I mean

$\{3\} \subseteq \Bbb R$ is a Lebesgue-measurable set

with measure zero

I mean, it helps that this is presumably coming from a problem set

So a simple example has a better than usual chance of working out nicely

@Semiclassical Do you agree with Rudi that QM should be QED above?

In classical mechanics you cannot solve the 3-body problem. In quantum mechanics you cannot solve the 2 body problem in relativistic quantum mechanics you cannot solve the 1-body problem and in QED you cannot solve the 0-body problem

So the 0-body problem is the one where the "quantum groups" come into play I believe.

or maybe I should just say "in a quantized setting" and leave it more vague

Not really, but quantum groups are beyond me. I think they show up in Physics in the context of exactly solvable models tho

Which I think tend to be many-body systems?

@Rudi_Birnbaum what do you mean by “cannot solve”? (i know little of physics)

So in that sense it might be right. But I don’t have a good sense for it

@LeakyNun: Good question, the same as in maths, how to invert the $\zeta$ function? Can we or can't we?

@Rudi_Birnbaum and in mathematics you cannot solve the quintic :p

where you cannot solve the "4" then?

@Rudi_Birnbaum well it’s not bijective i presume

maybe it is

or maybe you mean “closed form”

as in, an expression involving exponentials and logarithms

@not only: 3-body problem has "unstable" solution

s

From Wikipedia: “The term "quantum group" first appeared in the theory of quantum integrable systems, which was then formalized by Vladimir Drinfeld and Michio Jimbo as a particular class of Hopf algebra.”

michio jimbo

So for the three body problem there are no analytical solutions but even approximative approaches are spoiled by the limit of the knowledge of the boundary conditions

ok

In QM the n-electron problem is NP.

Even for 2 electrons its really hard to get good results, though there is nothing "chaotic" as in the CM 3-body one.

Ok, so it at least sounds like my original phrasing would be a bad idea (if it could cause a scientist to go "that's not quite right", then it is a no-go for this application)

You have operators with kernels that are everywhere non-continous.

There is such a subject as ‘quantum chaos’ but it’s a weird topic iirc

And worst is no one know how they look like nor there is a systematic way to approach them

That is called "DFT" density functional theory.

Quantum chaos sounds like hype

In relativistic QM you need high order perturbation theory to get the 1-particle problem half-ways ok.

And in QED you need that for the vacuum state.

Perturbation is nonsense

@Zee: How comes?

That is what the guys do all the time with their Feynman diagrams

In Cern and so on

Am talking nonsense but from a gut feeling type of thing

When ever I open a physics book , I can sorta understand the math of what’s going on

Until the perturbation part

Seems sloppy as hell

I'm sure it is :-)

I guess I will end up going with something like "quantum groups are of interest in mathematical physics due to their relation to quantum integrable systems"

Write an introduction?

Now I just need to make sure my formulation does not become much longer than what I already had, or I might go over the character limit

I mean for a maths paper?

@MikeMiller @BalarkaSen I forgot that the uniform spaces need to be Hausdorff in the statement of the theorem

All I know about quantum groups is the people who work in it love representation theory

@Rudi_Birnbaum Grant application. The summary needs to describe why the research is important in terms understandable by scientists who are not necessarily mathematicians

(and in fact, there are no mathematicians on the evaluation committee)

There is this story about the mathematician who found out that his results found an application. Then he immediately stopped his work on the topic. You guys should continue that way. Otherwise we are soon lost.

@TobiasKildetoft: Why prostitute yourself?

@Rudi_Birnbaum Sure, but we need to pretend when applying for grants that can also be applied for by people doing "real" stuff

@TobiasKildetoft: We too, but thats shit.

@Rudi_Birnbaum Because I have this weird craving for food sometimes

And this weird desire to be able to pay one's rent

yeah, that too

@TobiasKildetoft: Sure.

Is that not an issue which is discussed among mathematicians?

is what?

@TobiasKildetoft: If it really makes sense to justify the research with "applications".

I mean from side of the scientists

@Rudi_Birnbaum I think most mathematicians are content with math for the sake of math. But especially grants that are not paid with public money tend to require some more nods towards applications. Even more so when the applications are not sent out for individual expert review

@TobiasKildetoft: When I get grants to review I speak out against that.

At least the comments I have had from expert reviewers on similar applications have been along the lines of "this sounds good because it is good math, and that is all that matters"

@TobiasKildetoft: I would be happy about getting these :-) But of course I also see that it makes sense if one reflects ocassionally a bit about the "broader impact" of science.

Btw. Do you as mathematicians consider yourself scientists?

Hmm, good question. I rarely think about it. Sometimes I do and sometimes I don't.

@Rudi_Birnbaum I'm not a mathematician yet, but in the English usage of the word probably not, people seem to use "science" synonymously for "natural science" there, but I'd say in German, with "Wissenschaft" we also include stuff like philosophy, history etc. and then definitely yes

Simpler version of what I had before. Suppose I've got four unit 3-vectors $u^1,u^2,v^1,v^2$. Let me further assume that these two pairs of vectors generate distinct planes.

@MatheinBoulomenos I suppose that has the same origin as in Danish, "creation of knowledge"?

Vetenskap?

Or maybe more like "acquisition of knowledge"?

Or is it different in Danish

also hi all

@ÍgjøgnumMeg Videnskab in Danish, so almost the same

@MatheinBoulomenos idk if i interpret the english word science as consistently referring to natural sciences but i do it at least sometimes

Ah okay

@MatheinBoulomenos: Yes I like also that view, to see maths as "Geisteswissenschaft" which is not properly translated as "humanity". And the Union of "Naturwissenschaft" and "Geisteswissenschaft" as "Wissenschaft" as science. Most continental Europeans might agree there.

Suppose I project $u^1,u^2$ onto the plane generated by $v^1,v^2$ and vice versa. Will that leave the inner products $\langle u^i,v^j\rangle$ unchanged?

@ÍgjøgnumMeg

I want to say it does but I can't quite convince myself.

hmm

@Semiclassical: Unitary transformations?

I guess I don't need the vice versa.

@GFauxPas did you check out my question?

what question, no

These definitely aren't unitary, since the lengths of the vectors can decrease upon projection.

The one about a useless group

@Semiclassical: I see what you mean.

well semi maybe it helps that with projections P on Hilbert spaces you gave

@Rudi_Birnbaum But for the most part what matters (at least at the moment for me) is that math is generally grouped with the natural sciences (and sometimes engineering) when it comes to most grants.

@TobiasKildetoft I don't know if the suffix "-schaft" comes from "schaffen", but "Videnskab" is definitely closely related

(Px,y)=(x,Py)=(Px,Py)

is that useful

Only if i've got (Px,y)=(x,y) as well

plus, I don't expect all the inner products to be preserved

@MatheinBoulomenos: Its something the Germanic languages use to construct "abstracta" like landscape

I can certainly have the lengths of the vectors decrease upon projection

I see

@MatheinBoulomenos Hmm, actually isn't "wissen" more like "knowing" than "knowledge". So it becomes "the craft of knowing"

So the relation to schaffen is pobably only weak-

geo, link it

@GFauxPas the one about a useless group specifically the edit part

link it

@TobiasKildetoft wissen is a verb, too, but as a noun it's more like "knowledge"

ok

one thing I can do is Gram-Schmidt so that I can assume v1.v2=u1.u2=0 without loss of generality

I was talking with @ÍgjøgnumMeg about it

Also, something that took me a while to figure out why. The foundation I am applying for a grant from right now don't support research in human medicine and similar.

Turns out to be because there is a "competing" foundation that almost solely supports that.

looks like someone answered it for you already

@Semiclassical: Is it something like a rotation?

Rotations preserve lengths.

@MatheinBoulomenos @TobiasKildetoft @Rudi_Birnbaum there's an interesting project in England called Anglish that seeks to replace all non-germanic rooted words in the English language with Germanic equivalents

you never define $x$ geo

(I think it's an experiment, hopefully not some right wing crap)

@ÍgjøgnumMeg That sounds like it would sound hilarious

I feel like it's backwards to construct a group and ask whether it's interesting.

@ÍgjøgnumMeg: Saw it on youtube, quite interesting, yes!

If it's not interesting, then it being a group or not hardly matters.

@ÍgjøgnumMeg that stuff existed in the history of German some while ago (and I'm sure not only there)

@TobiasKildetoft it is quite funny, I think they call computers "reckoners"

a country is called a "rike"

@ÍgjøgnumMeg Well, if you study CS in Copenhagen, you will call them "datamater"

Reckoncraft!

That brings a different connotation to the phrase 'the hour of reckoning'

There were Germans who wanted to get rid of words with latin roots, so they suggested "Riechkolben" or "Gesichtserker" instead of "Nase" which comes from the Latin naso, middle-Latin nasus

"Reckoncraft was infound long before sparkwork." brilliant

lol

As an Italian I think it'd be far better to remove all Germanic words from English and use only words derived from Latin. Actually the best would be if everyone spoke Italian

Riechkolben hahaha

@Somethinh completely different: In modular forms, is there anything like the function is its self Fourier-transform?

I guess a long computing time would be 'the hours of reckoning'

@Rudi_Birnbaum do you fourier transform modular forms? I thought you only Fourier series them, as they are periodic

@GFauxPas I know I'm wondering about the edit because no one touched on that

so define it

@MatheinBoulomenos: I have no idea about the peculiarities but there is this transformation property wich looks like diffraction containing this (-1/\tau) ..

it's your question

@MatheinBoulomenos: So ot looks like the function is kind of inverted at this region around the origin. That reminded me in "diffraction" which is kind of FT.

some theories of diffraction look like Fourier transforms

Mellin transforms are used with modular forms to get a Dirichlet series

@GFauxPas what do you want me to define? X is a variable not equal to 1

well, okay, they all look like Fourier transforms, but the thing being transformed can be more interesting than a function

measures are where its at

@geocalc33 WHAT KIND OF VARIABLE IS $x$?

A real number? A complex number? A quaternion?

Real

A $p$-adic number?

A fish?

$2 + \text{fish}$

Real lol

@MatheinBoulomenos: $\eta(-1/\tau) = \sqrt{-i \tau}\eta(\tau)$

$\color{red}{\text{fish}} \color{blue}{\ \text{fish}}$

$f(\uparrow)=\downarrow$

@ÍgjøgnumMeg that makes perfect sense in $\Bbb{Z}[f,i,s,h]$

@Mathein nice

$\mathscr{fish}$

aw

but $f(\downarrow)=-\uparrow$

$\mathscr{f}$

where is mathscr

$\mathscr{FISH}$

$\mathscr{F I S H}$

hm doesn't work on lower case letters

$\mathfrak{fish}$

So. Metal.

$\mathfrak{FISH}$

ghoti

haha

@Rudi_Birnbaum I don't see the relation to the Fourier transform, but that's probably just me being ignorant about actual physical intuition for Fourier transforms

Anyway

Ah, now I remember, there are some functional equations that are derived with Poisson summation applied to something like $e^{-\pi \|x\|^2}$

mmm... Fish Summation Formula

@MatheinBoulomenos: So when you use instead of the argument the inverse argument that is one aspect of diffraction.

@XanderHenderson can you speak to my edit?

this should give you the functional equation for a theta series iirc

Modular forms are functions on the upper half plane right - how do you even define the Fourier transform ?

In the question

@loch The Fourier transform can be defined for pretty much any abelian group

@loch: With real functions its no big deal that you can have functions which are their own FT.

is there a verb missing somewhere in there, @Rudi_Birnbaum?

@Rudi_Birnbaum I think the functional equation for at least some Theta functions can be derived with Poisson summation and the fact that Gaussians are their own Fourier transform

@MatheinBoulomenos: cool :-)

this sounds like it would answer your qestion somehow. But Fourier transforming modular forms themselves doesn't make much sense, I think

a partial answer to what I had above

Suppose I do Gram-Schmidt and rescale so that $v^1,v^2$ are orthonormal

Not sure if I’m being dumb but the upper half plane is a group?

then $P=v^1\otimes v^1+v^2 \otimes v^2$ is a projection matrix

$ \zeta(s) $ to the power of 1/lnx only has one inverse for all s?

hi @loch

Hey@LeakyNun

@loch A modular form is not just a function defined on the upper half-plane---there is an underlying group structure, too

and we have $u^i \cdot v^j = u^i \cdot P v^j =Pu^i \cdot v^j$

@loch i’m free

the poster has been submittes

so $Pu^1,Pu^2$ will have the same inner products with $v^1,v^2$.

@XanderHenderson What group structure would that be?

i am now free... to do other maths :p

@Leaky congrats

I can then deduce the projection matrix $P'$ for the $Pu^1,Pu^2$ subspace and so obtain $P'v^1,P'v^2$ without changing the inner products

The upper half plane is not a group in an obvious/canonical way, but it's at least a homogenous space $\operatorname{SL}_2(\Bbb R)/\operatorname{SO}_2(\Bbb R)$

I don't know much about Lie groups

@XanderHenderson: Is it the "double periodicity", this group structure?

maybe you can Fourier transform functions on a homogenous space, too

they're not to be trusted

@Rudi_Birnbaum elliptic functions are the double periodic ones, modular forms are different

Oups :|

@Rudi_Birnbaum This is above my paygrade, but: modular forms on Wikipedia

@XanderHenderson Still not sure what abelian group you mean for there to be here

so, depending on what you are doing, $SL_2$

SL(2/Z)

@XanderHenderson comparing the group G={e^s/lnx}, (x is a real variable not equal to one, s is a real number), to the group K={zeta(s)^1/lnx} it looks like there is only one inverse for all s for group K unlike for group G

K isn't a group...

What type of group is K?

It's not.

a non-group

The only reason G was a group was because e^(x+y)=e^x*e^y

@XanderHenderson their transformation properties are related to $\operatorname{SL}_2(\Bbb Z)$ (they are not exactly invariant though, unless they have weigth 0), but they are not defined as functions on $\operatorname{SL}_2(\Bbb Z)$

There's no such statement for zeta(s) and therefore there's no similar group structure.

tumbling functions together in an arbitrary way does not a group make

@geocalc33: What I don't understand in the answer is, that he claims there are "many" group homomorphisms $\Bbb R \to \Bbb R^+$. I thought theres only the one (differentiable): e^x?

$2^x$ is also one

any positive base should work

@LeakyNun congrats!

@Rudi_Birnbaum $e^{ax}$

but yeah these are the only Borel-measurable ones

@Rudi_Birnbaum If you don't require it to play nicely with the topology, you get uncountably many, since once you fix one, you are looking at automorphisms of an uncountably dimensional vector spaces over the rationals

@MatheinBoulomenos: OK @TobiasKildetoft: cool stuff!

@TobiasKildetoft: So like each permutation of $\Bbb R$?

permutations need not respect the group structure

@MatheinBoulomenos: Oh yes!

Well, you do get a copy of the group of permutations of the reals, but not directly like that. You get it by fixing a basis as a vector space and permuting the basis vectors

(and this basis will have the same cardinality as the reals)

@TobiasKildetoft: How is that basis called?

Such a basis is called a Hamel basis

(the existence uses axiom of choice though)

@TobiasKildetoft: Hamel basis - cool! Choice never was an issue to me ;-)

@Rudi_Birnbaum zeta(s) •zeta(t) is not always zeta(s+t)

@Rudi_Birnbaum along the same lines, you get that $\Bbb R$ is isomorphic to $\Bbb R^n$ for all $n$ as groups (but not continuously)

oh huzzah for Mathematica's Orthogonalize command

@MatheinBoulomenos And to $\Bbb C$ as well!

@MatheinBoulomenos: Fascinating!

What is the issue with associativity, why is it so hard to see, when you know the operation? Is there any interesting answer to that?

I mean why is not every latin square a group.

I think it comes mostly from the fact that you have three variables

which is not an interesting answer

Yes you have to try 2*n^3

and how many of all latin squares are groups?

asymptotically? no idea

Guess thats quite hard to say.

I don't really think of groups as Latin squares with a special property

Thats maybe one of my mistakes ...

But it gives you a nice feeling for representations ...

colouring the square ...

How do you think about a group?

groups are interesting because they act in interesting ways on objects, an example is a representation which is an action of the group on a vector space in a linear way. But you have also other actions, e.g. the symmetric group acts by permuting elements in a set, the dihedral group acts on a square by reflecting and rotating etc.

from this viewpoint, we need associativity just because composition of maps is associative

@MatheinBoulomenos: Yes sure thats the bit of cathegory theory I got ..

@MatheinBoulomenos: And then cosets, what is your mental picture about them?

The integers act on the real line by translation and things that are invariant under that are exactly the periodic things (with period 1, say but we can act with any period we want)

@Rudi_Birnbaum cosets are orbits of the action of a subgroup on a group by left or right translation. Orbits in general can be quite intuitive, e.g. when you have the rotation group $SO(2)$ act on the plane $\Bbb R^2$, then the orbits are literally all circles around the origin

When we say "each set is determined by its elements" is it the same as "each set is determined by the property that defines which elements belong to it" ?

@Rudi_Birnbaum that's not really category theory

@MatheinBoulomenos: Oh :(

Its Chapter 0 ...

It's really just the definition of a group

@Rudi_Birnbaum that's an abstract algebra textbook (and a very good one!), with a decidedly categorical point of view, but not a category textbook, so not everything in that book is category theory

Yeah I agree with @Mathei that a group is really just the abstraction of "the collection of symmetries of an object"; in layman's terms, a symmetry means that an object looks the same even if you look at it from two different perspectives. Trying to formalize that, one (not unnatural) thing to say is that a symmetry is a transformation $f: X \to X$ that you can 'undo'; the layman's understanding above is that $f$ should be "changing perspectives".

eg if a pizza is rotationally symmetric, that is evidenced by the fact that you can rotate it 45 degrees, and then rotate back -45 degrees, and you're back where you started; the fact that the 45-degree-rotated pizza looks identical is what lets us write rotation as a transformation on the pizza

So he (Alouffi) introduced the category of all maps as the "set of all groups" or how was it?

(associativity is because composition of functions, or transformations, is an associative operation)

@Rudi_Birnbaum you're confusing some stuff there

The only magic in what I said is the word "transformation": what is X? and what is a transformation on X?

I don't reacall the details what was it.

?

you can take the class of all groups and make it into a category

@MikeMiller: With that part of group theory I am familiar with (even lecture about it...)

you need to define "arrows" between different groups, which you define to be the group homomorphisms

k well if that's nothing of value then i've got nothing more

but that's not really related to what a group is, you can do that with a lot of kinds of mathematical objects

@Mike No its of value of course, but I just wanted to say that I am fmailiar with that pictorial part of it! @MatheinBoulomenos: No it was something that started with functions/maps ...

I've got Aluffi here, I can look

(the book, not the person)

@MikeMiller: What I try to understand is how a mathematican looks at it, so to say the mental image of the abstract. I appreciate your descrption!

Ah, I see what confused you probably. In "Joke 1.1" Aluffi defines groups as a groupoid with one object

(I love that he has numbered this joke)

@MatheinBoulomenos: No there was more about it a bit later...

are axioms definitins?

definitions*

Yes, what else?

if - then in a definition is considered iff

Sorry for being blunt

so is an if-then in an axiom also an iff ?

no?

@Rudi_Birnbaum are you thinking about free groups?

I'm looking at the first chapter on groups right now and I don't really see what you mean

@MatheinBoulomenos: It was like here we have all the maps which you can apparently connect by "Hintereinanderausführung", and so we see tatata it must be all groups!

Was it some aspect of $\mathrm{Set}$?

@Rudi_Birnbaum the automorphisms of any object in a category are a group, and the group operation is composition (which for "concrete categories" like Set is actually composition of maps, yeah)

All morphisms from Set \to Set?

Yes

exactly!!

That was kind of my big Aha what cathegories could be finally good for ...

yeah, automorphisms are just a fancy way for symmetry basically. They are they are some morphisms that have a morphism in the other direction. If you have a category where you think of your morphisms as transforms (which you do for most standard examples), then this is exactly the formalization of what @Mike said

@MatheinBoulomenos, @MikeMiller: This is a very inspiring thought I think!

the special thing here is that we usually have morphisms in our category respect some kind of structure (else our category would not be very interesting), so the automorphisms will be respecting that structure as well, i.e. when you take the category of vector spaces, you get linear automorphisms, i.e. "change of basis" maps for your vector space

But now when you want more structure, like a semi-rings or rings. is there a way to do it like that?

@MatheinBoulomenos: Ok so its more like a by-product?

yes: you need to start with a category that has more structure as well. In general, the set of morphisms from one object to another is just a set, but sometimes it has more structure as well. Like for example, for vector spaces (or abelian groups) you can add two homomorphisms and get another morphism, thus the set of morphisms forms in this case an abelian group

You mean one for each of the two operations?

(and the composition of homomorphisms respects that as well in the sense that $f \circ (g + h)= f \circ g + f \circ h$ (here we need actualy homomorphisms, this doesn't work for set-theoretic maps) and $(f+g) \circ h = f \circ h + g \circ h$

@Rudi_Birnbaum no, currently I'm just talking about a category where you can "add" arrows between two objects

I see!