Mathematics - 2020-10-22 (page 2 of 2)

user2103480

15:00

@BalarkaSen speaks the mutant

@Anush yes, that'd be the set of functions from the empty set to {0,1}

Which is the power set of the empty set so {ø}

Balarka Sen

im a mere mortal

i can only apply the gromomorphism functor

Anush

@user2103480 {ø} makes sense, thanks

Edward Evans

For a representation $(\pi, V)$ of $G$, the span of $\lbrace \pi(g)v : g\in G\rbrace$ for some fixed non-zero $v \in V$ is a $G$-invariant subspace of $V$ and if $V$ is irreducible then this is all of $V$. What if I pick $v \in V^{K}\setminus{0}$ and take the span of $\lbrace \pi(g)v : g \in G/K \rbrace$?

Actually it's obvious

but I couldn't word it to myself

the action of $K$ fixes $v$ so the only elements that do anything non-trivial are those not in $K$ so one can span the whole of $V$ by $\pi(g)v$ with $g \in G/K$

aight cleared it up to myself thanks chat

user2103480

@BalarkaSen your knowledge is waaaay more useful. Foundations knowledge is nice to flex on mathematicians and philosophers alike, but hardly useful for anything remotely close to the "real world"

Balarka Sen

yeah i will switch to data science whilst all of you category theorists struggle to get tenure

thats what matters at the end

user2103480

15:08

Joke's on you, you'll be dragged into corporate bullshit and train linear regressions on steroids

Balarka Sen

lmao

geocalc33

Just get tenure and then do your own math research

user2103480

I'mma do a mathematical biology PhD and sit in some government bureau doing mediocre research for half your pay

That's my dream job

Alexandru Ionut

I'm an applied mathematician here going "wuuut" at all the previous comments

interesting how you don't hear the word "statistics" from the corporate types anymore

Edward Evans

I had a graph theorist lecturer during my undergrad who randomly started doing loads of research in design theory and then left to get a job in industry

Alexandru Ionut

15:11

it's all data science this, data analytics whatever

Balarka Sen

@EdwardEvans lmao

@Alexandru yeah data science is the new normal

Edward Evans

he was like

Thorgott

the only things I'm gonna apply are functors and my tears to the corner

Balarka Sen

big data

Edward Evans

quick publish some applied shit

Alexandru Ionut

15:12

ah yes big data another one

lol Edward

Balarka Sen

machine learning neural networks BIG DATA

Alexandru Ionut

lol lol

add "in a real business setting" at the end

user2103480

@EdwardEvans I respect that though. Graph theory requires a kind of creativity that I lack

Balarka Sen

@AlexandruIonut hahah

Thorgott

On Applied (\infty,1)-Topos Theory

Edward Evans

15:13

@user2103480 that but with any kind of mathematics

tbf

I could just

take a course on crypto and then be like "yeah number theory" and maybe get a job reading Hilary Clinton's emails or smth

Balarka Sen

lmfao yeah man easily

Edward Evans

Crypto is my plan B

user2103480

@EdwardEvans well yeah but I could have imagined doing set theory and a bit of logic till the end of days and I think I wouldn't have been too bad at it

Edward Evans

if I do manage to get a PhD position then I would probably have been in Germany for long enough to shed my British citizenship and take German citizenship and then work for the BND

I mean, post PhD

Balarka Sen

Edward Hackerman Evans

Edward Evans

15:16

@user2103480 I don't understand the appeal of set theory

Balarka Sen

cranks up those discriminants

to hack into people's mailboxes

Astyx

@BalarkaSen disruptive techonologies

Edward Evans

rofl

Hackerman

grow a mullet and buy some aviators

user2103480

The field combined being abstruse and weirdly intuitive for me such that, if it was still fashionable, I think that would have worked out better than other fields

@EdwardEvans yaaaayy....

Edward Evans

I just like the fact that number theory sounds retarded to the layperson

user2103480

15:18

😂

Edward Evans

like at least in other fields you have names that sound technical or smth

Thorgott

what doesn't

Edward Evans

but number theory sounds dumb

user2103480

Oh no I failed the corona test

Meaning that it's negative

Sike

Balarka Sen

Lol

Edward Evans

15:18

nice congrats

Balarka Sen

Congrats

Edward Evans

that goes on your zeugnis

5,0 in Corona

user2103480

fucc

Balarka Sen

@Edward

Edward Evans

you only get a 1,0 if you die

Astyx

15:18

People who see "number theory" on your CV probably understand "I like counting"

Edward Evans

@BalarkaSen ohno

user2103480

@EdwardEvans worth

Thorgott

will I not sound retarded when I go to a layperson and tell them I'm currently trying to write down the pseudo-2-adjunction describing the process of freely adjoining coproducts to a category

Edward Evans

Nah man you'll just sound like a nerd

Thorgott

even I think I sound retarded

user2103480

15:19

@BalarkaSen such a disheartening state of affairs

Astyx

Yeah, you'd be the tech guy that solves super complicated problems in films while speaking gibberish

Balarka Sen

lolol

user2103480

You sound retarded to a mathematician since you talk about categories

2

Edward Evans

hahaha then hit your keyboard really fast and change your PC theme to black background with green text

Astyx

hackertyper.com

Edward Evans

15:21

looool

amazing

Balarka Sen

Ok, throw me some numbers

►

Thorgott

lmao that's great

@user2103480 ooooof

user2103480

@BalarkaSen us

Edward Evans

@BalarkaSen lmfao

Balarka Sen

sorry to do it to ya

user2103480

15:26

@BalarkaSen if only TDA would actually be useful in business applications

Balarka Sen

yeah TDA is pretty darn cool

Astyx

TDA ?

Thorgott

ok, I successfully convinced myself that if $U\in V$ are two Grothendieck universes, the 2-category of all $U$-categories, functors and natural trafos is a $V$-category

Balarka Sen

im sure some companies will pay you for it though

Thorgott

back to the interesting stuff

user2103480

15:27

@BalarkaSen this talk might be interesting to you

Edward Evans

lol at "trafo"

user2103480

Benjamin Dunn - Toroidal topology of grid cell ensemble activity

►

Turns out TDA might be useful in biology

Balarka Sen

@Thorgott this might be interesting to you

data on a torus

manifold learnin

user2103480

tl;dr is that the topology of the pattern in which orientation cells in mouse brains fire might be a torus, consistently across different shapes of the environment

Balarka Sen

yeah i have seen this in Gunnar Carlsson's papers i think

its ok ill just do catastrophe theory and psychology

user2103480

15:31

It was known that this is the case for a simple square shaped environment

Thorgott

ahhhhhhhhhh, applications

user2103480

@BalarkaSen people dont want to admit it but the reason we are so bad at mathematical biology is that it's too hellishly complicated

Balarka Sen

yeah man we humans understand neither math nor biology

how can we understand math biology

medical science found a new human organ like yesterday

science is a scam

Astyx

@BalarkaSen Really ?

Balarka Sen

yeah

lol

Astyx

15:34

Got a link ?

user2103480

the based department

Balarka Sen

a fucking 2 inch big salivary gland up your throat

its in new york times

Thorgott

yeah, but have we discovered a new finite simple group

Balarka Sen

look it up

Daniel

I wonder if there is a more inference orientated application of TDA. Its seems to be quite static and boring

Astyx

15:36

Is it more than journalistic sensationalism ?

user2103480

Noice its true

Daniel

but nonetheless powerful

user2103480

@Astyx yup seems like it

Astyx

@Thorgott It turns out there's another group of order 6

Balarka Sen

lmao

imagine

Thorgott

15:37

damn, show me

Daniel

Using science, scientists have scientifically found a new organ thanks to science

8

user2103480

I've converted to balarkas and alessandros religion

Finite groups dont exist

Except for a point

Balarka Sen

@Daniel i imagine you can do something like that; i mean you can always say a certain data set is from a certain shape and then hypothesis test based on the persistent betti numbers

oh i see what you mean

yeah maybe you need to pick points from a distribution concentrated around your shape

idk how to say "points are from a given shape but upto a normal fudge"

Thorgott

actually, infinite groups don't exist

except $S^1$, I like $S^1$

Balarka Sen

numbers dont exist

-thorgott, 2020

finally a geometer

Rithaniel

15:41

Reject numbers, return to coming up with consistent axioms

Alessandro Codenotti

Woops sorry I had to leave for a moment

Thorgott

numbers? you mean isomorphism classes of compact objects in $\mathbf{Set}$?

Alessandro Codenotti

Is the universes stuff sorted out?

Thorgott

yeah, I convinced myself it works out, I think

Astyx

There's this french guy who claims to be the one to have solved the Collatz conjecture, and he posted like a 30 minutes long video explaining why he should be the one the honors go to

Alessandro Codenotti

15:43

@Thorgott good

Balarka Sen

collatz is the dumbest open problem in mathematics

Thorgott

now I'm trying to figure out the difference between biadjunction and pseudoadjunction

Balarka Sen

it looks like someone noticed it while making the next IMO exam

Astyx

The fun thing is he didn't bring a proof, he just says that he should be the one remembered as the first to solve it, but without giving a proof

Thorgott

I have solved the weak Collatz conjecture, which states that if you start with a natural number, half it if it's even and multiply it by three and add one if it's odd, you obtain a sequence of numbers

The strong Collatz conjecture crucially relies on this result

Astyx

15:48

lol

Thorgott

oh fuck fuck fuck, I think I actually have to work with 3-categories

someone save m

Daniel

ncatlab can be so silly sometimes. Is a 2-morphism simply morphism between morphisms?

Thorgott

I think that depends on how you define higher categories

Lucas Henrique

16:07

hey chat

Astyx

hi

Lucas Henrique

given that I have the invariant factors of an operator $T$ (i.e., if $V = \bigoplus_{j=1}^n C_T(v_j)$ as in the cylic decomposition and I know how the minimal polynomials $m_{v_j}$ look like), can I find a Jordan decomposition for $T$?

Daniel

Actually i was thinking the other day, is there an equivalent notion for graphs. I.e. one can get a graph from another by letting the vertices be nodes in the new ones, etc.

user2103480

@BalarkaSen easy. Just sample from the normal manifold distribution whose mean your manifold is

>implying this thing exists

But I've seen definitions of random manifolds, and would like to go through those some day

Daniel

Maybe TDA could tell you what samples you can't have given that you know a set of them.

user2103480

17:15

@BalarkaSen en.wikipedia.org/wiki/…. this seems to be a metric on riemannian manifolds

but I have no clue whether the corresponding space is separable. If yes that would be a first step for a manifold-valued random variable

For a hilbert space of manifolds, one could just apply the theory of gaussian measures on hilbert spaces

But how the hell do you define addition of manifolds

Thorgott

connected sum?

unless you want an actual group structure

user2103480

@Thorgott this

Alessandro Codenotti

Hmm I'm not sure about manifolds, but there are notions like "the (standard) Borel space of Polish spaces"

Thorgott

pass to the grothendieck group lol

Prateek Mourya

17:37

Can we apply binomial expansion on infinitesimal

Mike

Hello, I have a question about representing bounded linear functionals by signed measure with control statement.

What I've tried is to present the bounded linear functional Lambda by difference of two nonngative bounded linear functionals Lambda^+ and Lambda^-, but I failed to control the measure decomposition. Any ideas or thoughts? I may post this on the community if I have further progress. Thank you.

user2103480

18:05

whats the domain of lambda. just some normed vector space? banach?

Ah uh

f are continuous functions on [0,1]?

with sup norm

Alexandru Ionut

hi! any numerical analysts here?

Faust

18:25

uh

wtf is that

like analysis ?

Alexandru Ionut

numerical analysis

Faust

i googled it might be bale to help you out, whats the question

Alexandru Ionut

1

Q: Asymptotics for 2 Humbert series special forms

I have 2 Humbert seriers special forms that arise in a quantum physics problem. $a$ and $x$ are real. $$\phi_1(1+ia,ia,ia+3/2;1/2,ix)$$ $$\phi_2(ia,-ia,1/2;ix,-ix)$$ I want to find asymptotic expansions when $x \to \infty$ in order to compute precise approximations when x is sufficiently large.

Faust

i dont understand the notation of the maps lol.

Alexandru Ionut

Humbert series

In mathematics, Humbert series are a set of seven hypergeometric series Φ1, Φ2, Φ3, Ψ1, Ψ2, Ξ1, Ξ2 of two variables that generalize Kummer's confluent hypergeometric series 1F1 of one variable and the confluent hypergeometric limit function 0F1 of one variable. The first of these double series was introduced by Pierre Humbert (1920). == Definitions == The Humbert series Φ1 is defined for |x| < 1 by the double series: Φ 1 ( a , b , c ; x ,...

user2103480

18:31

I'd try using the faber-schauder system (a schauder basis for C[0,1]) to define the signed measure on [0,1]

Faust

Are you sure the first series is even defined when $|x|>1 $?

Alexandru Ionut

the special form I am concerned with has x=1/2 so it converges

sorry some confusion there

Faust

So the $x \to \infty $ is for the second series then>

Alexandru Ionut

it also converges

Faust

Sorry im not going to be able to help you lol. its intresting though what kind fo physics problem does it show up in?

Alexandru Ionut

18:34

Landau-Zener

Faust

Interesting ^^

Alexandru Ionut

19:11

@Ted Shifrin hià

rschwieb

19:26

Well this is a relief retractionwatch.com/2020/10/22/…

2

Thorgott

lol what

Tobias Kildetoft

@rschwieb That is at least something. Though I still find it highly improper that it was published at all

Any peer reviewer with the background to be asked to review the paper should have spotted the problem immediately

user2103480

20:11

ok, so if we find a finite measure \lambda such that \Lambda is L^2([0,1], Borels,\lambda)-bounded on the space of continuous functions.

Then, since the continuous bounded functions are dense in that space, we can extend \Lambda onto that L^2 space. This means that its extension is a bounded linear functional on that hilbert space and hence by riesz, there is a function g such that \Lambda(f) = integral over [0,1] of f(x)g(x) \lambda(dx)

Ted Shifrin

This guy must have a degree from Tromp University.

user2103480

Then we take g(x) as the density of the signed measure that we actually want

rschwieb

@TedShifrin Or even more likely, a university sporting his own name and distinguished by the fact all the staff and its most famous graduate are the same person.

Alexandru Ionut

sounds like my university

user2103480

And obtain that \Lambda(f) = integral over [0,1] of f(x)g(x) \lambda(dx) = integral over [0,1] of f(x) \mu(dx)

Ted Shifrin

20:16

Yeah, but his defensive pugilism based on delusion sounds like he learned from the master.

Hi, @Alexandru.

Alexandru Ionut

:)

user2103480

The problem is finding that finite measure \lambda for which |\Lambda(f)| \leq D*int_[0,1] |f|^2 d\lambda

Ted Shifrin

Dollar signs would be helpful.

rschwieb

@TedShifrin Maybe he's got a dayjob in the administration. Hard to understand how he'd stay employed elsewhere.

user2103480

@TedShifrin but latex formatting is not automatic here, is it?

Ted Shifrin

20:20

Well, he claims to be allied to Oxford.

user2103480

dollar signs make it readable for everyone who's got the formatting tool, but even less readable for the others

Alexandru Ionut

@TedShifrin I posted a question on the mainsite finally. Do you know any numerical analysts or special function people?

geocalc33

@user2103480 have you ever come a across a pdf that when integrated on it's support is an integral transform?

Ted Shifrin

Everyone here uses the tool. See the link on the upper right if you're on a computer.

Not here personally, @Alexandru.

Alexandru Ionut

I have am at an impass in quite a weird niche right now.

user2103480

20:24

@geocalc33 Uhhh i dont know much about integral transforms

geocalc33

$F(s)=\int_0^{\infty} x^{s-1}f(x)~dx$ is an example of an integral transform

I guess I'm wondering if there are any examples of known probability distributions, that are also well-known integral transforms...

Alexandru Ionut

20:39

*I am at an impass... (weird typo up there)

geocalc33

*probability distributions when integrated over their support rather

user2103480

20:52

@geocalc33 Can't you just define one exactly like that?

geocalc33

@user2103480 yeah, that's a good point, I'm just trying to come up with a good example

I need to choose an $f(x)$ s.t. the kernel is a known pdf

I guess $f(x)=x$ would work

then the kernel would be $x^{s}.$ And I could tweak the bounds of integration to be $0,1$

Lucas Henrique

21:12

is it possible that two real matrices are similar over $\mathbb{C}$ but not over $\mathbb{R}$?

geocalc33

good question

Ted Shifrin

@LucasHenrique real matrices?

Lucas Henrique

matrices with real entries

Ted Shifrin

The answer is no.

Lucas Henrique

my professor's book has an exercise asking for us two non-similar real matrices like this...

Ted Shifrin

21:22

Here's a hint. If $E$ is a field ext of $F$ can the gcd if $f(x),g(x)$ in $E[x]$ be different from that in $F[x]$?

Lucas Henrique

what i thought: similar over $\mathbb{C} \iff$ same Jordan form. if I could compare the real Jordan forms...

@TedShifrin nope

Ted Shifrin

The prof messed up.

Lucas Henrique

@TedShifrin I don't get it. :/

Ted Shifrin

You are rightl the prof is wrong .

ShazzaNic

Hi guys - wondering if someone can help me.
The number of items bought by people entering a shop is random variable X that has a geometric starting at 0 with mean 0.6. Find the value of the parameter p of the geometric distribution and write down the p.g.f of X.

So E(X) = 𝑞/𝑝 which can equal 1-p/p =0.6. I think the answer to p is 0.375, how is this answer worked out?

Lucas Henrique

21:30

@TedShifrin, I mean: how does that imply that they are similar?

ShazzaNic

never mind guys got it, thanks.

Lukas Heger

@LucasHenrique the question you're asking is equivalent to the following: if $L/K$ is a field extension, let $V$ and $W$ be $K[x]$-modules, finite-dimensional over $K$. Then does $L \otimes_K V \cong L \otimes_K W$ as $L[x]$-modules imply $V \cong W$ as $K[x]$-modules? Noting that $L[x] =L \otimes_K K[x]$, we may replace $K[x]$ by any $K$-algebra $A$. and ask the same question.
This is answered positively by the Noether-Deuring theorem: if $L \otimes_K V \cong L \otimes_K W$ as $L \otimes_K A$-modules, then $V \cong W$ as $A$-modules

Lucas Henrique

maybe I could use stuff easier than module theory

the rational form of a matrix

Leaky Nun

21:46

@LukasHeger Galois descent?

Lukas Heger

@LeakyNun there are no restrictions on $L/K$

Leaky Nun

aha

wie gibt's?

ich hab' nicht dich gesehen hier fur eine lange Zeit

Lukas Heger

mir ging es zwischendrin nicht so gut, aber jetzt wieder besser

if $L/K$ is finite, say of degree $n$, then there's a nice proof of Noether-Deuring: use restriction of scalars on the isomorphism $L \otimes_K V \cong L \otimes_K W$ to get an isomorphism of $A$-modules $V^n \cong W^n$. This implies that $V$ and $W$ have the same indecomposable factors with the same multiplicity by Krull-Schmidt

Leaky Nun

are $V$ and $W$ finite over $A$?

Lukas Heger

you need them to be finite-dimensional over $K$, even

Leaky Nun

21:56

this feels like rep theory

Lukas Heger

that's why they are of finite length and we can use Krull-Schmidt

you can call $V$ and $W$ representations of $A$ if you wish

of course $A=K[G]$ is a common application of this

Edward Evans

yooooooooooooooooooooooooooooooooooooooo @Lukas

Lukas Heger

hey @EdwardEvans

Edward Evans

Nice to hear from you man

Leaky Nun

@LukasHeger werdest du her öfter kommen?

Lukas Heger

22:01

hmm, mal schauen

Edward Evans

Bist noch im Studium oder bist du kurz so "ausgestiegen"?

Lukas Heger

ich bin wieder im Studium

Edward Evans

Okay freut mich :)

Was hast du dieses Semester vor?

Lukas Heger

meinen Bachelor endlich zuende bringen

ich muss noch so nervige Sachen wie Numerik oder Stochastik hören :D

Edward Evans

Hahaha ja gut, macht auch Sinn

Hab auch vor kurzem herausgefunden, dass ich mindestens eine Vorlesung "angewandte Mathe" hören muss, was auch immer das ist

Leaky Nun

22:05

was muss ich weissen, wenn ich in Deutschland PhD studieren will?

Edward Evans

Wahrscheinlich weniger als du jetzt schon weißt

Thorgott

o, welcome back

Ted Shifrin

@Lukas! We were worried about you!

Lukas Heger

hey @Ted

Leaky Nun

as in, what should I pay attention to

do's / don'ts

Lukas Heger

22:07

yeah I went underground for a while

Edward Evans

@Leaky probably best to ask @Alessandro as he recently applied, and was accepted, to a PhD program in Münster

Leaky Nun

danke

Edward Evans

Gerne

Ich werde ihn auch fragen, da ich mich auch dort bewerben werde lol

Leaky Nun

hast du einmal ein CEFR test getan? @EdwardEvans

Ted Shifrin

Glad to have you back! 😻

Edward Evans

22:09

@Leaky ja ich musste für die Zulassung zur Uni in Heidelberg eine Deutschprüfung machen

Leaky Nun

@TedShifrin hast du ein Katze?

@EdwardEvans was ist dein Niveau?

Edward Evans

Angeblich C2

Leaky Nun

nice

Ted Shifrin

Il y a longtemps j'avais plusieurs chats, mais pas maintenant.

Leaky Nun

warum nun nicht?

Edward Evans

22:11

@TedShifrin glad you just stuck to math chat then

Lukas Heger

it's possible to do a PhD in Germany without proving your German skills, though, there are some PhD students in Heidelberg with little German knowledge

Edward Evans

@Lukas I'll be taking p-adic Hodge theory this semester T_T

Lukas Heger

oh wow

that's challenging for sure

Edward Evans

yeah I'm a bit scared

I'm not sure what the content will be, but if it's some comparison between cohomology theories then I might be überfordert

The program isn't decided yet, so I guess I'll decide for sure when it's updated on Mampf

Ted Shifrin

Just stuck, @Edward?

Edward Evans

22:17

@TedShifrin wait, what's the question?

Ted Shifrin

@EdwardEvans That.

Edward Evans

does not compute

user2103480

@LeakyNun I think it's less of a terrible procedure than what I hear from the US and such

Leaky Nun

that's great

user2103480

Do you already have a master's?

If not, you could also apply for berlin mathematical school which is more like a normal PhD program

you can apply in either case, which I'll probably do as well, but you get a financial stipend with BMS*

*not blue mountain state

Edward Evans

22:22

Shouldn't you get like TV E-13 as a PhD student anyway?

user2103480

Yeh, but I mean during the master program as well

Edward Evans

ah I see

I applied for BMS and got denied

so fuck berlin

(not bitter)

user2103480

haha fair enough, corresponding to the PhD program that is more alike the US ones, the admission process is proportionally more annoying

I'm also just doing a normal master's

Edward Evans

sem

user2103480

Also, berlin doesn't have much in algebraic number theory, do they?

Edward Evans

22:32

errr I think I remember there being a reasonable number of number theory courses

but I think they are more focussed on arithmetic geometry or smth

I don't remember

they're dead to me

user2103480

hahahaha

Edward Evans

Spectrum of Advanced Courses

The advanced courses regularly offered include Abelian Varieties, Algebraic Stacks, Arakelov Geometry, Birational Geometry of Algebraic Varieties, Class Field Theory, Complex and p-adic Hodge Theory, Elliptic Curves and Cryptography, Moduli Spaces, Shimura Varieties, Theory of Automorphic Forms.

user2103480

https://agnes.hu-berlin.de/lupo/rds;jsessionid=997A1A6448222BBD22F31510A29653A5.angua_root?state=wtree&search=1&root120202=185123|180601|180709|175833&trex=step

HUs course catalogue is really solid this semester but not number theory, but they have serious geometry

Edward Evans

were you supposed to be at HU?

user2103480

nah, I'm at FU, but idc I do courses at all three as a nebenhörer

Edward Evans

22:37

ah nise

user2103480

FU has algebra III this semester with

Course contents: a selection of the following topics

properties of morphisms (proper, projective, smooth)
divisors
(quasi-)coherent sheaves
cohomology
Hilbert functions
the Riemann-Roch Theorem

and algebraic K theory

Edward Evans

algebra

user2103480

as another lecture

but that's really it

https://www.math-berlin.de/academics/courses/advanced-courses

I love berlin

2 advanced algebra classes across three unis

Edward Evans

I wonder why "Modular forms and applications" is in "Differential geometry, global analysis, and mathematical physics"

geocalc33

How can this be simplified? $\int_0^{\infty} e^{-x^s}dx+\sum_{n=0}^{\infty} \frac{(-1)^n}{n!} \zeta(-ns)$

Mike Miller

22:40

"Mathematical physics" just means math

user2103480

@EdwardEvans I think it stems from the professors specialization

Interests: enumerative geometry, quantum
algebra/topology/geometry, topological recursion,
integrability, moduli spaces, algebraic structures in
QFT and strings, random matrix theory, combinatorics, science outreach.

Mike Miller

"Science" "Outreach"

user2103480

take that, algebra

Edward Evans

lawl the single algebraic geometer at my previous uni insisted on his homepage that he was interested in mathematical physics and all of his research was pure category theory

I mean, the physics group at that uni was huge so maybe it was to get the job or smth idk

user2103480

to be fair he does the course on integrable systems which does sound pretty interesting

Description: The course is an introduction to the theory of classical and quantum integrable systems. The classical part is concerned with constructing solutions of (systems of) non-linear PDEs ; the quantum part is concerned with the 'explicit' diagonalization of (family of) operators ; in both case, their 'integrability' means that there are miracles making these seemingly complicated problems solvable. These miracles are closely related to the existence of many (hidden) symmetries. This applies to a variety of models that are relevant in physics, including examples of non-linear wave pro…

Edward Evans

22:43

the fuhhh

user2103480

yeh

Mike

@user2103480 Did you use Hahn Banach to complete the extension? And how to do you complete you last step-"finite measure \lambda for which |\Lambda(f)| \leq D*int_[0,1] |f|^2 d\lambda"?

user2103480

@Mike no no, I used the fact that if we have an operator that is bounded on a dense subspace of some larger normed space (bounded in the norm of that space!), that maps into a banach space, then you can uniquely extend it to a continuous operator on the larger space

since R is a banach space

and I dunno how to complete that step. Might also be the wrong way to tackle the problem

Mike

Okay. I might post this problem to the community. I applied Riesz theorem and Hahn decomposition to it, but I find it hard to use the control inequality. @user2103480

user2103480

You mean the boundedness of the functional?

Mike

22:50

I think your idea may be applicable, but I don't see how to complete the last step you mentioned.

user2103480

|Lambda(f)| \leq ||\Lambda|| * ||f||_sup

Mike

I think the proof should be two sided: One side is just
|Lambda(f)| \leq ||\Lambda|| * ||f||_sup (as you mentioned), but the other side should be harder.

user2103480

What do you mean by that? We are already given that inequality if I'm reading it correctly

and we need to use that to derive the rest of it

Ah I see that you already have a theorem that uses an idea, quite similar to what I was trying to do

From on of your questions

Geocrafter

I'm having trouble understand how this derivation works by integrating with bounds:
$\V=L\frac{di}{dt}$
$\\int_{-\infty}^{t} vdt = \int_{-\infty}^{t} L \frac{di}{dt}dt$

user2103480

You define \lambda([a,b]) = inf{\Lambda(f): f \geq indicator_[a,b]}

Mike

22:57

Yes, you are right.@user2103480

user2103480

i tried to take limits of ramp functions that converge to the indicator function from above, pointwise, but that is quite a bit uglier than the above definition

Mike

And I'll write a post regarding this question in the community. I'll send the link in this chat once I complete it.

Your method seems like proving Riesz representation for C[0,1] :)

user2103480

@Mike oh btw the way I wrote it is wrong, this needs to be |\Lambda(f)|^2, since else I'd need to take the square root of the integral

But tbh I'm too lazy to bang my head against trying to make mu([a,b]) = limit of \Lambda(ramp function_n) work :D it should work somehow, but maybe that direct way is inelegant

@EdwardEvans helene esnault at FU does arithmetic geometry

and has the most touching professor's website I've seen yet

Her husband passed away and I'd suppose that explains the bittersweet poems

Oh my, that Anna Akhmatova poem

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