UC Berkeley Math PhD Entrance Exam Question

Do you have what it takes to get a Math PhD at UC Berkeley? Watch this video and find out!
Topology Playlist: • Topology
Subscribe to my channel: / drpeyam
Check out my TikTok channel: / drpeyam
Follow me on Instagram: / peyamstagram
Follow me on Twitter: / drpeyam
Teespring merch: teespring.com/...

[@bprpfast]

This explains why I don’t have a PhD.

[@amritvignesh4434]

Lol bprp

[@AhmadAhmad-qx6fp]

Dr Peyam has been kind enough to let us peek into Berkeley Problem Set.. Tremendously appreciated!

[@hemonben9004]

Besides the math i really like his presentation

[@YaamFel]

What a nice problem. Honestly from the start seeing a countable intersection of nested compact sets my brain was screaming 'cantor's intersection theorem', so it was very satisfying when we finally got to that. Taking the family of nested closed balls centered at x0 really made everything fall into place nicely.

[@strmandola5484]

I'm Japanese. University in Japan exam is crazy. Please solve it.

[@adityadwivedi4412]

Can you send the question paper in english

[@tonyhaddad1394]

Plz send it

[@drpeyam]

Yeah send it :)

[@lanewang8387]

chinese student is now in the game

[@husseinolama]

Nice one! There is another way to do it though, that uses sequential continuity: Let x_n ->x in R^n, and assume that f(x_n)-/->f(x). First, the sequence x_n is contained in some closed ball B(x,r), and f(B(x,r)) is compact. By compactness, we can assume without loss of generality, that there exists a subsequence y_n of x_n such that f(y_n)->f(y) and y is in B(x,r). If y=/=x then there (by hausdorffness and the geometry of balls in R^n) there exists a ball B(x,e) that does not contain y. By condition (ii), if f(y)=/=f(x), then f(B(x,e/k)) does not contain f(y) for some natural number k. Finally, R-f(B(x,e/k)) is an open set containing f(y) and only finitely many f(y_n)’s A Contradiction! So f(y)=f(x), which shows that f(x_n)->f(x) :D

[@FT029]

Each individual step wasn't too hard to follow, but it must be difficult to come up with everything! I think this proof is very beautiful (usually you find delta based on some algebraic formula of epsilon, but here it's arguing that some N exists, and using delta < 1/N). The pictures on the side are also great for intuition!

[@dez-m]

Just got into my dream school, the college of engineering, at UC Berkeley! So excited for this to come!

[@drpeyam]

Omg congratulations!!!! It’s such a great school, I have fond memories of it!

[@korayduztas3558]

You should also prove this using topological concepts . I think that would apply to more general cases.

[@rohannuckchady2900]

wouldn’t this be slightly easier using the topological definition of continuity and related properties of compactness?

[@reyblais4858]

I’ve definitely said “WTF delta” but maybe not in the same context…

[@alexandersanchez9138]

12:20 "...So B^C is the closed ball." We get what you're saying, so it's OK. :)

[@johnflorio3052]

You remind me of my real analysis professor in grad school. I love this stuff....

[@tomatrix7525]

Love your videos Peyam! Good vibes always

[@fireemblem2770]

Good thing I'm not going to Berkeley for a PhD anytime soon, I got lost!!

[@babukaruppasamy2892]

I invent new method to find the prime numbers. It is alternative to the Eretosthenes's Sive method. But I have not studied mathematics properly. This is my 20 years struggle. What do I do? The great mathematics Society !.... Answer me. I am from Tamilnadu in India.

[@VIKASVERMA-nk8uw]

Sir love from India for u

[@nourdinespen8568]

Always the best ❤❤

[@japotillor]

Матрёшка :) (бабушка is grandmother)

[@drpeyam]

Thank you hahaha

[@japotillor]

@@drpeyam you're welcome

[@Kdd160]

75k, 25k to get the silver play button!

[@sagarhr1112]

love you sir you are amazingly incredible......

[@mathjitsuteacher]

Very nice exercise!

[@andreaalbano7741]

If the intersection of the nesting sequence of compact sets ( f(Kn) ) is a singleton then isn't it true that diam(f(Kn)) converges to zero and thus showing that f is continuous?

[@mathssolverpoint6059]

I am a graduate student and I could not understand and I am afraiding to my future

[@yiqin7906]

Me, I have my hope again. Thank you!

[@lacasadeacero]

Its not hard, but im not as enthusiast as time ago. Oh well, good for them.

[@guitar_jero]

17:45 for the grand finale

[@jonathanengwall2777]

What is the effect of a larger B in a real application? Will the absolute final boundary shrink?

[@AyushSharma-vf6ho]

Indians be like-Do problems on jee advance problems😂

[@adityadwivedi4412]

The math you see on jee is not the real math

[@frozenmoon998]

At least they did not ask someone to attempt to solve the Riemann Hypothesis :D, so it's doable, as long as that person loves maths as a whole.

[@adityadwivedi4412]

Why are you on tiktok?

[@AdityaKumar-ij5ok]

Lol to increase popularity, why else do you think a person from academia will decide to go there to confront many mind numbing videos, if interested also check Chris Sutherland, he too *(sighs)

[@drpeyam]

Why not? TT is really fun once you try it out

[@adityadwivedi4412]

@@drpeyam I am indian and it is not available in our country

[@rishabhbhutani5835]

For condition #1 at around 1:04, doesn't that imply that f is cont. on it's own because the image of a compact set to another compact is true iff f is cont? What is the need to do anything more?

[@drpeyam]

This is not true, consider f to be 0 if x is rational and 1 if x is irrational, then f is not continuous but it sends compact sets to compact sets

[@rishabhbhutani5835]

@@drpeyam Oh I see! I was confused because it reminded me of a theorem that you proved in your video on "Continuity and Compactness" essentially saying that if K is compact and f is cont. then f(K) is compact. I made a logically error in stating the theorem and thought it also implied the other implication. Thank you for the clarification; it is a very nice problem otherwise!

[@brendawilliams8062]

Thankyou

[@IoT_]

Not babushka , but Matryoshka 😂

[@dir2002usable]

"WTF Delta>0..." I am a little confused

[@subhasish-m]

It means what to find haha

[@eXfilPr4tik]

Errm I wasn't ready for the challenge

[@erzi3800]

Nice

[@VIKASVERMA-nk8uw]

Sir What course u are persueing plzz make sure about this.

[@drpeyam]

Analysis

[@sampurnanandsingh4564]

Sir what is use in daily life

[@drpeyam]

To get into UC Berkeley

[@adityaekbote8498]

Art is its own practicality

[@jjsmallpiece9234]

I gave up with maths when letters replaced numbers. You can't add or subtract letters ...

[@HDitzzDH]

Of course you can :)

[@cyrenux]

First

[@c1-math12]

WTF means what the fu** definition

[@dimageorgiev6801]

Babushka?

[@HDitzzDH]

Jesus christ...

[@VIKASVERMA-nk8uw]

Plzz make videos on engineering mathematics only

[@drpeyam]

Why, that is a ridiculous request

[@tomatrix7525]

Lol, why would he do that.... his audience are not predominently engineers, he himself isn’t, his channel isn’t built around engineering.....