What does a Complex Analysis PhD Qualifying Exam look like?

[kailysrat]

I haven't studied complex analysis in ~20 years but I may have some intuition on 5 for you. Basically if |f'(0)|<1 you wind up getting a contraction when you map a sufficiently small neighborhood around 0 onto itself around the origin. If |f'(0)|>1, then a sufficiently small neighborhood will map onto a set that contains the original neighborhood. When you iterate, eventually it will exceed the original boundedness of omega.

[torquencol]

I don't have this level of math but your videos are really relaxing

[orpheus]

Im not in the same field but I always find it inspiring to see other people studying and imrpoving in their field. This stuff goes way over my head. Im okay at the math I stopped at (linear algebra, calc 2 and discrete math) but Physics was always so hard for me to grasp. It's definitly not a subject that is intuitive for me. I was so happy to finish university physics II and leave that in the past lol

Ive got a couple exams/certs im studying for now and it can all feel so overwhelming at times. The amount of info and things to remember and look for. I find practing and building muscle memory is key.

Im in cybersecurity. I am taking the Certified penetration tester specialist test next month and the OSCP a few months later. I feel ready, pwning dozens of medium-hard lab machines but Im scared ill get stuck or get real nervous come time for the test and screw it up haha.

Anyways, thanks for the motivation and best of luck with grad school!!!

[deimosthewizard]

I love this channel, every time I watch a video I get like 10% of the things discussed which really entices me to start studying.

[JohnH]

I recently took a Complex Analysis preliminary exam, and I have a Real Analysis preliminary exam coming up in August. I want to say that I'm in the same boat you are, and I have so much more to say about these types of exams, but I'm guessing you have it more than I do! All the best in your studies!

[Adam-iooz]

For 4 I don't think you need to use Taylor expansion. You can find a contour in D(0, 2)\D(0, 1) where g does not vanish. Then for large n |z^ng| will be larger on this contour than the maximum of |f| on it. Interestingly, this argument wouldn't work on the unit disc and in fact if f and g are both identically 1 then 1+z^n has no roots inside the unit disc.
For 5, my instinct was the same as yours but I'm not sure how to argue via the Schwarz lemma. There is some a so that D(0, a) is inside Omega and which f maps into D(0, b) for some b. Thus, g(z) = b^(-1)f(z/a) is a map from the unit disc to itself (note that because we are working with discs to begin with we don't need the Riemann mapping theorem - i.e. the conformal map is just z mapsto z/a) and thus by the Schwarz lemma |g'(0)|=(ab)^(-1)|f'(0)| <1. But we don't know what a and b are so I'm not sure how to proceed.
Instead I think this is what the hint is getting at: consider the family of functions F=\{f^(n)\} given by iterating f (like f(f(f....(x)))). First, you can check via either induction or series expansion that d/dz f^(n)(z) at z=0 is (f'(0))^n. Now, because Omega is bounded, F is normal by Montel's theorem so it has some subsequence that converges uniformly on compact sets. It is a fact that for a uniformly convergent sequence of analytic functions their derivatives are also uniformly convergent, so we know that (f'(0))^n must converge which is only possible if |f'(0)| <= 1. Pretty tricky IMO!
Anyway, 5 is an interesting problem. Thank you for sharing.

[flamehours]

im 19, and my math is really bad . im talking (5th grader bad) . i might not understand any topics theorems or things you said and showed, but it is very motivational that you are you are so invested in mathematics.

[neerajnongmaithem]

I know nothing about math, but i still buzz in from time to time. I just find it comfortable to hear a man share his life in such a honest way, i can feel the effort he puts in his study, the struggles he have. Its like hearing a close friend sharing about their life

[EricaCalman]

In physics we have to take our own internal version of complex analysis but get to skip real entirely which was fine by me since if I never see another epsilon/delta in my life it'll be too soon. With the qualifier though that 'complex analysis' to me just means I can take a basic contour integral and use Cauchy's theorem to save myself some effort haha.

[furious]

i have no idea what is this but i find it enjoyable and relaxing lol

[yvonnesun]

literally have no idea what's happening, but your voice is soooo soothing, so I play your videos as background calming music when I do my *basic* calculus 3 homework

[Mehdi-qptz]

Hi! You've done a great job documenting your work. Have you ever thought of digitalizing everything and making it all available online for people to use even decades from now?

[milesman]

I think for number 5 you can let F be the infinite iteration of f and show that f_n=f(f(…f)) uniformly converges to it on compact subsets of Omega. Then suppose by way of contradiction that |f’(0)|>1. Then, by chain rule, |f_n’(0)|=|f’(0)|^n, thus |F’(0)| is infinite, a contradiction to conformality. I’m a bit iffy on the first step but I think it should be fine.

[tererenji]

Hey! For Q5, I think you can use algebra! take the group Aut(omega) for maps f preserving the group identity 0, and given that It's a group under composition, 1. differentials also form a group, 2. Aut(Omega) is isomorphic to SL(C) or SL(Z) (SL for boundedness, Z if it's not connected)!

[Marryatau]

Thank you I’m still learning real analysis though I’m glad to see others do more experienced math I’ll keep at it and I hope you do too!

[briang.valentine]

2. A Green's function can be constructed that uses the symmetry of the domain. The singularity at the point (o, i) can be sent to infinity. Then the mapping is analytic and conformal.

[Epilogue_]

Im a physics undergrad working on me thesis about quantum computation but i have to say ive always had a massive respect to mathematicians, even i got in love with one mathematician girl 😂.

[nazo]

I love how applying to a PhD is memorizing harder things than what you used to memorize for highschool.

[drewgonzales]

Cool stuff bro I look forward to pursuing pure mathematics and quantum physics..looks like fun.

[Steaks]

I’m nervous for your exam later this summer to be honest, like if it doesn’t work out what’s your plan? When you said the same guy was making half the test I got very worried for you