lol

I'm reading your last message

Hmm, this makes sense

But

If we think in terms of winding numbers

I'm not sure I've understood the definition of winding numbers fully yet

Isn't it the loop traced out by the points $F(\mathbf{x})$

Why should be shift the vectors anyway?

@BalarkaSen I think we need to get our definitions cleared first. $v(\mathbf{x})$ is a vector, right? What do you mean by "points" $v(\mathbf{x})$ ?

@BalarkaSen That sounds strange

I never heard that

Well, given that you follow the convention that it starts at origin and behaves like a position vector (which isn't really true in all of physics)

But then you guys don't care about physics :P

@BalarkaSen Okay, but that really sounds like a topology specific convention. I get it now, though

I was thinking of path traced out by the heads of the vectors when the tails remain in their original location

i.e. the boundary of the ball

That path would be just a point and not a closed loop around origin

@BalarkaSen It's not always possible to translate vectors in physics. For example you can't translate moment vectors ever (even if they are the difference of two moment vectors). Unless and until they are couple moments.

Translating moment vectors to origin would give you totally wrong results and make the field of statics/dynamics obsolete

@BalarkaSen What ?

@BalarkaSen Yeah, sure. I didn't refute that

Depends on what you define as origin. In statics and dynamics often origin is considered as different from fixed axis of rotation for making some calculations simpler.

Yup, I'm reading :)

@BalarkaSen Well, they are more interested in making buildings and flyovers earthquake proof :P Anyhow, yeah...this is interesting: "In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead displacement vectors, also called translation vectors or simply translations, between two points of the space."

I didn't know this term

Okay, I'm continuing reading nash sen then. Will ask you if I get stuck again. Cya!

@BalarkaSen Seems I'm again stuck :(. Page 6, Para 2: This means that $v(x)$ must have a $0$ somewhere... I don't follow how they conclude this from the previous sentence

Yes, that part I understood, because if you look at Figure 1.6, for all the vectors shown there, if you shift their tails to origin, then they (their heads) don't form a closed loop around it.

Now how does it follow that $v(x)=x-f(x)$ must be $0$ for some $x$ in the interior (rather on the boundary the circle C) ?

I'm trying to. Not sure why there can't be a jump in the winding number from 1 to 0, at some instant when we are doing the deformation gradually

@BalarkaSen I have some confusion. First of all, look at para 1 on page 6. It says we choose $\epsilon$ small enough so that C and C' do not intersect. If you start initially with $\delta B$ (i.e. map from boundary of $B$ to $C'$ inside $B$) then that is violated

Because in Figure 1.5 clearly C' is completely inside the bigger circle B

@BalarkaSen Okay, but I'm not able to relate this to the deformation logic you gave

@BalarkaSen Upto this sentence it is clear! Next: Why must $v$ vanish at that undefined point when $t=t_0$?

I need to re-think this thing once I guess

@BalarkaSen It's undefined :P

@BalarkaSen Yes, it's a integration of $1/z$ over the boundary of the curve at that instant of time when the index is undefined. Right?

$\frac{1}{2\pi i} \frac{1}{z}$ to be precise

That was what we used to define winding no. in the first place

Yes, corrected that

Well, if for some point $\mathbf{a}$ on the curve $v(\gamma_t)$, $1/\mathbf{a}$ were undefined. i.e. $\mathbf{a}=0$

That would just mean $\mathbf{a}$ and the origin coincide?

So, umm, ...

@BalarkaSen Right, yes. From here we need to conclude that $v(\mathbf{x})$ has a zero somewhere. Does $\mathbf{a}-f(\mathbf{a})=0$?

Nah, I'm not making sense

Phew, I think I get it finally. Roughly: As we deform $\partial B$ to $C$, we go through several curves $\gamma_t$ as time progresses continuously. Winding number is given by the integral over the curve formed by the heads of the vectors $v(x(t))$ generated by mapping all the points of $x(t)\in \gamma_t$ to $f(x(t))$.

At some time $t=T$ there must be a point $x(T) \in \gamma_T$, such that $v(x(T))=f(x(T))-x(T)=0$ in order to keep with with the fact that there is a sudden jump in winding number from $1$ to $0$, from $\text{index}_{\gamma_T}(v) = \frac1{2\pi i} \int_{v(\gamma_T)} \frac{dz}{z}=\text{undefined}$. So it basically means that $v(x(T))=0$,

implying that the point $x(T)$ on $\gamma_T$ is mapped to itself under $f$. That is there is at least one fixed in the ball $B$ (outside of $C$), which maps to itself under $f$.

Pheeeeeewwwwwwwwww.....so this is what topologists do all day :P

I like it though

@BalarkaSen Sure, but this is enough for today I think. I'll do something else now XD

Are you still doing those olympiad problems?

For CMI?

How's it going?

I see. I think the exam date's is in May. Lot's of time left, but good luck and keep practicing :)