right in the exams. We all get to suffer together.
Hey man! I am applying for an Astrophysics master's course. I do have basic understanding of the varied fields in it. Stellar physics, Observational, Computational, Theory of relativity. And my Maths is decent, that I am quite fine with Differential Equations. What else should I be aware of, before I join the course. Could you help me please?
@@ordenax That's about it, It's also very helpful to be good at latex as well as python/julia/matlab. A good understanding of physical mechanics and electromagnetics is also super valuable. This was the case for my programs.
@@zakaryjaynicholls9867 Yeah Physical Mechanics and Electromagnetism and even a basics of Quantum Mechanics I understand. To help with QFT. Sure, The maths would be advanced, but it is something I hope I can manage
@@ordenax In astrophysics, the math is really not that hard compared to doing graduate courses in actual math. So as long as you're good at single var calc/multi variable calc/basic ODEs/basic PDEs/numerical modelling/statistical modelling as well as numerical computing, you can probably learn what you need as you go along.
@@zakaryjaynicholls9867 Thank you for replying. Helped me. 👍
Algebra is by far my favorite subject, and this exam looks approachable, except for the field theory part. I really like that this connects to Analysis and Topology I believe we should not separate it, it's really exciting when it all makes sense finally why in school they make us do functions and equations in a certain manner because R is a field, and why we extend it to C to make it algebraically closed having them as integral domains because every field is sol, while this all happening in a topological space of collection of basis as domain and range. it's reassuring that every math discovery came out natural. I wish you a happy career wherever it leads you.
Bro just started speaking a different language
Thank you so much! Finishing calculus 2 this semester and starting Group Theory either 3rd or 4th semester (parents want me to leave off math for just Major courses in the Fall 2023). Truly shocked at how you wrap your brain around this, its like how I worked during High School Honors Geometry ("Pre-AP") but with confident whereas I was the study crammer before tests. Pursuing a Computer Science major but want to continue Math courses for a while since this was just my Freshman University year. Thanks again!
@@limefish865 Watches video about a graduate level math exam. Is surprised when people use mathematical language in the comments.
I have two actuarial exams this week. This video is oddly reassuring. Thanks for posting. The TH-cam algorithm has done something right.
The actuarial exam compared to this is like a Ford focus to Ferrari 😅. I used to learn a lot of financial mathematics subjects mostly in the Stochastic field but Algebra is mind boggling 😅
@@stochasticdifferentialeq.1393 yeah, I am a kindergarten compared to this guy, but it made me feel better. Also, I passed them both lol
In reality people with the passing mark on actuarial exam make 200K a year while guys who pass Algebra PhD Qual exam usually barely make any descend income and that is the fact. @@RegularSizeRick
Congrats on passing @@RegularSizeRick
@@RegularSizeRick Congratulations on passing both of your actuary exams. That's an excellent accomplishment.
My brain can't handle such questions as these
After seeing the questions on that test I will never again bitch about how hard my qualifying exam was for my Ph.D in Anthropology. LOL
Im using the Sheldon Alxler book for advanced linear right now. Its the first pure math course I've taken other than formal logic and Its super interesting to see how abstracted all the things I learned in my first linear algebra course were and I didnt even realize.
Yeah thats always a cool thing of learning higher math
I mean....it's not necessarily "advanced linear algebra" it's undergraduate linear algbera. Advanced lin alg would be something like Advanced Linear Algebra by Steven Roman
Congratulations on your accomplishments with graduate level Math. Not a lot of people are skilled at proof oriented kind of Math, pure Math, etc. I used to be a Math major. I changed my major from Math to accounting. The reason why I dropped out of being a Math major was due to being bad at proof oriented kind of Math. I'm very good at computational kind of Math, but I'm terrible at the proof oriented kind of Math that pure Math majors take. Math is computational, calculation based, etc all the way from Pre-K to College Calculus and College Differential Equations, but once you finish with College Calculus and College Differential Equations, most of the Math classes you take are more proof oriented than computational, calculation based, etc.
I struggled with courses such as Discrete Math, Topology, Abstract Algebra, etc. I got A's and B's in Math all throughout elementary school, middle school, and high school, and college in Math, but I did terrible in the proof oriented kinds of Math courses that Math majors take once they finish with Calculus and Differential Equations. Those proof oriented kinds of Math courses are no joke. Those proof oriented Math courses make classes like Calculus, Statistics, Intermediate Accounting, engineering, physics, organic chemistry, anatomy, pre-med, etc seem like mickey mouse classes.
Congratulations on your accomplishments in life. I wish you the best.
Sounds like if you had more grounding in a proof class (maybe a better instructor?), you could have done well in that field.
Now I know what a qualifying exam in Algebra looks like! Congratulations on passing it. 👍
Job well done I say.
Thank you very much!
My interests are applied math but I’m taking the required analysis courses right now… respect to anyone studying this at higher level because it is hard work
shouild i go for that shit?
i wanna do computers but back in highschool i tried figuring out how neural networks worked. Then i got hooked into math.
Now I feel like doing some math shit too in some combined degree.
Hopefully itl help me differentiate between other junior devs and hopefully i wont get stuck churning out fucking webdev soycode.
Is it interesting shit? What are some applications of the shit u learn?
@@honkhonk8009 im doing cs and math at uni. i wouldnt say pure math has an obvious application to cs. but its defo very fun and makes you stand out i reckon. itll also be much more difficult than your cs units so be prepared for that haha
I know two folks that graduated in degrees of applied math from respectable colleges, and both have been looking for work for over a year.
@@Castellante Same. It's quite discouraging. They ended up just becoming software engineers at startups (one of them quit shortly thereafter). The pure math career path seems brutal. It's one thing to do it for the love of it but I would hedge my bets in that regard with some more directly applicable studies
@@honkhonk8009 I recently graduated with a CS degree. Take as much math as you can, so long as you enjoy it. It will only serve to make you better at your craft.
Teacher: "Don't worry, the test won't be that hard."
The test:
Great channel! I'm also an analysis PhD student, working with operator means in C*-algebras. My abstract algebra skills are quite rusty. :D However one problem from the test seems trivial: P2 from the ring theory part. One can use the binomial theorem in commutative rings, and if x^m=0 and y^n=0 then it follows trivially from the binomial theorem that (x+y)^{m+n}=0. For the ideal property, we already proved that sum of nilpotents is nilpotent, and the product property is even easier. If x is nilpotent with x^n=0 and r \in R arbitrary then from commutativity we get that (xr)^n=x^nr^n=0r^n=0. Seems super easy compared to your analysis exam problems :O
Undergrad here! Maybe it’s just easy bc I’m taking galois this semester, but problem 3 from field theory is also super straightforward:
A we know a polynomial f has repeated roots iff deg(f)>1 and gcd(f, f’) != 1. Consider an irreducible polynomial with degree >1. Its gcd with its formal derivative must be 1 or itself. f’ is non-zero (characteristic 0) and lesser degree than f, so the gcd can’t be f and must be 1. Therefore f does not have repeated roots. This is true for all irreducible in our char 0 field, so the field is perfect
That is pretty much how it is done :)
Yeah. The identity seems to be unnecessary as well.
Thank you so much! Finishing calculus 2 this semester and starting Group Theory (i think if its allowed). Going to perform Group Theory for Undergrad either 3rd or 4th semester (parents want me to leave off math for just Major courses in the Fall 2023). Truly shocked at how you wrap your brain around this, its like how I worked during High School Honors Geometry ("Pre-AP") but with confident whereas I was the study crammer before tests. Pursuing a Computer Science major but want to continue Math courses for a while since this was just my Freshman University year. Thanks again!
Hey,
I'm also into C*-algebras. Are you going to YMC*A?
Please keep up these videos I love them so much
I love your video because it can be applied to life in general. That's how scary math is... If you have the capacity to imagine more abstractions, you are scratching the itch of existence.
Im not even near this level but its still super interesting, ive been watching for about a month now, maybe more, keep it up:)
Thank you for sharing this! Going to be going for a Ph.D. in Mathematics where I will be studying Algebraic Geometry. This is just getting me excited, yet nervous. Thanks again!
Damn, these videos are doing pretty well. Keep at it :)
This feels more like a bachelor level exam tbh
Especially the ring and group theory
Exactly
Enjoying your channel ... a great find. Cheers
These are great! We need a complex analysis exam now! :)
Here's a tough one: use Morera's theorem to show that the Gamma function is analytic on its domain of definition.
Really interesting vids!
Math undergraduate here(4th year),it is interesting compared to your Real analysis exam (i could not do much) your algebra exam is something i would be able to solve a few.
Linear section problem number 2,Group Theory problem number 2....not looking good so far but in Ring theory can do problems 2,3,4, and Field theory can actually do 1,2,3,4,5 except 4 c.
4 was actually an exam problem in Galois theory (Without C).
i found your channel just yesterday and dude i am obsessed with you ...i soo relate to u (i am an undergrad student studying electronics) ...love your videos ..hope to see more of your content in future ..keep it up ..ps would really like a entire day in a life type video
U studying in India or overseas?
@@extreme4180 india
what is the website that you mentioned that has the pool of algebra problems?
great content btw
that "i never like matrices" in 4:30 feels so deep hahaha. excellent content tho!
I like your channel very much. As a side note: I also learn English with you! You said that you didn't pass the exam with "flying colours." In German we say "mit fliegenden Fahnen" (with flying flags). All the best!
I'm surprised tbh, i'd expect it to be harder. Especially the field theory bit given that phd level field theory is so technical. Though i imagine the marking is very strict.
I think the test consists of Problems which look "easy" but you start the problem and get stuck halfway in...That happens to me always.
this is very reminiscent of my exams, considering I took group theory and rings and fields in second year
its scary how little I can recall immediately when looking back at questions like these
I'm sure relearning them wouldn't be crazy difficult and it wouldn't take long to click again but its always funny looking at previous content after a break and realising how baffling it is
It's pretty funny that PhD exercises are similar to my undergraduate Abstract Algebra courses from Universidad de Buenos Aires. I can share anyone who don't believe me, the only inconvenient is that you will have to translate it from Spanish to English with Chat GPT.
Watching this, I feel like what the rest of the Fantastic Four must feel like when Reed Richards goes on a rant.
Goodness! This is the kind of thing that stops me from more seriously pursuing a pure mathematics degree at any level. I have a BS in Statistics, and I picked up some things along the way, but I never got quite *this* deep into math. Whether I'm trying to read a paper from google results or the /r/math subreddit, everyone seems obsessed with talking about almost literally anything in abstract algebra terms. Groups, fields, rings, sets, magmas, abelian or not, commutative or not...I know what sets are, I know what operations are, and I know what commutativity, associativity, plus, and times are. (thanks to Google while composing this, I know distribution as well, but I never remember to count that with commutativity and associativity as an operational axiom or property.) But to me, this kind of qualifying test just looks more like a vocabulary test than a math test. I feel like if I knew the definitions of any words on that paper, or if the structures were given by operation and element properties instead of by special name, I could probably handle this. But I just cannot master the abstract algebra vocabulary, so I can't just know what identities, inverses, operations, properties, and axioms I have available when I only see the word. It's like if I try to learn Chinese. Even if I read about the radicals and practice drawing each one for a while, how will I ever comfortably remember what it means in the long term?
I'm disappointed not to see Pugh's Real Mathematical Analysis in the stack. I may be a bit biased, but I hear that's a solid work. :)
Keep up the good work!
I wish my qualifying exam was this easy, my school does 6 really hard problems where a 50% is a pass for theirs and they were absolutely brutal
Once I looked up the definition of ker, Im, and a plus inside a circle I figured out I-3 pretty quickly. Enough for tonight. When I took linear algebra 53 (?) years ago it was pretty much all matrices. I remember having an A going into the final on the first two quizzes but then slacking off the rest of the course and ending up with a B. Took grad level Real Analysis and got a D and although I had done enough to fulfill my undergrad major that was that. Eventually did an MBA - so easy, even at a top University. Just put in the time to go through the cases thoroughly and talk in class - and always come to a decision, any decision.
Hi, I'm a fellow grad student at a different institution (who likes algebra).
I can tell you how to do 5 on Ring Theory - take any unit u, and the conjugation map r --> uru^-1 is a ring automorphism. By assumption, this is the identity, so r commutes with all units.
Since every nilpotent+1 is a unit, nilpotent elements must commute! The rest is easy.
You know, I envy you for the fact that at your uni, your quals are not crazy hard (like they've been getting at my uni)! I wish you luck with your future research!
2:27 A trivial solution, to a problem, is usually something like 0 = 0
Love your content from Colorado ❤️ never got to pursue a math education formally, living vicariously through you. Would love to see what you’re curious about research wise
Surprised how in my second year of undergrad math and those exact 3 questions of the linear algebra section were on my example sheets, and in fact the same with some of the group and ring theory problems; not the easiest, but still very much doable! Gosh graduate level maths must be so stressful still
Yeah I was surprised how much of this I have seen. All of those questions must have been in the scope of my first three semesters in my bachelors in mathematics. (I probably would have failed the Ring and Field theory parts)
Yeah that's the point of a qualifier lol
I've extensively applied linear algebra (quantum mechanics), group theory (X-ray structure analysis), field theory (applications on differential geometry), and you're beginning to see ring theory being applied in a lot chemistry/molecular physics especially dealing with aromatic structures. We skim over the algebras but dear all that is unholy, seeing this type of mathematics in the forms of proofs... that's a completely different animal. Thank you for studying such subjects. Physics and chemistry are heavily indebted to the mathematicians.
it's bitter sweet...they are vile creatures for introducing reminder theory and modulus...
This is much easier than the algebra qualification we had in the mid 1980’s. Any old timers like myself think’s so? That group theory looks almost first year undergraduate…. No offense btw.
This exam is pretty easy compared to the ones in other universities even today.
I was wondering the same thing lol, why did it look so elementary?
"they test material not taught in class" any else get violent urges when they heard this?
imagine this is your homework and you try to find the answers on chegg
Am I missing something with the 4th group theory question, because it seems too trivial to me:
Suppose g is in G_b, so g*b = b. By transitive action, we have b = h*a for some h in G.
So g*(h*a) = h*a, then (gh)*a = h*a, then left multiply by h^(-1) and use group action property to get (h^(-1)gh)*a = a.
So h^(-1)gh is in G_a = {1}, so h^(-1)gh = 1. Rearrange, or notice that 1 is trivially central, to get g = 1. Any element of G_b is 1, and 1 is trivially in G_b, so G_b = {1}.
This feels like the sort of thing you would see as just an example when you first learn about group actions, so like first year (or earlier) group theory - the entire proof is just basic use of the properties of a group action and rearranging group elements, so most of the steps are so trivial that ordinarily I would just leave them out and write the trivial fact: if b = h*a then G_b = G_a^h (where ^h means conjugation by h).
Yes, it really is trivial. Idk, the entire exam seems to be fairly trivial to me, but then again, I've taken much more advanced classes in algebra. The only one that I thought was somewhat nontrivial was #5 on Ring Theory, but I've figured it out!
@@heartpiecegaming8932 what about group theory q1? I don't know how to approach that one.
Best of luck with everything! Remember, the more intelligent you are, the more you doubt yourself and your capabilities. Keep going!
I am also taking PhD Math qualifying exam this Midyear term. Thank you for these sample problems. I would try to solve them.
Your choice looks really reasonable for me that I would pick exactly same problems if I was taking it lol
I'm not sure if I could get correct but at least some problems look more comfortable
As a BS Chemistry for 25 years, I have much respect for the Mathematics scholars. That stuff is HARD!
It’s esoteric, abstract, and thus hard to stay interested in for many people. Not necessarily very complex. The rules for many mathematical structures are often quite simple, and yet it results in so much interesting/useful results and connections.
Science such as chemistry is the study of finding and using models, almost always formulated strictly in mathematics, for very specific situations. These specificity can actually make things much more complex than the generality sought out by more abstract math. The objects of interest are more tangible though, so that compensates for it again.
This literally looks like I'm reading another language. I have NO idea what any o fit means but it's still neat
I struggle with understanding field theory. ☹ Your time to completion sounds reasonable. Congratulations on passing.
I admire your discipline.
I remember that on college ( I studied economics) we had this kind of exams, there were horrible and intense because each paper that I used it was a must to put my name and enrollment number.
goddamnit. I've been graduated for several years... almost forget everything
That looks like british university undergraduate maths
Question 4 of group theory looks incredibly easy to me : if b = g a then G_b = g G_a g^{-1} so the result is quite obvious
Q3 of linear algebra too, you can just use the Kernel Lemma : if P and Q are relatively primes polynomials and f is a linear map then Ker(PQ(f)) = Ker(P(f)) + Ker(Q(f)) and the sum is a direct sum
@@valentinmassicot1725 I don't understand any of what you wrote but can you just prove that the sum is direct (take v in both, v = Tu = T²u = Tv = 0) and then the sum of the dim of the sum is the sum of the dim which is dimV due to rank-nullity thm?
As an undergraduate student going into his fourth year, doing a math minor I’m happy that I can do all the group theory questions in this video after taking a course on group theory lol
I, uh, managed to get a B in Algebra 1 in high school. So…there’s that.
Interestring, I didn't had too much trouble with the analysis test you showcased, but I don't even understand most of the Algebra problems
I would much rather be in your position lol
@@PhDVlog777 I guess I need to find a good Algebra book to fill my gaps
Glad I changed my major from math after the first year. Life is too short.
Just a highschool student getting tips and tricks on how to pass even if you can't pass with flying colors
Hey! I am a PhD student in pure math who is preparing for qualifying exams also. Is there any way you could link the website you mention at 2:49 ? Please don't if you would get in trouble with your program for some reason, but this would be immensely helpful. Thanks!
Sam
hey could you share the link with me too?
hey could you share the link with me too?
hey
I’d just cry.
Terrence Tao solved this in his sleep
can't believe I passed all this shit
I woulda probably chose the same problems too.
Oh my I remember that linear algebra problem! I think it was a homework problem - and I took grad Linear over 35 years ago
3rd question in linear algebra we had it in homework in algebra 2 . 1st year of college 🗿
Abstract algebra is badass!
10:24 "I remember that was a homework problem and I didn't study it too much because I didn't like the problem, and then it showed up on the test" so fucking real this happens to me all the time lol
This might as well be written in a foreign language because I don't understand any problem at all
This sounds exactly like me going over the ODE exam I just took 💀
I'm used to 2 to 5 problems in an exam. One or two in a midterm. 10 sounds like overkill.
I took abstract algebra by my own choosing, as an elective, what a huge mistake that course was brutal.
Pleasantly surprised I could do the linear algebra q3 in like 5 minutes. Couldn't get the first 2 but hey i didn't learn jordan form so :p
2:53 - What is the website called?
I have a stupid question. So P4 in the Rings section looks easy enough if we can use the fact that if R is an integral domain, then so is R[x]. (The result follows immediately if we can use that fact without proof: For any p(x) € R[x], if deg(p)>0, then it cannot be a unit--e.g., f(x)=x does not have an inverse in R[x] because 1/x is not in R[x], and similarly with higher order polynomials--and accordingly, f(x) is a constant polynomial and thus is in R.) But do you have to write out the proof on such an exam that R being an integral domain implies that R[x] is too, or can you just assume that without proof?
Also--thanks for sharing this! Best of luck.
I am pretty sure that you do not need to reprove Noehter's theorem and just assume it. But you dont need it. Look, if f(X)=aX^n+... has degree n>0 and you multiply it by some polynomial h(x)=bX^m+... of degree m - it is enough to look at the coefficient at X^{n+m} which is ab, and which is non-zero unless h(X)=0. Thus fh can not be 1.
@@danielmaxhoffmann1917good point
Haven't done much abstract algebra tbh (having a masters in probability theory (and mathematical finance)), and have only scratched group theory surface a bit out of curiosity.
Group theory question 1 is doable
Analysis > algebra all day baby
They're pretty standard questions. I thought it was much more challenging.
Thanks for the cool video! I also wondered what website you were talking about, containing all the problems. Any way you could link the website?
as an undergrad, wtf is going on?
Lmaooooo. Felt. This used to be me, and now my qual is in like 4 days 🙈😭
i just started my masters degree and have done lots of algebra and algebraic number theory, this looks pretty doable
As a music major this exam looks like a different language
And I’m sure most math majors would say same thing about music theory
this stuff you do in in undergraduate algebra at a decent university
Please, can you share the website they pull the problems?
I've studied Abstract Algebra in depth several times in my life, but it never remains in my head.
Next time I might try to read a book on classical algebraic number theory. I hope my interest in number theory helps me on this alergy.
What is the website at 2:50 you mentioned that has a lot of extra problems?
Do you know what website they pull problems from? I've never heard of this.
Yeah.. that’s why I’m going into applied stats for grad school
Is 2B basically that since the action is transitive, the stabilizer of b is conjugate to the stabilizer of a, and the only thing conjugate to the trivial subgroup is the trivial subgroup? A lot of the other ones looked pretty hard but i think this one looked pretty approachable to me
Haven’t seen you other videos but how do you support yourself through the intense studying? Did you have time for a social life? Have some sort of balance is probably a good idea?
I am a GTA, so I am employed by the university. As far as social life is concerned, if they don't work in the math department, then I don't speak to them lol
I wish I could understand math...so embarrassing not to make sense of any of these...but its too late in life all the best to the test takers
Just wanted to say your videos are super awesome. I’m pure math PhD student as well in analysis. Im currently working on some ongoing research with my advisor on groupoid cross products over Fell bundles (advisor is an operator algebraist, and im hoping to become one myself lol). Which subfield of analysis are you interested in?
I am still figuring this out, I did research in the past on Cantor sets in the complex plane and measured Hausdorff dimension. So I will hope to do more with this but ultimately it will be whatever my advisor thinks I should do
The projection exercise in linear algebra is trivial...
What's that magical website?
III 2. (i) is the binomial theorem?
I'm self-taught and wondering if I could pass an algebra exam... the answer is definitely "no". Still need to study harder 😅
what is the website you were talking about at the start