How important is natural talent for getting into a PhD program?

Natural talent plays a role, but hard work and practice are crucial for success in advanced mathematics.

What influenced the speaker's interest in math?

The speaker's father helped with schoolwork and the speaker had a strong interest in science, especially during elementary school.

What challenges did the speaker face in college?

The speaker struggled with proof-based mathematics and found it difficult to transition from computation to proof understanding.

How did the speaker prepare for the PhD program?

The speaker increased their reading and practice, leveraging past mistakes to improve in their PhD studies.

What is the significance of the geometry course result mentioned?

It relates to a known theorem about volume and cross-sections of convex bodies and poses a potential PhD problem.

Natural Born Talent vs Practice

00:28:46

https://www.youtube.com/watch?v=AqK1YwBy1MI

Summary

TLDRThe speaker reflects on their journey in mathematics from K12 to PhD level, emphasizing the mix of natural talent and rigorous hard work over the years. Starting with a supportive upbringing, they developed a love for science and later math, excelling initially without realizing the need for deeper understanding. Various educational stages revealed increasing challenges, especially with proof-based mathematics, highlighting critical shifts where practice and study became more vital than innate ability. The speaker eventually managed to adapt to the demands of their PhD program, recognizing the importance of diligence and the transition from talent to skill cultivated through consistent effort.

Takeaways

👨‍🏫 Early support from family influenced education.
🔍 Interest in science led to a passion for math.
📐 Natural talent initially masked need for deeper understanding.
📚 Success in college required consistent practice.
🎓 Proof-based mathematics posed significant challenges.
📖 Reading textbooks improved comprehension in grad school.
🧠 Transition from talent to skill through hard work.
🔄 Teaching experience enhanced understanding of proofs.
🦠 Engaging with complex problems prepares for research.
💡 Growth through perseverance is key to mathematical mastery.

Timeline

00:00:00 - 00:05:00
The speaker discusses how natural talent and hard work contribute to success in mathematics, particularly in pursuit of a PhD. They reflect on their K12 years, noting how their father's involvement in education and personal interests in science fostered early enthusiasm for math, culminating in algebra appreciation by eighth grade.
00:05:00 - 00:10:00
Transitioning into college years, the speaker attributes initial success in calculus to a combination of talent and passion. However, they later encounter challenges with proof-based math courses that highlight limitations in their independent study skills, leading to a shift in perspective about their abilities and ultimately changing their major from math to environmental science.
00:10:00 - 00:15:00
The master's program sees a rekindling of interest in mathematics, prompting acceptance into a math-focused graduate program. Although they struggle with advanced concepts, they gradually adapt through practical engagement and attending classes that enforce understanding beyond surface-level, ultimately becoming a teaching assistant which enhances their learning experience.
00:15:00 - 00:20:00
In the PhD program, the speaker emphasizes the essential role of diligent reading and practice over innate talent. They observe improvements in their problem-solving abilities, realizing that mastery comes with time and sustained effort. The understanding gained during this period becomes a foundation for navigating complex mathematical theories and concepts.
00:20:00 - 00:28:46
The speaker concludes by sharing their experience with a geometry theorem in relation to a potential PhD problem, underlining the challenge of advancing in mathematics, where consistent practice and engagement with literature are crucial for significant academic achievements.

Mind Map

Video Q&A

How important is natural talent for getting into a PhD program?
Natural talent plays a role, but hard work and practice are crucial for success in advanced mathematics.
What influenced the speaker's interest in math?
The speaker's father helped with schoolwork and the speaker had a strong interest in science, especially during elementary school.
What challenges did the speaker face in college?
The speaker struggled with proof-based mathematics and found it difficult to transition from computation to proof understanding.
How did the speaker prepare for the PhD program?
The speaker increased their reading and practice, leveraging past mistakes to improve in their PhD studies.
What is the significance of the geometry course result mentioned?
It relates to a known theorem about volume and cross-sections of convex bodies and poses a potential PhD problem.

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00:00:00
a few days ago I got a comment on one of
00:00:03
my YouTube videos asking me how
00:00:06
much skill like a natural born talent
00:00:10
that you have helps you get to a PhD
00:00:12
program versus how much of it is act
00:00:15
just hard work and and doing doing
00:00:17
homework problems and stuff like that
00:00:19
like how much do you attribute to
00:00:21
natural-born Talent versus honing a
00:00:24
skill and I wanted to go in a little bit
00:00:28
more depth about
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my personal upbringing with and my
00:00:34
skills in math and how much of it
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started out as just natural-born talent
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and when that transitioned over to
00:00:41
insane amounts of
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practice so I divided it up into five
00:00:47
categories the K12 years uh the first
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two years of college the last two years
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of college which really that should save
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five because it took me five years to
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graduate college
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sadly and then we have the master's
00:01:06
program and then my PhD
00:01:08
program so we'll start with
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K12 um growing up my dad was pretty
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involved in my school work uh I always
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had him helping me with stuff that I
00:01:23
struggled with in school and helping me
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understand certain Concepts if the if
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the topics were too tough because my dad
00:01:32
uh was at one point a middle school
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science teacher science and math so but
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by the time I was in school he had he no
00:01:41
longer did that but he still had that uh
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teacher mentality so I was kind of like
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wherever my my teachers at school fell
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short you know he was able to fill in
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the details so I had that helping me and
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I also had the fact that you know I was
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kind of interested in science and
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general General it wasn't necessarily
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math but I had an interest in you know
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the
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planets I was really I really wanted to
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be an astronaut when I was very little
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and I would read
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encyclopedias about that type of stuff
00:02:13
and you know I was also interested in
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dinosaurs not so much earth science I
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was really interested in chemistry too
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but I wouldn't really say I was that
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interested in math but I did love the
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sciences and I you know when I was in
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really Young Elementary School I think
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that also contributed to it so that was
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my own personal interest helping me and
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then by the eighth grade that's when I
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started doing algebra like college
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algebra and then in high school we
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started doing interesting math I'll call
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it so I think 8th grade was when I
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really got interested in math and I was
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always wanting to do the next hardest
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thing you know I remember a lot of
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people struggling with algebra one but I
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liked it I thought that the problems
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were interesting and I remember solving
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the quadratic formula for the first time
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in eighth grade towards the end and I
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was I found that very satisfying how
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that formula was
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dered so the fact that I was very
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interested in it and I did have natural
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it was mostly Natural Born talent I
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think my brain was just built for that
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type of stuff in high school so I I had
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a reputation for being really good at
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math and science English not so much I
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was not good at reading and writing
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unfortunately cuz didn't like reading
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nonfic well I liked reading non-fiction
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like if I was going to read a a book
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about the state of Utah then that was
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way more interested interesting to me
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than reading a book like Harry Potter or
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something I don't know so in high school
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I kind of breezed through everything you
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know trigonometry was nothing I put in a
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little bit of effort for trig but if I
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got too lazy I'd fall behind but for
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trig you know it was easy
00:04:00
calculus one they don't even do like I
00:04:03
think they briefly touch on the
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anti-derivative in senior year of high
00:04:07
school if you take their calculus course
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but at that point I was taking college
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courses
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and we they go in farther depth with
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Calculus so for K12 I would say that
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Natural Born Talent kind of saved me and
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it gave me false sense of security of
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how good at math I was um and the first
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two years of college didn't really help
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either because during those two years I
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took Calculus 1 2 and three
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and those classes
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were simple to understand but they
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needed practice but I had so much
00:04:46
interest in calculus and I felt like I
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was doing something amazing by being
00:04:51
able to solve those C problems in the
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Stuart book that I would just stay up
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late at night and practice you know I
00:04:58
would do all the homework problems and I
00:04:59
would look and else and see what's in
00:05:01
there and see if I could try it I
00:05:02
remember I tried doing an application
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problem with Newton's law of cooling I
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believe it was like how long can I leave
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I can of pop in the freezer before it
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you know bursts I think I remember doing
00:05:14
that by by myself so I loved it and also
00:05:18
I took um differential equations here I
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remember differential equations and
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calculus 3 being tedious but not hard
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like it was really easy to make a
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mistake AK and get the wrong answer so I
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was annoyed with those classes because I
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understood the concepts like what was
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going on and how to solve problems but
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it would take like it would take me like
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an hour to make sure that I didn't screw
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it
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up um but I did both of those classes
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and I took math proofs and Elementary
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linear
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algebra I did those two classes as
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independent studies I think if I took
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Elementary linear algebra as you know a
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normal
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class then uh I wouldn't have struggled
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strg with it but I was not good at
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independent studying I relied too much
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on the teacher to tell me and explain me
00:06:05
things I couldn't read math books
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because I didn't like reading and the
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same was true for math books but for
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math
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proofs
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um he the instructor gave me a lot more
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atttention than he should have you know
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he was a handh holder which I was very
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thankful for but it kind of you it it's
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good in the moment but you pay for it
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later and when I took the class I ended
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up getting an A in it but it did not
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prepare me for University level math
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because that was the year three and five
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when you know that was proof-based
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mathematics that's where real variables
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reels its um ugly head and abstract
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algebra is in there too so I would say
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years so k12's Natural Born Talent years
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1 and
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two again Natural Born talent but also
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just an enthusiasm for the subject
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because I I would do it
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voluntarily but but then when I got to
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University during these These Years the
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the proof-based mathematics just
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absolutely wrecked me you know
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computation was no big deal like if
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there's no proofs involved
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then I I didn't have any problem with it
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but the proofs just like proving de
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Morgan's law I remember that was a
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homework problem we had once and I
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remember doing it but it was just it it
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didn't I didn't have any good feel for
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it and because I didn't have a good
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understanding of proofs I thought that
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if this was what mathematics was like at
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the end of your University
00:07:36
degree then it wasn't for me I also had
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some bad teachers in there unfortunately
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during these years um so that's why I
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left mathematics I thought it was too
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hard for me and I got my degree in
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environmental science but I had friends
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that stayed in the program and I kind of
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was jealous this is kind of not a good
00:07:57
reason to do a degree but because of you
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know how much I loved math during these
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years and how much I walked away from it
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during these years and I had friends
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that finished it it made me really want
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to go back and get a degree in
00:08:12
mathematics because it felt like the
00:08:14
thing I should be doing you know I do
00:08:17
what I think I should do well okay I was
00:08:21
going to say I do the things I think I
00:08:23
should do not the things I want to do
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but that's absolutely not true I'll play
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video games one night if I really don't
00:08:29
want to do my 4A analysis homework so I
00:08:31
got to keep myself
00:08:32
honest but uh during the master's
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[Music]
00:08:37
program I applied to a couple math
00:08:40
degree math programs but also some
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environmental science
00:08:44
programs and I got accepted into a math
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Masters and my friend was going to go
00:08:49
into that program as well so that's part
00:08:52
of the reason why I did it cuz I really
00:08:55
wanted to get back into
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mathematics so uh
00:09:00
I entered the master's program they made
00:09:02
me take real variables and abstract
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algebra at the undergrad level because I
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didn't have these years in math so I
00:09:11
that was my first year and my second
00:09:13
year is when I took measure Theory and
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you know abstract algebra instead of
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modern
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algebra which is the just the graduate
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level version and I remember at first it
00:09:23
took me a while to get used to it but I
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had to look up a lot of stuff on the
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inter internet to get comfortable the
00:09:31
book the books that we had were good I
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will say that the books that we had
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during these years was good except for
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maybe the Zigman Weeden book I know I
00:09:40
use that book a lot these days it's just
00:09:42
it's hard to look up information in that
00:09:43
book but to comment on how much my
00:09:46
talents were helping me here it was very
00:09:49
little I I had to look up a lot of
00:09:52
Solutions on the internet and I felt
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guilty about doing that because it just
00:09:57
feels like I'm just copying what other
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people are doing doing but I had to
00:10:00
reason with myself like look I'll borrow
00:10:03
the information now but I'll get
00:10:07
comfortable with it later I had a I
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think my probability Theory
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instructor also said something like that
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when I took his class during these years
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because he says well as long as it you
00:10:19
know you do it not knowing how by just
00:10:22
watching me but then it becomes yours
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you know no one starts out running
00:10:26
marathons you have to crawl first so
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there's like a transition the math
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proofs is the big barrier and I ended up
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taking like I say I take math proofs
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like four times because what I did was I
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took the independent study I got to
00:10:42
University couldn't do real variables so
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I dropped it and I signed up for their
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math proofs program so that was my
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second attempt and then my third attempt
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was when I took a computer science
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course which was called discret
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structures and I really didn't need to
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take it
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because it was kind of U
00:11:02
lowlevel it's math proofs for computer
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scientists so that was the third time I
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took it and because I did well in it
00:11:11
because it was my third time uh the
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instructor asked me if I would be a
00:11:14
teaching assistant for the class and I
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said yes so the teaching assistant
00:11:18
position was the fourth time I took math
00:11:21
proofs cuz then at that point I was
00:11:22
telling other people how to prove stuff
00:11:25
and it was because of that I became a
00:11:27
lot more comfortable you know that's
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where practice picks up
00:11:31
with with doing
00:11:34
a what was I saying I just distracted
00:11:36
myself that's where practice overtakes
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natural-born talent and because of that
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it helped me succeed in the master's
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program cuz I could do I could do a lot
00:11:46
of problems by myself you know but there
00:11:49
were more problems that I just had to
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look up cuz it's like where would you
00:11:53
come up with this trick you know
00:11:55
sometimes I'll see that like where would
00:11:56
this come up I was also not doing myself
00:11:59
any favors by not reading the textbooks
00:12:01
this was the big issue I would just use
00:12:03
class notes and sometimes look stuff up
00:12:05
in the book if I needed them what I was
00:12:07
not doing is what I ended up doing in
00:12:10
the PHD program so let me move on to
00:12:12
here the first year of the PHD program I
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took abstract algebra real analysis and
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complex
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analysis and I did not take very good
00:12:22
notes but I was more comfortable reading
00:12:25
books and it was because I took bad
00:12:27
notes and because I knew that how tough
00:12:30
that qu was going to be I was like oh
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crap I really do need to study harder
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because this stuff is not intuitive to
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me at all I'm really struggling here you
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know I thought Masters was hard but PhD
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was like they're going another step and
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that's when I started reading like a
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madman I just started reading all these
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not all these books you see but a lot of
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these books you see I started reading
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and taking notes over you've seen my
00:12:56
other YouTube videos and so in the PHD
00:13:00
program I would say natural-born Talent
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plays almost no role it plays some role
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but it plays a role for everyone because
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every single person that's in the PHD
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program you know during at their high
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school they were probably the top math
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student in their High
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School probably maybe not everyone but
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when you when you see what kind of math
00:13:23
that they're doing at PhD like the
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homework problems they can do and the
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papers that they read and how many books
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they read They're comparable to me so
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when I saw you know maybe that first
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year that I wasn't doing as much as I
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should have been doing and I was also
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afraid of that test that's when I
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started it became all practice and
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eventually you know those all those
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years of practice they add up and then
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you just see the solution to a lot of
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these measure Theory problems like over
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the Christmas break I took this book
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with
00:13:57
me I took this book home with me and I
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opened it up to like chapter
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3 cuz I was interested in seeing like
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how obvious cuz when I was in the
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master's program and I had this book I
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looked at these problems I was like uh
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oh my gosh I can't do any of these
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problems but then I was looking at these
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you know problems over the break and I
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read through it I was like oh that
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doesn't really seem as bad as I thought
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it would you know you you start to see
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things after years of
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practice like number 10 for example look
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how simple 10 is show that the measure
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of the Union Plus the measure of the
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intersection is equal to the sum of
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measures you know I remember a problem
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like that would have given me a headache
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cuz it's an easy statement right even
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when I was a master like okay that's an
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easy statement but how do you prove it
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you know that's you that was back when I
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didn't have as much skill as I have now
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but after 5 years maybe even six years
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now of just reading all these measure
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Theory books you know I can I look at
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that and go like oh I know what to
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do
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so eventually practice becomes Talent
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it's not natural Bor talent but you know
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you do you do get better and you see
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yourself get
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better so and it helps you it helps you
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because uh you need that when you're
00:15:21
trying to work on Research problems
00:15:24
which is kind of what I want to do now
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so let me talk about this result that we
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did in my geometry course so this is a
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known result it's a I'm not sure which
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books it it you see it in I think it's
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called alexandrov's theorem I'm not sure
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but here's what the the theorem
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says you have an origin symmetric convex
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body K in
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RN and it says that for any unit Vector
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in the direction of the sphere so this
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is the n minus one sphere it's just
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think of the unit sphere and RN but
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you're missing the inside of it you're
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just looking at the Shell so XI is just
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a direction of length one whichever
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direction you pick then the N minus1
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volume of your body K intersected with
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the plane that's orthogonal to that
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Vector so this volume this n minus1
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volume which think of it as area if n is
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equal to
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3 then this is just the ukian bow in R3
00:16:36
with hollow inside then the area of the
00:16:40
cross-section that you slice if it's if
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K is a potato you just slice the potato
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in any way if the area of the potato
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chip that you create is absolutely
00:16:48
constant meaning that it's independent
00:16:50
of your well uh I guess it'd be your
00:16:55
your direction XI well okay it has to be
00:16:58
con Conant regardless of how you cut it
00:17:00
so your area is always constant then the
00:17:03
conclusion is that K must be some ball
00:17:06
meaning it's a your body K is the
00:17:10
ball because balls no matter where you
00:17:13
slice them through the center they're
00:17:14
always that Circle you know what I mean
00:17:17
so the area of that circle is always
00:17:18
constant regardless of how you cut cut K
00:17:21
so this is a theorem it tells you
00:17:23
something about a body K if you know
00:17:25
something about the area of the
00:17:27
cross-sections and here is the proof of
00:17:30
it now I am cheating a little bit here
00:17:32
because this proof uses a another result
00:17:35
a Lemma if you want and I don't have the
00:17:37
proof of the Lemma so we talked about it
00:17:39
at a later date so he gave us you know
00:17:42
the fruits and vegetables as he says it
00:17:44
and then he had to explain where the
00:17:47
where the LMA comes from but here's how
00:17:49
it's proved so we use this function
00:17:51
called row K row K is called the radial
00:17:55
function and the radial function is
00:17:57
defined as the maximum of Lambda such
00:18:00
that Lambda time your anglea is an
00:18:03
element of K so Theta is a direction
00:18:06
that you have in your cross-section and
00:18:09
you pick so if I were to draw a picture
00:18:11
it's easier to explain with a picture so
00:18:13
let's say that this is your
00:18:16
cross-section and here is the center you
00:18:19
pick a direction let's say it's Theta so
00:18:22
this is my D my my Vector Theta and the
00:18:25
radial function says okay it's the
00:18:27
Lambda
00:18:29
that takes you until you hit the
00:18:31
boundary so the
00:18:34
length of this Vector that you have to
00:18:37
multiply by Theta Lambda
00:18:39
Theta that length is what the radial
00:18:41
function
00:18:45
is now we look at the area of the
00:18:48
crosssections think of the
00:18:50
three-dimensional case cuz it's just
00:18:51
easier that way so the N minus1 volume
00:18:54
of K intersected with your plane which
00:18:55
is just the area of the potato chip is
00:18:58
by definition equal to the integral over
00:19:01
that Subspace that you have the
00:19:03
orthogonal Subspace of the
00:19:06
characteristic function of KP DP where
00:19:08
this is the differential of the
00:19:11
two-dimensional leg Le measure it just
00:19:14
Returns the area of the shape that's all
00:19:15
it does the the
00:19:18
cross-section and then you pass through
00:19:21
uh polar coordinates so bipolar
00:19:23
coordinates we're looking at the radius
00:19:25
and this D Sigma which represents the
00:19:29
uh it's essentially equivalent to hold
00:19:32
on let me make sure I don't screw this
00:19:33
up it's basically the
00:19:37
measure instead of area like in the in
00:19:40
the coordinate space you have area like
00:19:43
this but in the in polar coordinates you
00:19:46
have more of like you're measuring how
00:19:49
much area you get when you rotate I'm
00:19:51
not explaining that very very well but
00:19:53
it's just polar coordinates that's all
00:19:54
it is you're passing through a polar
00:19:56
coordinates when you go from here to
00:19:57
here
00:19:59
so you get this characteristic function
00:20:01
times of the radius time Sigma and then
00:20:04
this is your Jacobian that sits out here
00:20:07
when you do a switch change of
00:20:09
variables and when you integrate from 0o
00:20:12
to Infinity you really only go up to the
00:20:15
value of your radial function because
00:20:17
once you pass once you get outside of K
00:20:20
you just get zero so it doesn't make any
00:20:23
sense just to go up to it only makes
00:20:25
sense when you go up to row K
00:20:28
so one way one way that we can write
00:20:30
this is I've explained it kind of here I
00:20:32
wrote kind of sloppy here but the
00:20:34
integral of x times the characteristic
00:20:36
function of e is just integral of f over
00:20:38
e that's all that I do here so I can
00:20:41
move this guy sort of you know so to
00:20:43
speak up here by just writing row K
00:20:46
Sigma and then it just becomes an
00:20:48
integral that you can evaluate so just
00:20:51
add one to the exponent divide by the
00:20:52
new exponent when you add to the
00:20:56
exponent one you'll get nus1 which is
00:20:59
where this comes from if you plug in
00:21:01
zero you get nothing if you plug in this
00:21:03
you get this and then you have to divide
00:21:06
by n minus one by pulling it out here
00:21:08
and then you just get the singular
00:21:10
integral so this is what we have this is
00:21:13
what the two-dimensional or not two-
00:21:16
dimensional but the N minus one volume
00:21:18
of the cross-section
00:21:20
is okay so notice that when you
00:21:25
integrate over the sphere intersected
00:21:28
with the orthogonal Subspace that's your
00:21:32
chip that you want to use your uh radial
00:21:35
function on so this is the
00:21:37
chip uh this is by definition the set of
00:21:41
all Sigma these vectors Sigma such that
00:21:44
if you take the dotproduct of Sigma with
00:21:46
that normal Vector you get zero because
00:21:49
that's just what it is everything all
00:21:52
vectors in this Subspace here are
00:21:54
orthogonal to
00:21:56
XI so you have this definition
00:21:58
definition and we also know that row K
00:22:01
is even how do we know that it's even
00:22:03
because K is origin symmetric so it's an
00:22:06
even function which shows up important
00:22:09
later and then for the next line he kind
00:22:12
of uh you know my instructor kind of
00:22:16
changes variables to S uh what is this s
00:22:20
yeah uh which is okay but it got you
00:22:23
know it's kind of confusing cuz he throw
00:22:24
Sigma Theta and S and XI all over the
00:22:27
place too many Greek letters is what I'm
00:22:29
saying so you take this integral I move
00:22:32
it down here I just rewrote it in a nice
00:22:36
way and what do you know we know that
00:22:40
this is equal to a constant so I'm
00:22:43
skipping this line and going down here
00:22:45
it's equal to a
00:22:46
constant because we said it was in the
00:22:48
very beginning this volume is equal to a
00:22:52
constant
00:22:53
and one way that you can write this is
00:22:56
you can write this Con as the integral
00:23:00
of that constant
00:23:02
nus1 divid the measure of this space
00:23:04
here because if you integrate I mean if
00:23:07
we just pull this stuff out here what
00:23:08
are you going to get here you're going
00:23:09
to get the measure of this space here
00:23:11
and the measure of this space will
00:23:12
cancel with this number n minus one will
00:23:14
cancel with this and you're just left
00:23:16
over with the constant and it's a
00:23:18
complicated way of writing it and the
00:23:19
question is why are we writing it this
00:23:21
way and the reason we write it this way
00:23:24
is because if you push this inside the
00:23:26
integral and you combine this nus one
00:23:28
with this constant you'll just get a
00:23:30
different constant which we denote as
00:23:32
constant
00:23:33
Tilda so all of this just to say that
00:23:37
this integral is equal to 1 n-1 *
00:23:41
integral of a
00:23:43
constant
00:23:46
okay next
00:23:50
page if you combine the two integrals by
00:23:53
subtracting the left side from the right
00:23:55
side you will get why is my paper C
00:23:58
that's bugging
00:23:59
me if you push this if you move this
00:24:03
integral to the other side and then
00:24:04
subtract and combine under a single
00:24:06
integral you'll get this equivalent
00:24:08
statement that this function minus this
00:24:11
constant and if you integrate you get
00:24:12
zero and that has to be true for every
00:24:15
Theta that you choose in s again we
00:24:18
switch from
00:24:20
Theta I think there's a lot of change of
00:24:22
variables here but it all makes sense I
00:24:25
think the notation was a little bit
00:24:27
sloppy but it's fine and this is where
00:24:30
the trip comes from so I said we needed
00:24:32
a Lemma or a corer we called it a corer
00:24:34
in class is that we need to invoke the
00:24:39
funk heck theorem I think is it's called
00:24:41
which says the following if you have a
00:24:43
function that is continuous on the shell
00:24:47
of the sphere you know the outside part
00:24:50
such that if the integral of f of over
00:24:54
this orthogonal Subspace is equal to
00:24:57
zero and that's true for every
00:24:59
Theta if your function is continuous and
00:25:02
it has this integral property then it is
00:25:05
enough to say that f is an odd
00:25:08
function so I don't prove that here we
00:25:10
just use it we actually covered it in
00:25:11
class
00:25:13
yesterday um this correl says that this
00:25:18
function is e uh not even but um
00:25:20
continuous definitely CU row is an even
00:25:22
function raising it to n minus one is
00:25:24
even subtracting constant is still even
00:25:27
and we're saying that this integral is
00:25:29
equal to zero for every
00:25:33
Theta so that means that this function
00:25:35
must be odd so this function is odd but
00:25:39
row K is an even
00:25:41
function which means that if you take an
00:25:45
even function raise it to n minus one
00:25:47
it's still even and if you subtract
00:25:48
constants still even so this means that
00:25:51
you have this function inside is
00:25:54
simultaneously even and odd which means
00:25:57
that the function function has to be
00:25:58
identically equal to zero CU that's the
00:26:00
only function that I believe is both
00:26:02
even and
00:26:03
odd and if it's identically equal to
00:26:05
zero all you have to do is algebra just
00:26:07
move the constant over take the N minus
00:26:09
one root and you'll see that the radial
00:26:13
function itself is constant and if the
00:26:15
radial function is constant then the
00:26:17
only possible shape that you get is the
00:26:19
bowl so K must be a bowl and that's the
00:26:22
end of the proof it's kind of slick I
00:26:25
thought I thought this theorem would
00:26:26
take a lot longer to prove
00:26:28
but you do use this little trick here in
00:26:30
the coral so that kind of shortens it up
00:26:33
and why did I show this it's because
00:26:35
there is a potential PhD problem I say
00:26:38
that because my instructor asked you
00:26:42
know it's a famous
00:26:45
problem that if you solve this then it's
00:26:47
for sure worth a PhD so my ears perked
00:26:49
up I was like I want PhD so here's how
00:26:53
it's
00:26:54
related you have a body that that's
00:26:57
origin
00:26:58
symmetric and it's a convex body so far
00:27:02
everything's the
00:27:03
same and we say that for any Vector any
00:27:07
unit normal Vector that we have we say
00:27:11
that the
00:27:12
volume except this time we're looking at
00:27:15
nus
00:27:17
2 of the boundary of K
00:27:21
intersected with this guy is
00:27:26
constant and then the question
00:27:28
is is
00:27:33
cable so this is the well I shouldn't
00:27:35
put theorem because we don't know what
00:27:38
it is yet so uh
00:27:43
problem so all this is saying how this
00:27:45
is different from the previous problem
00:27:47
is that you said the areas of the
00:27:49
cross-sections were always constant
00:27:50
regardless of how you sliced the potato
00:27:53
but now you're just looking at the
00:27:54
perimeter of the cross-sections and
00:27:56
you're saying those are constant and
00:27:58
this is a lot difficult or this is much
00:28:01
more difficult because you don't have
00:28:03
the radial function trick in this one
00:28:06
there's no con there's a convolution
00:28:08
that's involved in the previous theorem
00:28:11
that's embedded in the corollary that
00:28:13
you can't uh use here and that's what
00:28:16
makes this problem difficult and then in
00:28:18
class he said if you solve it you
00:28:20
essentially that's that's worth a PhD to
00:28:22
him I don't know if I'll be able to do
00:28:25
anything with this but it intrigued me
00:28:27
enough to where I wanted to look at it
00:28:30
more
00:28:31
so most likely won't be able to do
00:28:34
anything with it but that's one of the
00:28:37
things you do in a PhD program you just
00:28:38
look at a bunch of problems hopefully
00:28:40
you can make progress on something
00:28:42
anyway this video is way too long so I'm
00:28:44
going to call it the quits here