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
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
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!!
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!
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!
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
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).
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
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 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.
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.
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)
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!
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.
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
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 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!
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!
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
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!
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
Wow, such a test, though I excelled in Trig in college 40+ years ago, would give me nightmares. My trig tests were easy compared to this, I even challenged my college's algebra program by taking their final so I would not have to enroll in those classes, which were prerequisites (always dreaded that word in those days) for Trig. I was a liberal arts student, with the goal of learning business and teaching in that field, which I eventually did on the customer and guest service, and systems side. Trig completed my math requirements in addition to the courses I challenged and I loved Trig especially the way our teacher taught it. It was funny, six years after I left college I was taking the airport bus from Napa to San Francisco, on my way to a trip to Washington DC. My former college Trig teacher boarded our bus bound for London, what serendipity. I aced all of his trig semi-finals and finals and he'd return the tests to me with red ribbons around them, embarrassing me in front of some students who were math majors unlike me, and I knew them as the sharpest minds from my grammar and high school. Like "Good Will Hunting", when it came to trig, or tutoring my friends in calculus, I could "just play" though over the years, all that stuff fell out of my brain so I could leave space in my brain for enjoying retirement, and my career allowed me to retire early just doing odd projects since if asked as a consultant. I am drawn to videos like yours because of the intelligence you have in sharing mathematics, a skill my father also had, and sharing how those going for their PHD in this field of study are tested. My daughter chose to major in liberal arts like I did, so she can teach and travel like I did, and she embraced mathematics in the way I did, embracing the practical courses. My fav math course when I was "yiddle" was learning how to count US money. I thought "Wow, this is what math is for, I love it already!". I loved math even more when I learned algebra, I loved balancing equations and inequalities due to the simple way my dad, who was a physicist, taught me, that all balancing equations was about was keeping a scale of weights balanced--it helped me in Chemistry too, I loved balancing Chem equations seeing its similarity to Algebra, especially in advanced chem. Thanks for sharing this, I will have to watch your other videos. Math is such a wonderful language that is part of all of us.
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 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.
The exercice 3 in linear algebra has something to do with projections right ? The minimal polinomial for a projection is X^2 - X and a consequence for the projection is V = kerp + Imp . That's the first thing I thought
Thank you for discussing this material with us. I wish you the best for having chosen mathematics as your profession: this subject is my life, love and joy. I only wish that God gave me more brain power to invest.
Ive taking Abstract Algebra 2, about 5 years ago, and I'd say 80% of this looks quite familiar. How much more past Abstract Algebra 2 would you say this content is?
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.
I finished my masters in pure maths a short while ago and I am already working. I haven't done any algebra at the end of my studies. Can you link the website with all those problems you mentioned at the start because I don't want to become super rusty at algebra as I actually enjoyed much of it.
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?
Im a comp sci student, these videos motivate me to keep pushing in my math classes even though i slightly hate math. 1 question, what is the end goal for you? You obtain a phd, then look for a job aligned with your qualifications? The amount of effort put would rival med school. Or is it just the nature of curiosity that keeps you wanting to learn more? I myself find it difficult to find purpose, im primarily doing it for money. Again thanks for making these videos, it keeps me pushing seeing someone else also grinding out their education especially in math. Hope you the best.
You don't hate math, you just don't perceive it from physics reference, which makes it absurd. It is like trying to prove immagination is true. You can't you need phyiscs point of reference to make it dance with the math. Hope it helps you, have a nice one
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.
the first question seems off to me. Am I stupid? Any diagonal matrix commutes with any other. But I cant write diagonal matrices with different entries on the diagonal as polynomial in an nxn Jordan-block, right?! easiest example. Let A be a 2x2 Jordan block for Eigenvalue 0(a 1 top right, rest 0s). And let B be diag(1,0), so a 1 in the top left. Clearly A is a Jordan block and they commute, but B is not a polynomial in A…
I wonder why the questions are all pretty much first-second year undergraduate abstract algebra ? Since you do a PhD after doing graduate type of math why isn't there any graduate-level math ?
I am an undergraduate math student who is very interested in researching pure math at the graduate level. Would you mind sharing how much time you had to complete this exam and where you are completing you research?
For this exam we were given a total of five hours, but most finished well before then. This test took me about 90 minutes. Also I would prefer not to disclose my university at least at the moment, but a determined viewer colder probably figure it out lol
At the masters level, I did research involving Cantor sets in the complex plane and measured Hausdorff dimension. I might switch to geometry. Honestly, whatever gets me my PhD I will be happy with lol
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
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
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!!
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.
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!
Please keep up these videos I love them so much
Enjoying your channel ... a great find. Cheers
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
I have two actuarial exams this week. This video is oddly reassuring. Thanks for posting. The UA-cam algorithm has done something right.
Damn, these videos are doing pretty well. Keep at it :)
Im not even near this level but its still super interesting, ive been watching for about a month now, maybe more, keep it up:)
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).
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
what is the website that you mentioned that has the pool of algebra problems?
great content btw
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
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.
Now I know what a qualifying exam in Algebra looks like! Congratulations on passing it. 👍
Job well done I say.
Thank you very much!
I admire your discipline.
I am also taking PhD Math qualifying exam this Midyear term. Thank you for these sample problems. I would try to solve them.
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.
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
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?
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!
My brain can't handle such questions as these
Oh my I remember that linear algebra problem! I think it was a homework problem - and I took grad Linear over 35 years ago
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.
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'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 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!
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?
as an undergrad, wtf is going on?
Lmaooooo. Felt. This used to be me, and now my qual is in like 4 days 🙈😭
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
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
I struggle with understanding field theory. ☹ Your time to completion sounds reasonable. Congratulations on passing.
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
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
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.
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.
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.
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
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
Abstract algebra is badass!
Best of luck with everything! Remember, the more intelligent you are, the more you doubt yourself and your capabilities. Keep going!
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
Wow, such a test, though I excelled in Trig in college 40+ years ago, would give me nightmares. My trig tests were easy compared to this, I even challenged my college's algebra program by taking their final so I would not have to enroll in those classes, which were prerequisites (always dreaded that word in those days) for Trig.
I was a liberal arts student, with the goal of learning business and teaching in that field, which I eventually did on the customer and guest service, and systems side. Trig completed my math requirements in addition to the courses I challenged and I loved Trig especially the way our teacher taught it.
It was funny, six years after I left college I was taking the airport bus from Napa to San Francisco, on my way to a trip to Washington DC. My former college Trig teacher boarded our bus bound for London, what serendipity. I aced all of his trig semi-finals and finals and he'd return the tests to me with red ribbons around them, embarrassing me in front of some students who were math majors unlike me, and I knew them as the sharpest minds from my grammar and high school.
Like "Good Will Hunting", when it came to trig, or tutoring my friends in calculus, I could "just play" though over the years, all that stuff fell out of my brain so I could leave space in my brain for enjoying retirement, and my career allowed me to retire early just doing odd projects since if asked as a consultant.
I am drawn to videos like yours because of the intelligence you have in sharing mathematics, a skill my father also had, and sharing how those going for their PHD in this field of study are tested.
My daughter chose to major in liberal arts like I did, so she can teach and travel like I did, and she embraced mathematics in the way I did, embracing the practical courses.
My fav math course when I was "yiddle" was learning how to count US money. I thought "Wow, this is what math is for, I love it already!". I loved math even more when I learned algebra, I loved balancing equations and inequalities due to the simple way my dad, who was a physicist, taught me, that all balancing equations was about was keeping a scale of weights balanced--it helped me in Chemistry too, I loved balancing Chem equations seeing its similarity to Algebra, especially in advanced chem.
Thanks for sharing this, I will have to watch your other videos. Math is such a wonderful language that is part of all of us.
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 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
i just started my masters degree and have done lots of algebra and algebraic number theory, this looks pretty doable
2:27 A trivial solution, to a problem, is usually something like 0 = 0
Teacher: "Don't worry, the test won't be that hard."
The test:
The exercice 3 in linear algebra has something to do with projections right ? The minimal polinomial for a projection is X^2 - X and a consequence for the projection is V = kerp + Imp . That's the first thing I thought
You can prove it directly. Any x in V is in kerT + ImT and their intersection is 0.
Do you know what website they pull problems from? I've never heard of this.
goddamnit. I've been graduated for several years... almost forget everything
Thank you for discussing this material with us. I wish you the best for having chosen mathematics as your profession: this subject is my life, love and joy. I only wish that God gave me more brain power to invest.
Are a lot of these problems from Dummit and Foote? Cause I vividly remember doing them last year in my Abstract Algebra course.
Ive taking Abstract Algebra 2, about 5 years ago, and I'd say 80% of this looks quite familiar. How much more past Abstract Algebra 2 would you say this content is?
It’s about the same. I haven’t taken anything last graduate level algebra
3rd question in linear algebra we had it in homework in algebra 2 . 1st year of college 🗿
I'm used to 2 to 5 problems in an exam. One or two in a midterm. 10 sounds like overkill.
I’d just cry.
What is the website at 2:50 you mentioned that has a lot of extra problems?
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...
I woulda probably chose the same problems too.
Can i get solution of exercise 3 and 5 in group theory section please today, since tomorrow i have an exam in it
What's that magical website?
imagine this is your homework and you try to find the answers on chegg
This literally looks like I'm reading another language. I have NO idea what any o fit means but it's still neat
I finished my masters in pure maths a short while ago and I am already working. I haven't done any algebra at the end of my studies. Can you link the website with all those problems you mentioned at the start because I don't want to become super rusty at algebra as I actually enjoyed much of it.
I think the last "algebraic" course I have taken was algebraic topology, but that was basically just group theory :/
Yeah.. that’s why I’m going into applied stats for grad school
2:53 - What is the website called?
Please, can you share the website they pull the problems?
Glad I changed my major from math after the first year. Life is too short.
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?
How do you get copies of these tests?
I can read the first page instructions...how many marks for that?
Watching this, I feel like what the rest of the Fantastic Four must feel like when Reed Richards goes on a rant.
Can you pute the file in a Format pdf in description,thank you
I barely passed stats last semester as an undergrad…what in the world is this 😵💫
If you do more than 10 problems how do they choose which ones are counted towards the score?
What's the time allocation?
What university is this abstract algebra PhD qualifying exam taken from?
Im a comp sci student, these videos motivate me to keep pushing in my math classes even though i slightly hate math.
1 question, what is the end goal for you? You obtain a phd, then look for a job aligned with your qualifications? The amount of effort put would rival med school. Or is it just the nature of curiosity that keeps you wanting to learn more? I myself find it difficult to find purpose, im primarily doing it for money.
Again thanks for making these videos, it keeps me pushing seeing someone else also grinding out their education especially in math. Hope you the best.
You don't hate math, you just don't perceive it from physics reference, which makes it absurd. It is like trying to prove immagination is true. You can't you need phyiscs point of reference to make it dance with the math. Hope it helps you, have a nice one
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.
Do you suggest wanna book where i can findthis type of problems and theorems
When you refer to homework what do you mean?
Cool!
The projection exercise in linear algebra is trivial...
What is G' in question 1? What do the primes mean here? I've never seen this notation.
It denotes the commutator subgroup (or derived subgroup) of G. Extra primes mean derived subgroup of the derived subgroup and so on.
Group theory question 1 is doable
at 2:50 what is that website called?
the first question seems off to me. Am I stupid? Any diagonal matrix commutes with any other. But I cant write diagonal matrices with different entries on the diagonal as polynomial in an nxn Jordan-block, right?! easiest example. Let A be a 2x2 Jordan block for Eigenvalue 0(a 1 top right, rest 0s). And let B be diag(1,0), so a 1 in the top left. Clearly A is a Jordan block and they commute, but B is not a polynomial in A…
A and B don’t commute in your example.
@@hakerfamily shouldnt diagonal matrices commute with all other matrices?!
oh, they dont, you are right, I am stupid
What college do you attend?
I wonder why the questions are all pretty much first-second year undergraduate abstract algebra ? Since you do a PhD after doing graduate type of math why isn't there any graduate-level math ?
I took abstract algebra by my own choosing, as an elective, what a huge mistake that course was brutal.
I am an undergraduate math student who is very interested in researching pure math at the graduate level. Would you mind sharing how much time you had to complete this exam and where you are completing you research?
For this exam we were given a total of five hours, but most finished well before then. This test took me about 90 minutes. Also I would prefer not to disclose my university at least at the moment, but a determined viewer colder probably figure it out lol
@@PhDVlog777 Thanks for sharing! That's understandable to not share where you are studying. Tha k you anyways
Excuse me
What kind of Linear Algebra and Abstract Algebra textbooks for qualifying exam ? Bachelor or Master degree ? Mathematics are interesting
This might as well be written in a foreign language because I don't understand any problem at all
What area of Analysis do you work in? :)
At the masters level, I did research involving Cantor sets in the complex plane and measured Hausdorff dimension. I might switch to geometry. Honestly, whatever gets me my PhD I will be happy with lol
May I ask which university this is?