Sounds like some kind of associative ring generated from a set

where $f \circ$ are unary operations for each elementary function $f$

and the usual $\cdot, +$ for binary operators

While that will ensure you will get everything since composition and finite arithmetic operations of elementary functions is still elementary, to characterise each equivalence class may not be straightforward as it it akin to determining whether $fffgff = fgggfffg$ for example

Also, since one can exponentiate iteratively, the general term in this set may be something unwieldy like:

$$e^{e^{e^{ax^2+bx+c}}} + e^{e^{dx^5+ex^4+fx^3+g}} + \cdots$$

the basis for the R-module of elementary functions may be uncountable

as the basis of the vector space of all polynomials is already countable to start with

Consider the Hankel matrix (math.stackexchange.com/questions/3065461/…). How does one show it satisfies the given formula for the determinant? If one sums over any column or row, one gets a column/row of equal entries, namely 1/2n(n+1). What would be the next steps to derive the terms $n^{n-2}$ and $(-1)^{1/2n(n+1)}$ that together with 1/2n(n+1) make up the product of the formula?

Correction. Not $(-1)^{1/2n(n+1)}$ but $(-1)^{1/2n(n-1)}$.

hey @ÍgjøgnumMeg

Hi @Mathein :)

How's it going

pretty well, thanks

and yourself?

when are you moving again?

Yeah not bad, my rent subsidy got approved today which is nice

Moving in 5 days :P

just let me know when you're here and have time, then we can meet up

Cool will do! I'm not going to Bonn anymore since it clashes with my moving in plans etc.

First thing I need: Leberkäse und Döner

that's easy to get

you can even get Leberkäse at the bakery right next to the math building

and Döner in the supermarket right next to the math building

mathematikon?

yeah

Nice :) the dreeeam

I matriculated the other day

nice

just waiting for some post from Heidelberg I think

as soon as you matriculated you can apply online for the language course if you want to do one

Well I don't have any information from the uni yet

like a student number or anything

and then when you're here you need to go to the office of the Sprachlabor

language courses are first come, first serve

so you want to do that rather soon

Fair, maybe I will

just to be safe

if you want to do a language course, ofc

This might seem random, but within two years I will be applying to various places for a doctoral position. Some place that I have had some contact is with Universitat Wien. I was thinking whether I have to know German to apply for doctoral programs as I will be finishing my masters and applying

I could just leave it until next semester, I don't want to be überfordert

yeah sounds like a good plan

@SayanChattopadhyay depends on the place really

Heidleberg, Bonn?

in HD, you need German for bachelors and masters, and no German for a doctoral position. In Bonn you don't even need German for your masters (and neither for the doctoral position)

HD= Heidelberg lol

@Sayan I think it's safe to say you should at least know SOME German if you are going to study in a German speaking country

but it's still nice to know some German

But often a doctoral thesis will be written in English anyway so

even if you don't need it officially

Yeah but like knowing at a very good level is going to be difficult since I already have lots of math to do, write a solid thesis and US is too expensive

@MatheinBoulomenos That's cool

even some basics can help you in your everyday life

you're not expected to do math in German as a PhD student, I know some PhD students and postdocs here at HD who don't know much German

Well, if you're living there, you'll pick up the language one way or another.

I think you're fairly safe when applying for a PhD but make sure you find out first lol

Getting ahead of the curve will just make life a little more livable.

Cool, thanks folks

@Rithaniel depends. I heard from people who only speak English at work/university and only meet up with other foreigners and they pick up 0 German

or maybe not 0, but only $\varepsilon$

Well, I'm honestly surprised.

I don't like the notion of Graduate programs, like most US universities have graduate programs rather than doctoral positions, thats also one reason why I do not want to apply to places in the US

I will have to do all the stuff again for two years in a graduate program

@SayanChattopadhyay what kind of math do you want to do for your PhD? Maybe I can even point you to some profs here at Heidelberg

@Rithaniel actually I also found quite a lot in Austria that Turkish and Balkans people would remain in their own circles, and only be able to speak $\varepsilon$ German even after 40+ years in the country lol

but if you make the effort then yeah, it's easy to pick up a language in that country

@MatheinBoulomenos Right now I am not so sure, but broadly in geometric analysis/mathematical physics

I would have figured learning the local language would happen, even if passively. Though, I suppose I hadn't considered that you could be sufficiently insulated to avoid it.

Aye, most people would use their children to carry out errands that involved speaking German

@ÍgjøgnumMeg lol

@Mathein I'm actually most excited to try out the cycle commute atm

hahaha

@MatheinBoulomenos Wow

@SayanChattopadhyay these are the two I can think of. geometric analysis is not that huge at Heidelberg, but maybe you find something you like

Hello, I know we aren't supposed to post main site questions here, but I need to verify my answer urgently and hence I posted it here. Any help would be really appreciated.

These are really nice, @MatheinBoulomenos, thanks.

np

i heard my field

Ryan pokes his head out of the savannah like a meerkat

hi @Ryan

lol

@Mathein god my spoken German is so rusty

well I keep forgetting words and accidentally using the wrong articles lol

@schn there’s no closed-form for determinants of Hankel matrices in general. There is a closed-form for the special case of a circulant matrix, of which the matrices in that answer are an instance

@Semiclassical Okay. Any ideas of how one would derive the formula for that special case?

@ÍgjøgnumMeg you'll be fine

@SayanChattopadhyay u like geometric analysis?

Heidelberg has very European geometric analysis

what kind do you like

@Mathein sure :) It just sucks when I'm speaking on the telephone and I'm like "... können Sie das bitte wiederholen, ich habe Sie nicht gehört (übersetzung: Ich bin zu dumm um Sie zu verstehen)"

in before @Ryan suggests to go to Russia to do geometric analysis

lol

Hey @Alessandro

Hi @ÍgjøgnumMeg

wassuuuup

Not much, what about you?

Also not much, just doing loads of paperwork for the move lol

@schn look up circulant matrices. They all get diagonalized in the same way

When will that be again?

I'll also be traveling back to Germany soon

5 days from now :P

but I won't be going to Bonn since it clashes with my move in date and stuff

interested to see who my room mate will be rofl

Good luck

Are you in a university dorm or did you find a flatshare?

It's a private dorm, so only students can live there but it's not rented by the university

I'll share a flat with one other person

we'll share a kitchen and bathroom and "living space" but have separate rooms :P

I see

That seems to be a common arrangement for student dorms in Germany

Orly, here I think most student halls have like 6 people in a flat sharing a kitchen

I think it's 3-4 in Bonn but I'm not sure

Fair :) We will be the first people to live in this building, I hope they haven't f*cked up anything

lol

Why would you go to Russia to do geometric analysis

I would suggest other places in Germany....

@RyanUnger To work with Perelman of course

No one works with Perelman

I know

(removed)

@RyanUnger I have done very little of geometric analysis to say I like specific things in it, but for instance I like global analysis (Fredholm operators, indexes and stuff), the analysis coming from Riemannian geometry (constructing manifolds with different sectional curvatures and everything, minimal surfaces) but things that I have done the most are integrable systems and gauge theory stuff from the physics point of view and I am really interested in looking at all the mathematics behind it.

So yeah its all over the place

But analysis and geometry seem to be the thing I seem to like the most

@RyanUnger that was my cat lol

@Sayan: Global analysis is analysis on manifolds. The topics you listed are functional analysis.

Oh I see, I have been told this is geometric analysis by people here so I thought that.

@TedShifrin this stuff falls under geometric analysis in some countries

Geometric analysis in America is a word invented by the NSF

If you're doing Atiyah-Singer index theorem, sure ...

He said GLOBAL analysis, BTW.

He did say that and global analysis falls under geometric analysis in certain countries

NSF ?

National science foundation

Anyhow ...

These guys get to decide the name of math fields?

I think Math Reviews made up a lot of the classification. I don't know who came first.

I think just geometric analysis

It’s a relatively new field and they needed something to call it

It’s political

GA is the most specialized NSF department

I always thoughts naming different branches comes from some first paper on it.

Yau politicized everything, I suppose.

@RyanUnger Expected that.

The sad bit is I don't have many people who do this stuff here. Ramanan visits sometime and I believe he will take our Global analysis course but that's that.

Everyone's either commutative algebra or some number theoretic algebraic geometry.

I don't know why but everyone seems to be doing number theory in India. I don't see many geometers

(I'm so sorry)

@TedShifrin Yau credits himself in his book

The NSF asked him what to call his research

Well, of course.

I mean the term GA makes perfectly good sense when you're doing PDE on manifolds coming from Riemannian (or other) geometry.

Its complicated to decide what I should do for my masters thesis. I don't want to commit to a lot and not be able to do all that. I also want it to be good.

Masters is different from doctorate, obviously. Pick something you're interested in learning and get as far as you can get. You're not expected to do original research.

I was thinking something along the above mentioned topics. I need to give a proposal by the end of this year.

And then there will be regular seminars within each three months where I will be asked to present what I am doing which will go on till I finish my thesis.

That's excellent experience, to give seminar talks for people less expert than you.

Yeah, I have to distill it to an audience who has no clue about anything even close to the topics mentioned. So that will make me understand the bare bones of stuff.

@TedShifrin To do more the analysis and geometry stuff about gauge theory and integrable systems, do you know of any texts I can follow? Naber has a book but he only mentions this stuff in small footnotes.

Not integrable systems, but do you know Blaine Lawson's beautiful lectures?

Nope, I will have a look

Anything he writes is wonderful.

Why is $r = \cos(\theta/2)$ symmetric about the Y axis?

A curve is symmetric about y axis if $(r, \theta) = (-r, -\theta)$ or $(r,\theta) = (r, \pi- \theta )$

But none of these is true for the above curve?

@Archer What is $\text{cos}(\frac{\pi}{4})$ and $\text{cos}(-\frac{\pi}{4})$, by your understanding?

@Rithaniel same $= \frac 1 {\sqrt 2}$

@Rithaniel: That's symmetry about the $x$-axis. I agree with @Archer.

Usually what I have seen in polar coordinates is the the conventional x axis is theta and y is r

Wait, are we counting the $x-$axis as the vertical axis or the horizontal axis?

@Rithaniel horizontal. And yes I know it's symmetric about X axis ... I dont see why its symmetric about Y.

@Rithaniel Its (r, theta) in this case

However, this curve is symmetric about the $y$-axis, so we need to work a little harder.

Ah, well then I am confused. Are we mixing polar and Cartesian coordinates?

@Rithaniel We are making a polar curve on XY plane

Alright, then this is a circle of radius $\frac{1}{2}$ centered on ($\frac{1}{2},0)$?

No, it's not.

That's $r=\cos\theta$.

Ah, right.

Its more involved than that. It loops on the y axis

Indeed, plotted it just now.

(Here I thought this was just a basic trig question)

I think what makes it tricky is that you have positive r in some places and negative r in others

Oh, duh, it's obvious. Look at what happens if you put in $2\pi - \theta$ for $\theta$.

@TedShifrin $\cos(\pi - \frac \theta 2) = - \cos (\frac \theta 2) \ne r$

Nice

No, it's $-r$ at the angle $-\theta$.

That means you win.

@TedShifrin how is the angle $-\theta$ ??

Because $2\pi-\theta$ is the same angle as $-\theta$.

right.

This argument can be taken to show that $r = \cos(n\theta/2)$ is symmetric around $Y$ for even $n$

No, odd $n$?

Nope

I think so atleast

after drawing a few

Well, $r=\cos\theta$ is not symmetric about the $y$-axis.

Yeah

if X is uniform on [0,1] and Y is exponentially distributed with lambda 1, what is P(X+ Y <= 1)?

Are these random variables with the same domain?

@TedShifrin the exponentially distributed random variable is between 0 and infty and the uniform one is between 0 and 1

But you are going to sample $Y$ only on $[0,1]$?

no

So you sample on $[0,\infty)$ but the probability density of $X$ is $0$ on $[1,\infty)$?

So, at any rate, why are you asking the question? Why can't you do it?

Any hints are appreciated. (math.stackexchange.com/q/3369372/642262)

Hey @Ted

Hi @Mathein

how do I deal with being interested in too many things? Having a broad interest sounds like a good thing, but there are just to many things I want to learn about

LOL. I always had broader interests, too, than the typical bear.

I think it's fine to be like this until you start writing your PhD thesis.

okay

I was always interested in parts of math where lots of different parts of math intersected/overlapped.

yeah that's true for me as well

In my BA thesis, I have algebraic geometry (classical and also stuff where schemes are essential), complex analysis, I guess some "baby" complex-analytic geometry (though I could do more if I have time), p-adics, measure theory and obvously lots of number theory

I know it's not your cup of tea, but NT uses almost every other branch of math, it's amazing

I hope that I can avoid stacks, my advisor said I can make some extra assumptions if that simplifies things

working with moduli stacks would be the most "natural" setting I guess

but then I would have to do étale cohomology of Deligne-Mumford stacks... I think I prefer the case where the Deligne-Mumford stack is actually a Riemann surface lol

I really know nothing about stacks. ... I remember back in the late 80s or early 90s I taught a year-long differential geometry grad course and one of the grad students was working in number theory. He was convinced it would be useful to him in the future to learn some of the stuff in my course, even though his adviser told him repeatedly not to "waste" his time.

hmm, I guess as a grad student you have to be wise with how you use your time

I also am facing the question of how to deal with having too many interests. I was hoping to read some "classical algebraic topology" papers from the 1970's and stuff (Milnor, Bott, Atiyah etc.), but at the same time modern stuff in algebraic topology like spectra and topological modular forms is quite far away from that I guess

And also I need to read up more on diff top/diff geom and algebraic geometry anyway

but knowing differential geometry doesn't sound useful for a number theorists, but it depends on the type of NT you do

I am reminded of that 3blue1brown video about how a topology result can give insight into a combinatorics puzzle.

if you want to do global Langlands, there's always the Archimedean places to worry about, which means that you have to do things over $\Bbb R$ and $\Bbb C$. Given that you use a lot of p-adic geometry for the stuff that happens over $\Bbb Q_p$ etc. especially in modern approaches, it seems very reasonable that you can utilize diff geo for stuff that happens over $\Bbb R$ and $\Bbb C$

I even went to a conference talk where they used complex-analytic geometry for $\Bbb C$ and p-adic geometry together to get a global result for varieties over $\Bbb Q$

so it doesn't seem unreasonable to do diff geo as a number theorist

but maybe it's not the most helpful course you could take

@Ted is it okay if I email people I just know from a conference with questions that they might now the answer of? I'd like to keep in contact with a few of those people

Okay quick question, say I have two functors $T, F$ from categories $\textsf{C}$ to $\textsf{D}$ with objects $A$ and $B$ in $\textsf{C}$ and $G$ and $H$ in $\textsf{D}$ and suppose I have a natural transformation $\eta$ between $T$ and $F$. And suppose I want to show that $\eta : A \to G$ has some nice property. If it turns out that $\eta : B \to H$ satisfies that same nice property and $A \cong B$ and $G \cong H$ (though through different maps)

would it imply that $\eta : A \to G$ the desired property?

@Perturbative I don't think your question is well-defined. What do you mean by $\eta:A \to G$ etc.?

you can have to objects $A$ and $B$ in $\textsf{C}$ and then look at $\eta_A:T(A) \to F(A)$ and $\eta_B:T(B) \to F(B)$

I think that's what you mean

but then the rest of your question doesn't make much sense

do you have a specific example in mind?

Hi

Hi @nbro

@AkivaWeinberger I woke up from a dream and I was at home. Then I woke up and I was at my hall of residence.

@MatheinBoulomenos Yes you're correct that's what I mean

Are you familiar with Serre classes and mod $\mathfrak{C}$-isomorphisms?

Hi all

If you are then I can explain the example properly

Hi @ÍgjøgnumMeg

(lol wrong ping)

lol

@Perturbative I know what a Serre subcategory is if that helps

I have a basic geometry-related question. Suppose I have a pose $P$ of an object, represented by the x and y coordinates and the orientation, given by an angle $\theta$. Consider another pose $Q$. How can I find the coordinates x and y and the orientation of $Q$ with respect to $P$? Apparently, we can represent all these poses in a homogenous space, then use a homogenous matrix to transform $Q$ to the $P$ space. Can someone shed some light on this? My geometry skills are quite rusty right now.

Wonder what various kinds of mathematics are relevant in Iwasawa theory lol

(referring to your previous comment that it "depends what kind of NT you do")

Note that both poses P and Q are given, so I have their values in a "global space".

I know what a homogenous representation of a vector is. You just append 1.

@Perturbative so if you have an abelian category $A$, then a full subcategory is called a Serre subcategory if for all SES $0 \to X' \to X \to X'' \to 0$, then $X$ is in the subcategory iff $X'$ and $X''$ are

is that the same thing as a Serre class?

I could just give the definition real quick. So a Serre class of abelian groups $\mathfrak{C}$ is a collection of abelian groups such that if we have a short exact sequence $0 \to A \to B \to C \to 0$ then $A, C \in \mathfrak{C}$ if and only if $B \in \mathfrak{C}$

that's just what I said

expect in an abelian category instead of $\mathsf{Ab}$

so yeah, I'm familiar with that

and also with the fact that you can define the quotient category

@Mathein Oh I don't know enough category theory to know that your defn and mine were equivalent

np

Okay so then (just for my own clarity) if we have two abelian groups $A$ and $B$ and a morphism $f : A \to B$ then we say that $f$ is a $\mathfrak{C}$-isomorphism if $\ker f \in \mathfrak{C}$ and $\operatorname{coker} f \in \mathfrak{C}$

Hello. Suppose $g\in L^\infty (\mathbb R^n)$ is a continuous bounded complex valued function. If $g(x+y)=g(x)g(y)$ then I want to say $g(x)=e^{\varphi x}$ for some complex valued functional $\varphi$ on $\mathbb R^n$. Is this right?

Never mind. If you have a pose $P=(x_P, y_P, \theta_P)$, you can build a homogenous matrix, which you can multiply with any other pose $Q=(x_Q, y_Q, \theta_Q)$, so that to move this pose to the coordinate system of $P$.

@Arrow do you suppose that $g \neq 0$? That's obviously necessary

@MatheinBoulomenos yes, sorry.

In fact, the multiplication with the homogenous matrix corresponds to the application of a translation and a rotation to the coordinate system of $P$

@Perturbative I'm with you, continue

If it helps if we have two group homomorphisms $\alpha, \beta$ and if two of the maps $\alpha, \beta, \beta \circ \alpha$ are $\mathfrak{C}$-isomorphisms, then so is the third

Also $\cong$ above means a standard group isomorphism not a $\mathfrak{C}$-isomorphism (though it turns out to be a $\mathfrak{C}$-isomorphism anyway)

@Perturbative my guess would be no.

Let $\mathsf{C}$ be a discrete category with two objects $A$ and $B$. Define $F=T$ via $F(A)=T(A)=F(B)=T(B)=\Bbb Z$ (we don't need to specify morphisms as $\mathsf{C}$ is discrete). Let $\eta$ be the natural transformation given by $\eta_A=\mathrm{id}_{\Bbb Z}$ and $\eta_B=0$

as $\mathfrak{C}$ we can take the zero Serre class, so $\mathfrak{C}$-isomorphisms are just regular isomorphisms

Ahh okay a quick counterexample

Thanks for that!

you're welcome

Also if you don't mind me asking, what are Serre subcategories useful for?

@Perturbative just a moment, writing an answer to @Arrow's question atm

Sure np

From what I read I think Serre invented Serre classes of abelian groups to prove generalizations of theorems like the Hurewicz and Whitehead theorems in algebraic topology, apparently they were "tailor-made for spectral sequences", and Serre subcagetories seem to be a generalization of Serre classes of abelian groups

this is just beautiful! no uniqueness theorems of differential equations needed!

@MatheinBoulomenos thanks! How about refining to say $\varphi$ is the restriction of a complex functional on $\mathbb C^n$?

that's not really a refinement: any functional on $\Bbb R^n$ can be extended to $\Bbb C^n$

note that the bounded assumption is superfluous, as the proof shows

but you can use a very similar argument to show the corresponding statement for holomorphic functions $\Bbb C^n \to \Bbb C$

@Perturbative Serre subcategories are nice because you can take the quotient category modulo the Serre subcategory

Yes, I wanted boundedness for something unrelated (applying the linear algebra Riesz representation theorem to relate to Fourier transforms).

@MatheinBoulomenos at any rate, very elegant! Thanks again!

@Arrow sure, it was a fun problem

@ÍgjøgnumMeg probably lots of alg geo and NT

lol

p-adic analysis

yeah, true

p-adic geometry

p-adic everything

it looks like something I'd be really interesting in studying, but we'll see what happens over the next couple of years lol

@Perturbative so here's a cool example: let $R$ be the category of $R$-modules for a commutative ring $R$ and let $S \subset R$ is a multiplicatively closed subset. Then the category of $S$-torsion modules is a Serre subcategory and one can show that the quotient is equivalent to $S^{-1}R$-modules

this stuff shows up in algebraic K-theory

there I have seen Serre subcategories before

@Rithaniel Maybe "Choose one right-inverse for $f$ - call it $s_0$. Then the set of right-inverses for $f$ is $s_1+{\rm Ker}(f)$."

Dunno exactly what they want

@Mathein I guess in the "Spezialisierungsmodul" this is the kind of thing one gets to study

yeah

"ANT 3" kinda thing

yup

Exciting :)

it depends a bit on the prof what kind of Spezialisierungsmodul he offers

yeah sure, fingers crossed for Venjakob

our prof just asked us what we wanted to see and I said global CFT lol

lol

nise

the proofs for global CFT were the most difficult proofs I have completely worked through so far

What is so difficult about them?

@Mathein: Sure, of course you can email people you met at a conference. Remind them of your conversation so they can get local coordinates on you :P

I mean why is CFT considered so infamously difficult? lol

Hi, I'm no set theorist but isn't this basically asserting the continuum hypothesis is true?

For reference $c$ is the cardinality of the reals

@SirCumference I'm not set theorist, but no

@ÍgjøgnumMeg probably just how long they were and how many non-trivial subarguments they contain

No, CH says that $2^{\aleph_0}=\aleph_1$

fair

@TedShifrin I was thinking of asking someone to apply for another conference I'm going to apply to as well

$\mathfrak c=2^{\aleph_0}$ is a theorem (as proved in your screenshot)

I need to have my networking game to be on point lol

(presumably networking is not that important yet)

@AlessandroCodenotti crud, welp ignore me

You shouldn't ask a faculty member "to apply." Rather, inquire politely if he/she is planning on attending, since you are so planning. :)

oh yeah "asking" was way too strong for what I meant, sorry

What is CFT?

class field theory

oh, duh