Arguing with a math PhD friend be like (unscripted, unedited)

I asked a math Ph.D. @drpeyam to explain the topologist's sine circle (topologist sine curve) from the book "Real Analysis" by Charles Pugh. It is the set of all (x,y) so that x=0 and the absolute value of y is less than or equal to 1 or y=sin(1/x) for x belongs to (0,1]. This is a very interesting curve because it is path-connected but not locally path-connected.
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[blackpenredpen]

The book is "Real Analysis" by Charles Pugh

[thalesdaviddomingues]

Fact: 99% of discussions between mathematicians ends when one states an really elegant looking argument and the other one just pretend to understand in order to not look dumb.

[xissel]

"You don't need to understand it to teach it, you just have to sound confident enough that it's true"
I felt that

[LuisFlores-mujc]

I have never felt this attacked before. As an algebraist, the number of times I have told my friends to "just identify them" is uncountable.

[valizeth]

PhD in mathematics: Do woosh and you have a circle
Absolutely brilliant

[yazito]

You know you've reached the peak of maths when you can explain a concept using sounds effects from a free trial video editing software, some donuts & coffee, and most importantly, CONFIDENCE.

[Hyparbeem]

"It's implicit in the definition"
-"Fine, whatever."

Welcome to math!

[tejas]

"Take this, do whoosh, and then you have a circle."

Explains so much and yet so little 😂
This video was so fun to watch

[strayorion]

I wish I could do a PhD and keep my happiness just like Dr.Peyam

[bohanxu]

"but why are those two points the same?"
"because I identify them to be the same point."
I don't know why, but this just cracks me up... maybe because I would have the same exact confused reaction (as a young physics student) before I took some abstract math classes.

[pneujai]

Dr Peyam is always the happiest man I have ever seen

[oxrosdragon]

Dr. Peyam's enthusiasm for math never fails to amaze me 😂

[Elektrolite]

I think the problem with drawing a circle to represent identifying the two end points suggests that there are points in the set being graphed which lie on that circle, where in fact when you reach one end point of the graph you "teleport" to the other end point.

[isaacdeutsch]

I know I'm a mathematician because I was getting frustrated too I was like "we just mod out by the equivalence relation. the circle is no big deal they're homeomorphic"

[awflister]

"Do whoosh and you have a circle" really makes me think what the hell is actually going on in the heads of PhD people

[ablation]

Had Dr. Peyam as my differential equations and calculus 3 Professor. Really great at teaching!

[ultimatedude]

I'm not sure I'm interpreting this correctly, but here's how I'd think of it: Instead of the domain being [0, 2π), think of the domain as being ℝ mod 2π. This way, in a very real sense, those two points are equivalent, because 0 ≡ 2π (mod 2π).

[zihaojiang]

I had Charles Pugh as my real analysis teacher when he was visiting our math department that year. It was a very challenging but enjoyable experience.

[beatoriche]

Note that the function defined by f(x) = x*sin(1/x), a close relative of this one, may be used to construct a simply closed curve that has a point not reachable from the inside using straight lines. By "reachable from the inside using straight lines, " I mean that you can use a sequence of finitely many line segments not intersecting the curve to reach a given point on the boundary. But if you consider f(x) = x*sin(1/x) on, say, [0, 1] (continuously extending the function through f(0)=0), do the same thing with cosine but rotated ninety degrees (both curves reach the main diagonal of the coordinate system infinitely many times but never shoot beyond it because cosine and sine are bounded by the number 1, and the choice of sine/cosine ensures they only intersect at 0), and then just use straight lines to connect everything as necessary, you get a genuine simply closed curve – there is no self-intersection or anything –, and yet you can not reach the origin from its inside. That is one of my favorite counterexamples relating to the topology of the Euclidean plane.

[bruceypotato]

I only took up to Calculus II in undergrad, so I understand NOTHING of what he is saying, but goddamn, that was a hilarious interaction