The h Bar

In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the Lax equation. Lax pairs were introduced by Peter Lax to discuss solitons in continuous media. The inverse scattering transform makes use of the Lax equations to solve such systems. == Definition == A Lax pair is a pair of matrices or operators L ( t ) , P ( t ) {\displaystyle L(t),P(t)} dependent on time and acting...

The original Lax paper gets those matrices by trial and error, fair enough, that's one example which takes ages but at least something

The other examples e.g. NLS make me want to cry :p

0celo7

@Semiclassical every schoen and yau paper

G. Bergeron

@bolbteppa You find a solution of the Yang-Baxter equation

bolbteppa

I've never seen YB brought up, hmm

Nonlinear Schrödinger equation

In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to Bose-Einstein condensates confined to highly anisotropic cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted...

Always stuff like that, a mess, or they just give you matrices out of thin air, e.g.:

G. Bergeron

8:16 PM

arxiv.org/pdf/hep-th/9612046.pdf

bolbteppa

e.g. page 45 for sine-gordon here

damtp.cam.ac.uk/user/md327/ISlecture_notes_2012.pdf

Semiclassical

the problem i always had with integrability stuff was how hard it was to find a source which explained things well

bolbteppa

Yeah, these notes look really crazy haha

G. Bergeron

I suggest reading old Baxter's book

bolbteppa

The Babelon book is insane too

The exactly solved models book is crazy

Semiclassical

8:18 PM

i was about to say

bolbteppa

It's all crazy

Semiclassical

Babelon is mad

There's a set of notes by Dubrovin which I always wanted to make work

bolbteppa

Yeah them notes look good

The book I linked to ages ago, Baker Abelian is like a baby versio of Dubrovin, and it's like 1000 pages long

Algebraic geometry like nobodies business

G. Bergeron

Are you familiar with the maths, by themselves?

@bolbteppa The one by Baxter?

I thought it was cute and nicely introduced 8\

bolbteppa

The maths is all too crazy, I think integrability requires a different kind of thinking

The Das book is pretty good

This is the only book I've found a way to derive the Lax matrices in, but it's pretty convoluted and requires some very clever choices

G. Bergeron

8:22 PM

This, for me begin to put a toe in the crazy territory: bookstore.ams.org/cbms-85

bolbteppa

amazon.com/Integrable-Models-Scientific-Lecture-Physics/dp/…

G. Bergeron

He solved the XXZ chain directly in the infinite case by making use of a quantized affinely extended universal enveloping algebra of sl(2)

bolbteppa

You can see the contents in that, it's pretty nice, the only real book that's made sense so far

I am stunned at how complicated the math is yet they actually solve problems

G. Bergeron

I used some results which I tediously verified to calculate some cross-section for neutron scattering in the end

Yes, the expressions for the overlaps you obtain are in terms of humongous contour integral that takes a page to write, the results of which you then need to integrate numerically directly in C cause everything else is too slow bweuh!

bolbteppa

Apparently fermions make things easier in this realm, e.g. Ising is more straightforward, need to write it up though

G. Bergeron

8:28 PM

But the nasty maths is mainly because there are quantum algebras of symmetries which means a lot of the things that make everything nice in undergrad physics disappears

Emilio Pisanty

-2

Q: Is terahertz radiation new?

Regarding the future applications of terahertz technology, I thought we already knew the entire electromagnetic spectrum. In fact we do not, terahertz lies between microwaves and infrared radiation. The future applications of this is astonishing. For example, faster communications and next genera...

radiation

This was a middling question. But then, that last sentence.

Absolutely golden.

G. Bergeron

Right now I'm calculating expressions for the matrix elements of the SU_q(3) quantum group in the symmetric representation and a lot of tools I want to reach to do not work anymore...

@bolbteppa That seems old school!

Also, symplectic geometry can rapidly become tangled as it is a relatively new branch of mathematics

Semiclassical

I think that the phrase "symplectic geometry" is a bit deceiving

in that it's not just "symplectic manifolds"

G. Bergeron

ok symplectic topology, if you prefer

Semiclassical

Sure, I wasn't meaning to quibble

Bernardo Meurer

8:31 PM

@EmilioPisanty Can you give me a hand with Bibtex?

Semiclassical

just to say that it's tough going

G. Bergeron

yeah

Semiclassical

The math subject which tantalized me for a while were Riemann-Hilbert problems

but jeebuz

that's rough

bolbteppa

I think the logic is, starting from the Lax equations, you can derive the zero-curvature conditions, and then by expanding out the zero-curvature conditions with 2x2 Pauli matrices you can derive all the basic Lax matrices by some simple choices, it's in chapter 12 of that book, the question is why those choices, but it seems very unified

Quantum groups, my god, another mountain

Semiclassical

Too many mountains.

bolbteppa

8:34 PM

This stuff is like the 'combinatorics' of theoretical physics I'd say

G. Bergeron

@bolbteppa It's quite nice in how the ''quantization'' procedure is rather the same as the one applied to phase space to obtain QM but now applied to the dynamical symmetry algebra

Semiclassical

The other subject I've had an interest in is resurgence theory, mostly because I would like to understand nonperturbative physics

bolbteppa

The non-genius normal mathematicians stay away from combinatorics :p

G. Bergeron

@bolbteppa LOL a few in my families are combinatorists...

bolbteppa

@G.Bergeron I tried to learn about them without knowing the serre relations you deform were a lie algebra thing :(

Semiclassical

8:35 PM

The only parts of combinatorics I feel like I trust myself on are the parts I can reduce to generating functions

G. Bergeron

@bolbteppa Yeah, the Lie theory is often too nice... We get spoiled by it

bolbteppa

and the q deformation stuff is madness, but I recently found a q-calculus book which q's baby calculus so if I read it I'd say I'd love quantum groups :p

G. Bergeron

@bolbteppa It's the stuff of my daily research

Semiclassical

a pity that most of the resurgence theory papers I've seen just go over my head too fast

bolbteppa

Overall this stuff is cool, it's been a real challenge to make any sense of it at all, and am still basically nowhere

G. Bergeron

8:37 PM

@Semiclassical That I'm not that familiar with

bolbteppa

How much lie theory had you studied going into this, e.g. did you go through Humphreys or something

Semiclassical

The nLab page is a decent summary for once: ncatlab.org/nlab/show/resurgence+theory

G. Bergeron

@Semiclassical Except that most talks in special functions conference seems to rely on that :P

Semiclassical

yeah

G. Bergeron

@Semiclassical nLab is like the least pedagogical resources, in general

Semiclassical

8:38 PM

hence the caveat :P

G. Bergeron

@bolbteppa A had a graduate course and a lot of picking up things on the way

There is a lot I still don't know

Semiclassical

I think what I understand of it can largely be summed up as "how to make asymptotic series useful"

G. Bergeron

I'll read it up!

bolbteppa

The idea of divergent series meaning something was sheer magic to me for a while

G. Bergeron

OK, now I need to go back to work... On those q-deformed groups!

bolbteppa

8:40 PM

Good stuff

Semiclassical

I mean, the usual issue is that asymptotic series work as approximations but shouldn't be pushed past that

G. Bergeron

See you

Semiclassical

this stuff is in the spirit of getting past such obstructions

later

bolbteppa

I'd love to know Hardy's book on divergent series

There is simply too much to know

Semiclassical

yeah

the more you know, the more aware you are of your own ignorance

(the opposite holds as well, as I think most of us have experienced)

bolbteppa

8:43 PM

The price of such knowledge is trying to do a tiny bit on some topic every now and again apparently :p

I will finish up the derivation of these stupid matrices someday and put it on an overleaf

Emilio Pisanty

8:57 PM

@BernardoMeurer I at a bad keyboard

But shoot

vzn

@G.Bergeron any opinion on Verlinde?

Bernardo Meurer

@EmilioPisanty How would you make an entry for:
1. https://law.stanford.edu/publications/the-future-of-copyright-in-a-digital-environment-summary-of-discussion/
2. https://archive.nytimes.com/www.nytimes.com/books/99/04/18/specials/rushdie-address.html

Oh, wait, 1. is ez

just 2.

Actually 1. isn't easy

rob

@EmilioPisanty Oh, good question. What seems to have happened is that two moderators added the "book recommendation" banner (about forty seconds apart) so the post ended up with two banners. The fix was to hit "remove banner" (which removed them both) and put the banner back.

Bernardo Meurer

@rob o/

Emilio Pisanty

9:34 PM

@BernardoMeurer yeah, that'll need to wait for a better keyboard, likely tomorrow morning

eulB

howdee

Emilio Pisanty

@rob ah

@BernardoMeurer but that one looks reasonably easy

Try as a template the reference HHG Tutorial in my thesis

What style are you using?

lılostafa

10:13 PM

Why doesn't this get any answers after 5 days? :/

eulB

don't people get asked to contribute chapters by the editor?

lılostafa

@eulB Are you the former Blue here?

@eulB Yes, I believe the editor contacts them first

ACuriousMind

@lılostafa Probably because no one wants to generalize there - I suspect people have very different views on this and there's no single "professional" opinion.

eulB

@lılostafa no?

@lılostafa yeah so there's your answer

lılostafa

@eulB What?

What if an unknown researcher who happens to be the editor of a book contacts you to write a chapter for a book published by Springer?

@ACuriousMind This is something that I really need advice for

eulB

10:28 PM

@lılostafa then do it if you want?

what's the big deal here?

it's not a paper, you're writing something for a book, so it doesn't matter if the editor isn't well known as long as the publisher is.

0celo7

@ACuriousMind I showed my advisor the error and he reacted with glee because I have to something harder to fix it :(

Secret

11:53 PM

Last night dream involve 4 weird expressions:

(Cont here instead)

$-(xz)^2 z(forgot)dz-zdz^*$

$z(forgot)dz-zdz^*$

$(xz)^2 z(forgot)dz-zdz^*$

$-z(forgot)dz-zdz^*$

According to that weird blog in the dream, the author basically create 4 types of complex numbers. After looking at them the first thing that reminds me of is the multivalued magnetic potential of magnetic monopoles. A cutscene then showed a hemisphere seen from the top, coloured in 4 different colours correspond to the 4 quadrants defined by the above equations

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