@EdwardEvans No, but I had a seminar with only 6 people

It was expected to be very small though, so the talks were organized in groups of two so that we could have any even number of people and make it work

I'm probs gonna have a seminar with only 4 people next semester

Howdy, @Thor and demonic @Alessandro.

Hi

On which topic @Thorgott?

on manifolds homeomorphic to the 7-sphere

and hi Ted

That would be a good seminar

@psitae you can have two separate axiomatic systems where in one you prove a proposition "P", and in the other you prove "not P".

two

Yes, thanks

I don't really see the law of contradiction as an axiom though

isn't it more to do with propositional logic?

Axioms are just assumptions. You can assume it if you want. But I think it's deducible from other common axioms.

@Alessandro okay cuz I really wanna do this seminar

rofl

but there's 13 talks and I really don't wanna prepare 2

is it an easy deduction?

what other the axioms?

Also hi @Ted and @Thorgott

@psitae quod.lib.umich.edu/u/umhistmath/AAT3201.0001.001/…

*3.24

what page?

117

nvm

I can't just jump into that. I don't understand those symbols

oh shit it's principia mathematica

isn't that a failed endeavor because of Godel?

It's failed in the sense that it is not complete.

It's not failed because it proved something wrong.

@EdwardEvans That's definitely a risk

Lol in that case I'll just take the first talk and the 12th

since I'm already preparing the material for the first talk hahaha

Taking the first talk of every seminar is the most efficient way to farm credits haha

Better do it well then

yeah that's the plan

lool

the last number theory seminar went really well and that was basically a special case of this seminar

as in, the result

I'm afraid of asking more details

lol last semester's seminar was basically local Langlands for GL(1)

(also known as local class field theory)

Ah ok, I've only heard the name

btw do you have any news from Lukas?

i haven't heard anything from him for many months

I keep meaning to ask his bachelor supervisor but I'm scared of him hahaha

I see

@TedShifrin Any ideas for a (super) elementary proof that an isometry $\mathbb R^2\to\mathbb R^2$ is surjective?

Everything I can think of draws from linear algebra or properties of isometries (like decompositions as reflections).

What's your definition of isometry?

d(x,y) = d(Tx,Ty)

These are necessarily affine-linear, are they not?

Yes, but I don't want to use that.

That can be proven elementarily, though

I need better mileage than that.

If that's a car, I need a bike.

Or a horse-drawn buggy

Clearly, such a mapping is injective, hence by invariance of domain the image is open. From isometry, it also easily follows the image is closed. Since $\mathbb{R}^2$ is connected and the image is non-empty, the map is surjective. :P

Thorgott, that's a Boeing 747-8 VIP.

Yes, @anakhro. I think i even have this as an exercise in my algebra book. Use circles.

it's the first manned flight to space

Triangulate, as it were.

circlificate

You had that in your abstract algebra book? Or your linear one?

Abstract. I had a section on isometries of the plane (to strengthen complex numbers skills).

Truly a nice feature

Isometries of the plane were the first place I learned about abstract groups.

I was proud of making up this question. I think a few students may have done it for extra credit.

I am working on some slides for a presentation to some highschool students about isometries. And I realized that the book I learned about isometries from assumed surjectivity.

Odd.

Truly.

I was just at a loss because I have so far presented isometries as something they already knew and were familiar with vis rotations, translations, and reflections from elementary school.

But the surjectivity part seemed to be something you couldn't think of an isometry without, but I couldn't get something that fit nicely with my presentation. I didn't want to go to great lengths just to prove surjectivity.

My argument is pretty easy.

I think I just thought of a way to make it easier.

I assume it is something along the lines of taking three points in the range, then pulling back circles?

Oh wait, but you can't guarantee that the circles are contained in the range.

Well you can pull back the points and just intersect the circles of those radii

In the general setting you definitely want surjective to be part of the definition otherwise you get isometric embeddings

Of course.

@Alessandro @Balarka youtube.com/watch?v=g00XxIBIDFw