Mathematics

3:01 PM

what I think is that there's no need to put that much emphasis on what 'mathematics' is, but rather on the things that people are actually studying, e.g. number theory or other stuff - then i feel that you might not be so conflicted over this.

if you forget about the word 'mathematics' - and perhaps just observe that some people care about number theory, some people care about topology etc. then i think the problem that these things may not be relevant to reality is not that big of a deal (at least imo) - after all a lot of people (in other fields - even!) study things that don't quite …

Mike Miller

@loch I think that just ignores an independently interesting question of "what is mathematics?" because there definitely is some similarity between the practices in different fields

Bill Thurston has a beautiful essay on something like this called "on proof and progress in mathematics"

Balarka Sen

mathematics sucks ass

thats my view on it

loch

@MikeMiller that is true - i should've probably worded what i said in a better way.. i guess perhaps what i really wanted to say is just that people study things they find interesting..

Kasper

Okay, but now, I find studying football video games interesting and beating oponents in ellegant ways really interesting, but nobody seems me to allow to study this on a university.

Is finding something interestig enough?

loch

3:18 PM

i think the practical answer is that when enough people do - then yes

Leaky Nun

@loch he's still going on

loch

@LeakyNun differentiability?

Leaky Nun

yes

loch

lol
it's important!

Leaky Nun

Is |x| differentiable at x=0?

very interesting

:thinking: hmm this is a hard question

Nûr

3:25 PM

how are mathematics useful in biology or in agriculture?

'sophisticated' math

loch

what do you mean by sophisticated?

Nûr

I speak about what we study in university and more. It is obvious that school mathematics is useful everywhere.

Leaky Nun

pretty sure PDE is useful for biology

and agriculture as well

just dynamical systems in general

Balarka Sen

thats ODE not PDE

loch

my friend had to learn dynamical systems for a class she took (called systems biology) - so there's that

and also i think they were supposed to learn what 'fourier transforms' are (I'm not sure if they understood what it is - but at least that was part of the syllabus..)

so basically stuff you see in applied math

Semiclassical

3:31 PM

Eh.

Balarka Sen

predator prey dynamics is the basic example

Semiclassical

Depends on what system you're talking about

I'd take predator-prey to be systems of ODEs

Balarka Sen

systems of ODE = matrix ODE

Semiclassical

Fair.

Balarka Sen

I do not actually know what a dynamical system given by a PDE means

Semiclassical

3:32 PM

Well, a wave equation would be an example

Take a string and pluck it.

Balarka Sen

Why is that a dynamical system?

Semiclassical

Because the time evolution is governed by the initial condition? I guess if you want that to formally be a dynamical system in the usual sense, you'd discretize the time and ask how the state of the string at one instant determines the state of the string at the next instant

I think in agriculture the chief relevance of math would be in the wider context of operations research

and stats

Balarka Sen

I understand dynamical system to be a state space where you are iterating a function (in discrete time; in continuous time you have a flow)

What is that state space and what is that function here?

Semiclassical

The state space would presumably be the configuration of the string, and the function would be the time evolution operator

Balarka Sen

So why don't you just have a ODE given by the time derivative of the time evolution operator on the configuration space of the string?

Semiclassical

3:39 PM

well, for one, a configuration of the string isn't a number. it's a function.

But, let me be precise in the case I know well.

Nûr

Translation of a speech from Grothendieck: For this reason, I personally abstain,
as much as possible to participate in this type of activity. I
would like to clarify the reason why at the beginning I interrupted my
research activity: it was because I realized that there was
problems so urgent to resolve regarding the crisis of survival that it
seemed to me madness to waste forces on doing research
pure scientist. By the time I made this decision, I thought
to spend several years researching, acquiring

Balarka Sen

So you'll have an ODE on a function space. I feel like it's a classical mechanics thing where you pass to the cotangent bundle.

Nûr

it's interesting I think

Balarka Sen

But I also could be talking bollocks

Semiclassical

Take the Schrodinger equation $i\partial_t \Psi = \hat{H}\Psi$ where $\hat{H}=-\frac12 \nabla^2+V$

where I'm assuming for simplicity that the potential is time-independent

then the solution to that PDE can formally be expressed as $\Psi(\vec{x},t)=e^{-i t\hat{H}}\Psi(\vec{x},0)$

Balarka Sen

3:42 PM

Ahh, ok. So that is an ODE on a function space

Because $\hat{H}$ is a linear operator on a function space

Nûr

There is the complete conference there, but it's in French. There is a link to the script so you could translate if you want to understand: archive.org/stream/Allons-nousContinuerLaRechercheScientifique/…

Alexander Grothendieck: Allons-nous continuer la recherche scientifique? (CERN, 27/01/1972)

►

Semiclassical

sure. but you're still solving a PDE

(things get more annoying if you have a time-dependent potential.)

Balarka Sen

Yeah I get it. I was confused because my mental model for a dynamical system is an ODE and I was trying to reunite your example with that model

Semiclassical

mmkay

Balarka Sen

The conclusion being, my model isn't wrong, but your statement isn't wrong either

The latter being the more practical view

Semiclassical

3:45 PM

this is for a linear PDE, mind

not sure wth you'd say for a nonlinear one

Balarka Sen

Ah. :big shrug:

Semiclassical

i suspect the same kind of statement works, but now the time-evolution operator isn't linear

Balarka Sen

Gotcha

Mike Miller

@loch I didn't mean to phrase that hurtfully, if I did - I really just wanted to recommend Thurston's paper

loch

@MikeMiller oh no i didn't take it that way at all!

i just like using excessive ... (assuming that's the reason you interpreted the tone of what i wrote the way you did)

anyway i'll have a look at that paper at some point when im procrasinating

Mancala

3:58 PM

Can anyone clarify me the definition of orientation form of a smooth manifold $M^n$?

I'm reading Lee's book "Introduction to smooth manifolds" (second edition) on pg. 381, and I am confused about how I verify that a differential $n$-form on M is a form of orientation.
https://math.stackexchange.com/questions/2811524/help-to-understand-orientation-form-definition

Mike Miller

That is indeed why @loch

Leaky Nun

@loch how do you study ag without doing any groups

loch

im not sure if you meant how do i not know much about algebraic groups, or if you're asking hypothetically how woudl someone study ag without knowing what groups are

if it's 2 i don't know

that seems like a bad idea

Leaky Nun

and if it’s 1?

loch

4:13 PM

if it's 1 - for example vakil / hartshorne doesn't do anything on algebraic groups (which is kind of where i learnt most of the stuff)

i will eventually have to learn it though (and probably soon)

Leaky Nun

and Galois theory?

loch

Its not strictly necessary for eg most of the stuff in Hartshorne - but I think people should know galois theory just because I think people expect you to know it if you're doing anything related to algebra lol

mercio

how can you breathe without doing any groups

6

Balarka Sen

like this

breathes in

breathes out

Simple

Daminark

@BalarkaSen but that was a Z/2 action on your lungs!

applications of math to bio

@BalarkaSen and yeah Z/2 is a simple group

Maximus

4:26 PM

I am trying find an upper bound for the following integral, I know the expression is less than $1$, and I would prefer to find a bound that looks like $\frac{1}{2^{f(n)}}$. Where $f(n)$ is any function in $n$.

The integral is: $$\frac{1}{r^n} \int_{||\vec{u}|| \le n 2^n C} e^{-\pi ||(\vec{x} + \vec{u})/r||^2} d \vec{u}$$

$n \in \mathbb{N}$, $\vec{x}$ and $\vec{u}$ are vectors in $\mathbb{R^n}$. $C$ and $r$ are just constants. $||\vec{x}|| \ge \sqrt{n} r$ and $r > 2^{2n} C$

Balarka Sen

@Daminark

Maximus

I've tried using the reverse triangle inequality $-||\vec{x}+\vec{u}||^2 \leq -(||\vec{x} - \vec{u}||)^2$ but unable to separate some terms

Balarka Sen

@MikeMiller Hey

I'm trying to compute some spectral sequences

MatheinBoulomenos

Hi @BalarkaSen @MikeMiller @Daminark @AlessandroCodenotti

Mike Miller

Hi

MatheinBoulomenos

4:42 PM

I'm stumped by something which should be an easy homological algebra exercise

Maximus

@MikeMiller Hey you've helped me in the past and I much appreciate it. If you have the time would it be possible to take a look at my question above? I have been stuch on that problem for about a week now

MatheinBoulomenos

I'm quite sure that it's true (but maybe it's actually wrong)

@MikeMiller So if I have a chain map between two positive chain complexes of vector spaces over a field that induces the zero map on each homology group, is the map homtopic to the zero map?

Mike Miller

@Mathei No

Daminark

Get @Mathein

MatheinBoulomenos

oh, so it's wrong, then no wonder that I couldn't prove it

Daminark

4:45 PM

*Hey

MatheinBoulomenos

I thought you wanted to write "get rekt"

because I make wrong homological algebra claims

@MikeMiller do you have counterexample?

Mike Miller

Oh but over a field idk I still guess no but it would be harder for me to assert so easilt

I shouldn't be doing this right now though, sorry

MatheinBoulomenos

no problem

Alessandro Codenotti

Hi @Mathei

Looks like I need to brush up my German, I got into Bonn for my Master!

MatheinBoulomenos

@AlessandroCodenotti Wow, Gratulation!

Bonn soll ziemlich gut sein

ein Freund von mir studiert da

Alessandro Codenotti

4:51 PM

@MatheinBoulomenos Ich habe auch dass gehört

MatheinBoulomenos

@AlessandroCodenotti Weißt du schon was du machen willst? Logik? Zahlentheorie? Algebra? Oder erst mal von allem ein bisschen?

Alessandro Codenotti

Logik und Mengenlehre, natürlich!

MatheinBoulomenos

Ah, okay, da kenne ich mich nicht aus

Ich verwende einfach locker das Auswahlaxiom für echte Klassen ohne Skrupel zu haben

Alessandro Codenotti

Hmm, I'm confused by this sentence

Daminark

Making wrong homological algebra claims is quite the crime tbf

MatheinBoulomenos

4:57 PM

I use the axiom of choice for proper classes casually without a qualm

@Daminark we still don't know if it's wrong

Alessandro Codenotti

@MatheinBoulomenos you monster!

MatheinBoulomenos

yeah

I wonder what large cardinal axiom I'm implictly using by doing that

Daminark

Oh I know, I was just saying that it was fair to believe I said "Get rekt" at the time given that we thought it was wrong, since such an act is horrendous

MatheinBoulomenos

I agree

@AlessandroCodenotti is that a cardinal sin to a set theorist?

Alessandro Codenotti

Or an ordinal sin?

famesyasd

5:02 PM

are rings assumed to be associative?

MatheinBoulomenos

yes

famesyasd

thanks

MatheinBoulomenos

@Daminark yeah the only way to atone for such a misdeed would be hours of pious diagram chasing while praying to the algebra gods for forgiveness

@AlessandroCodenotti I guess when you make a mistake and end up with $1=0$ at some point, that's a Reinhardt cardinal sin

Daminark

@MatheinBoulomenos agreed

Alessandro Codenotti

Groans...

Tobias Kildetoft

5:38 PM

@AlessandroCodenotti Congrats on Bonn

You can do some representation theory with Stroppel

ÍgjøgnumMeg

Love this

Alessandro Codenotti

6:10 PM

@TobiasKildetoft I have to look how it works exactly because logic, number theory and algebra are in the same area and you can only take that many credits in your main area so I don't know if I'll have any left after all the logic and set theory courses

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$Mathematics$ Mathematics