This question is quite general, I'll write just my own point of view, and hope others add more to get a complete picture.
0) Let me quote A. Kirillov himself:
"In conclusion I want to express the hope that the orbit method
becomes for my readers a source of thoughts and inspirations as it has
been for me during the last thirty- ve years"
Sorry I cannot find another much more colorful quote from him, where he says something like
orbit method not only produced results on many principal questions of representation theory, but gives informal guidance how to invent new and new results.
1) What is the context of orbit method and why it is related to mathematical physics.
Orbit method as a particular case of quantization ideology.
I think orbit method should be seen in the context of quantization and roughly speaking its relation to the mathematical physics is that orbit method is particular example of "quantization program". (Well, there some other relations with integrable systems, but they are not so central, imho).
Let me try to explain this in more details. Consider universal enveloping $U(g)$,
it is an algebra with generators $e_k$ and relations $[e_{k}, e_{l} ] = c^{i}_{kl} e_l $,
let us insert parameter $h$ here: $[e_{k}, e_{l} ] = h c^{i}_{kl} e_l$.
For $h=0$ we have just the commutative algebra - denote it $S(g)$, for any non-zero $h$ this algebra is isomorphic to $U(g)$.
Now let us look at $h$ "very small", i.e. we can formally take $h^2=0$, what will see from
the structure of non-commutative algebra $U(g)$ reduces to commutative algebra $S(g)$
"plus" a Poisson bracket on it.
So the moral is that
Non-commutative algebra, when non-commutativity tends to zero is commutative algebra with Poisson bracket. (It is called classical limit).
The big goal of quantization program
is try to express everything about non-commutative algebra in terms of
Poisson algebra. The reason is that Poisson algebra is something more simple than non-commutative algebra.
In particular we can be interested in description of irreps of non-commutative algebra.
What are the corresponding objects for the Poisson algebra ?
Answer - symplectic leaves.
Observation: the symplectic leaves of $S(g)$ (classical limit of $U(g)$) are exactly the coadjoint orbits.
So this puts the orbit method - in more general framework of quantization. Where one may hope to describe the irreps of quantized algebras via symplectic leaves in Poisson algebra.
The problem is that such construction does not always exists neither for $U(g)$, nor for general quantum algebras, at least it does not exist in some simple sense, and big activity is to understand what are the borders between true statements from fakes, it is neverending activity.
It is worth to remark that such point of view on orbit method is not the original one, but emerged later when quantization theory begin to develop.
The basic question of representation theory are calculating characters, induction-restriction, tensor products. The natural question: what are the parallel constructions
in the Poisson world (i.e. symplectic geometry) ? how representation theory questions can be answered with the help of symplectic geometry ?
There are ideological answers to these questions and again it is neverending game to make "ideology" to theorems or to counter-examples.
Some MO-questions with more details on quantization: Q1, Q2.
2) What is naive quantization without metaplectic correction. (It is better to call it correction (imho), but not quantization, but people use both).
So, our basic wish is to construct a irreducible representation of $U(g)$ (or more generally of some quantized algebra $\hat A$).
The "naive recipe" is the following:
1) take symplectic leaf
2) Consider algebra of functions on it and split it into two halves "P-part" and "Q-part
3) The representation space is ----- all functions of "Q"-variable,
and representation is constructed as: Q-variables act as multiplication operators,
while "P" acts as $\partial_Q$.
Now, what I mean "split in two parts", informally you should think as follows Darboux theorem says that symplectic form is $dp\wedge dq$ is appropriate coordinates, so
you have these "p" and "q" as my splitting. The problem is that Darboux is local result,
and you need something more complicated to make it work, look at the word "polarization" for more information on that.
3) Towards metaplectic correction.
In the previous item I wrote that naively we should take "half of functions" on the orbit(symplectic leaf) as a Hilbert space.
This actually a point to be corrected.
We should not take "functions", but should take "half-forms".
The simplest motivation is that we want to have a Hilbert space, so we need to have an inner product, but there is no canonical one on the functions.
But if we take half-forms: $f(q) \sqrt{dq}$ we have the canonical inner product:
$\int fg \sqrt{dq}^2 = \int fg dq $.
So the metaplectic correction is story is how to introduce these "half-forms" into a business in an appropriate way.
Some time ago we exercised with quantization of sphere $S^2$ and argued that in general
this should be consistent with the Duflo isomorphism.
4) Finally to your main question: "quantization of coadjoint orbits".
Well, sorry, I cannot say much. The general ideology here is that we should take
a coadjoint orbit and try to construct irrep. Kirillov done it in 1962 for nilpotent groups, for solvable much progress achieved later. For semisimples - generic orbits - no problems, but for many orbits it is impossible (or at least in some naive sense impossib). I do not know what is the current state of art.
It might be there are some particular classes of orbits where some people think
that one can indeed construct irrep and it is in reach and good topic for a paper of a PhD. It might be that ideas by Ranee Brylinski Geometric Quantization of Real Minimal Nilpotent Orbits can be somehow developed...
But I do not know much about it, my impression that all left open problems are quite difficult and technical, I would not start this as a PhD.
Any case I would ask David Vogan or Jeffrey Adams (he is sometimes on MO).
By the way have a look at D. Vogan's REVIEW OF “LECTURES ON THE ORBIT METHOD,”.
Any way good luck !
Parabolic bundles were introduced in the 70's by Mehta and Seshadri in the set
up of a Riemann surface with cusps. They were trying to generalize the
Narasimhan-Seshadri correspondence on a compact Riemann surface (between
polystable bundles of degree $0$ and unitary representations of the
fundamental group). In the non-compact case, they were able to determine the
missing piece of data - partial flags and weights at each cusp. They
established what is now called the Mehta-Seshadri correspondence. Then they
proceeded to study the moduli space.
Mehta, V. B.; Seshadri, C. S.
Moduli of vector bundles on curves with parabolic structures.
Math. Ann. 248 (1980), no. 3, 205–239.
https://link.springer.com/article/10.1007/BF01420526
Since then, the definition of a parabolic bundle has been clarified (tensor
product with the initial definition is not really computable for instance) and
generalized. This is a long story starting with C.Simpson, I.Biswas, and many
authors. The upshot is that given a scheme $X$, a Cartier divisor $D$, and an
integer $r$, there is a one to one tensor (and Fourier-like) equivalence
between parabolic vector bundles on $(X,D)$ with weights in
$\frac{1}{r}\mathbb Z$ and standard vector bundles on a certain orbifold
$\sqrt[r]{D/X}$, the stack of $r$-th roots of $D$ on $X$. So one can turn your
question in: why are these orbifolds natural ? They were first introduced by
A.Vistoli in relation with Gromov-Witten theory. They also turned out to be
related to the section conjecture (rational points of stack of roots are
Grothendieck's packets in his anabelian letter to Faltings). So parabolic
sheaves - and stack of roots - are ubiquitous. They are also very strongly
related to logarithmic geometry.
Best Answer
In response to the second question (which I interpret as asking for math models of spider webs as they appear in Nature): There exist several distinct types of spider webs. The most common type, the orb web of araneids, has been modeled in Simple Model for the Mechanics of Spider Webs (2010).
A key property of the orb web model is that the web is free of stress concentrations even when a few spiral threads are broken. This is distinctly different from usual elastic materials in which a crack causes stress concentrations and weakens the material.
The model highlights the mechanical adaptability of the web: spiders can increase the number of spiral threads to make a dense web (to catch small insects) or they can adjust the number of radial threads (to adapt to environmental conditions or reduce the cost of making the web) – in both cases without reducing the damage tolerance of the web.
Left panel: Construction of the orb web described in the cited paper.
Right panel: Naturally occurring orb web (Wikipedia).