MATH6040: Spring 2021, Week 10

April 12, 2021 in MATH6040, MATH6040 - Night | Leave a comment

Differentiation Test

Test Tuesday 20 April 19:30 to 20:45. Five questions, one for each of the five sections in Chapter 3.

You can find old (e.g. Chapter 3) videos on my YT channel here. (Links to an external site.)

One more video that is actually Chapter 3 material (and the recording was a little messed up so no live writing… a new version will be in the Week 11 lectures.):

Week 10

Lectures

Revision of integration and integration by parts.

Here are last year’s lectures of the same material:

I recorded a lot of this material before in a live lecture: press here to watch live version (Links to an external site.) (84 minutes).

Exercises

How much time you put into homework is up to you: of course the more time you put in the better but we all have competing interests. Please feel free to ask me questions about the exercises. 

Try:

  • p. 165, Q.1-13
  • p. 171, Q.1-5

Additional Exercises: p. 171, Q. 6-9

Submit work for Canvas feedback by Sunday 18 April for video feedback after Monday 19 April.

Outlook

Week 11

Perhaps of the order of 1.5 hours of lectures on completing the square (Links to an external site.), and work (Links to an external site.).

Week 12

Perhaps of the order of 1.5 hours of lectures on centroids (Links to an external site.) of laminas and centres of gravity of solids of revolution (Links to an external site.).

Weeks 13 and 14

I will continue to provide learning support.

Assessment Schedule

Week 11  – 25% Differentiation Test (Zoom Tutorial in Week 10, after Easter)

Week 14 – 25% Integration Test (Zoom Tutorial in Week 13)

Study

Please feel free to ask me questions about the exercises via email. I answer emails every morning seven days a week.

Student Resources

Please see Student Resources (Links to an external site.) for information on the Academic Learning Centre, etc.

MATH7016: Spring 2021, Week 10

April 12, 2021 in MATH7016 | Leave a comment

Assessment Corrections

Have finally been completed. Please check your grade and if you have any queries please don’t hesitate to email me.

Assessment 4

Written Assessment 2 will not be entirely unlike Written Assessment 1. The Week 12 Zoom will be Monday 26 April 15:00 instead of Tuesday 27 April.

25% Written Assessment 1, based on Weeks 6-10, so everything from p.74 to 106.

It will be a one hour assessment, but I am going to give ye 15 minutes grace, as well as 15 minutes to upload. The test will run therefore from 09.30 to 11.00, Tuesday 27 April. It is open book — you can use your manual, any Canvas materials, as well as Excel/VBA.

40% of the marks will be for Section 1.9

30% of the marks will be for Section 2.1

40% of the marks will be for Section 2.2

Academic Dishonesty will not be accepted and suspected breaches, such as communication with others during the assessment, will be pursued in line with this policy (Links to an external site.) (Links to an external site.). 

Week 10

Lectures

 This (Links to an external site.) explains the connection between the Heat Equation and the Jacobi Method approximations to the Laplace Equation

There is a longer, more in-person (old, last year) version of the above material. How different they are I am not too sure.

Theory Exercises and Q & A

Q&A on Tuesday 13 April as normal: p.104, Q.1-3

Week 11

No more lectures, no more labs: you will focus on VBA Assessment 2. Q&A on Tuesday.

Week 12

No more lectures, no more labs: you will focus on Written Assessment 2. Q&A on Monday.

Study

Study should consist of

  • doing exercises from the notes
  • completing VBA exercises

Student Resources

Please see Student Resources  download  for information on the Academic Learning Centre, etc..

MATH7021: Spring 2021, Week 10

April 12, 2021 in MATH7021 | Leave a comment

You have to put time to work on Chapter 3:

  1. because without doing so you could be very, very lost on 35% Assignment 3, and
  2. because if you are going into Level 8 Structural Engineering it will be assumed that you are competent with the Chapter 3 material

Assignment 1 & 2 Corrections

These have finally been released. Apologies for the delay. Any queries do not hesitate to contact me.

Week 10

Lectures

No lectures: all MATH7021 time to be invested into Chapter 3 and Assignment 3. See Week 11, below, if you have completed Assignment 3 and are hoping to look ahead.

Exercises

As I send this, there are two slots for feedback over Easter with deadlines of today, 5 April, for feedback, tomorrow, 6 April, and 12 April for feedback 13 April.

If you are up to date on the Week 7 exercises, put some time into:

  • p.127, Q.1-11

Additional exercises p.130, Q.12-19.

Submit work for Canvas feedback by Monday 19 April for video feedback after Tuesday 20 March.

Weeks 11 and 12

Weeks 11 and 12 will be given over to Chapter 4. Perhaps of the order of 2 hours of lectures on Double Integrals (Links to an external site.) in Week 11 and of the order of 2 hours of lectures on Triple Integrals (Links to an external site.). Students who have already submitted Assignment 3 can look at the old lectures. The material is exactly the same… for the sake of of doing honest work I will be rerecording for Weeks 11 and 12, but if you want to get ahead you can watch these:

Week 11:

Week 12:

Re: Section 3.5 Systems of Differential Equations (Links to an external site.); important for next year, won’t be examined this semester but will get ye the notes and lectures before the end of year (so as to have them as a reference for next year).

Assignments

35% Assignment 1 on Chapter 1 — due end of Week 5, 28 February.

15% Assignment 2 on Chapter 2 — due end of Week 7, Sunday 14 March

35% Assignment 3 on Chapter 3 — due end of Week 11, Sunday 25 April will be released soon

15% Assignment 4 on Chapter 4 — due end of Week 13/14/15, (7/14/21 May) tbd

Student Resources

Please see Student Resources (Links to an external site.) for information on the Academic Learning Centre, etc.

Some problems in compact quantum groups

April 8, 2021 in Quantum Groups, Research, Research Updates | Leave a comment

After completing a piece of work, I like to record some things that I would like to work on next. The previous time was a little over 11 months ago, and was heavily geared towards random walks on quantum groups. The need at that time was to start properly learning some compact quantum groups. I made a start on this: my plan was to write up some notes on compact quantum groups… however these notes [these are real rough], were abandoned like the Marie Celeste in May 2020. What happened was that I had a technical question about the construction of the reduced algebra C_r(\mathbb{G}) of continuous functions on a compact quantum group \mathbb{G}, and the expert who helped me suggested some intuition that I could use for quantum permutations… this kind of set off a quest to find a better interpretation for quantum permutations that started with this talk, then led to this monstrosity, and finally to this paper that I am proud of (but not sure if journals will concur).

So anyway some problems and brief thoughts. I tried and failed to resist the urge to use the non-standard notation and interpretation used in this new paper… I guess this post is for me… if you want to understand the weird “\varsigma is a quantum permutation” and “\varsigma\in\mathbb{G}“, etc you will have to read the paper.

No Quantum Alternating Group

So the big long crazy draft of has some stuff about why there is no quantum cyclic or alternating groups but these are arguments rather than proofs. A no-go theorem here looks as follows:

A finite group G<S_N has no quantum version if whenever \mathbb{G}<S_N^+ is a quantum permutation group with group of characters of C_u(\mathbb{G}) equal to G, then \mathbb{G}=G.

I know the question for A_N is open… is it formally settled that there is no quantum \mathbb{Z}_N? Proving that would be a start. A possible strategy would be to construct from G<\mathbb{G} and a quantum permutation like \varsigma_{e_5}\in\mathfrak{G}_0 a character on C_u(\mathbb{G}) that is in the complement of G in S_N. The conclusion being that there can be no quantum permutation in C_u(\mathbb{G}) that is C_u(\mathbb{G}) is commutative. There might be stuff here coming from alternating (pun not intended) projections theory that could help.

Ergodic Theorem for Random Walks on Compact Quantum Groups

From April 2020:

My proof of the Ergodic Theorem leans heavily on the finiteness assumption but a lot of the stuff in that paper (and there are many partial results in that paper also) should be true in the compact case too. How much of the proof/results carry into the compact case? A full Ergodic Theorem for Random Walks on Compact Quantum Groups is probably quite far away at this point, but perhaps partial results under assumptions such as (co?)amenability might be possible. OR try and prove ergodic theorems for specific compact quantum groups.

What I would be interested in doing here is seeing can I maybe use the language from the Ergodic Theorem to prove some partial results in this direction. The analysis is possibly a little harder than I am used to. What I might want to show is that if \phi\in C(\mathbb{G}) is an idempotent state (I am not sure are there group-like projections \mathbf{1}_{\phi} lying around, I think there are maybe here), and p_{\phi} its support projection in \ell^{\infty}(\mathbb{G}), that the convolution of quantum permutations \varsigma_1,\varsigma_2 such that p_\phi(\varsigma_i)=1, that p_\phi(\varsigma_2\star \varsigma_1)=1. This would almost certainly require the use of a group-like projection. Possibly restricting to to quantum permutation groups we would then have a good understanding of non-Haar idempotent “quasi-subgroups”:

A quasi-subgroup of a compact (permutation?) quantum group is a subset of \mathbb{G} that (for C_u(\mathbb{G})) contains the identity, is closed under reversal, and closed under the quantum group law. A quasi-subgroup is a quantum subgroup precisely when it is closed under wave function collapse.

The other thing that could be done here would be to refine two aspects of the finite theory. One, from the direction of cyclic shifts, in order to make the definition of a cyclic coset more intrinsic (it is currently defined with respect to a state), and two to further study the idea of an idempotent commuting with something like a “finite order” deterministic state (see p.27).

Maximality Conjecture

For N\leq 5, there is no intermediate quantum subgroup S_N<\mathbb{G}<S_N^+. That is there is no comultiplication intertwining surjective *-homomorphism \pi:C_u(S_N^+)\twoheadrightarrow C_u(\mathbb{G}) to noncommutative C_u(\mathbb{G}) such that there is in addition there is another such map \pi_C:C_u(\mathbb{G})\twoheadrightarrow F(S_N).

This well-known conjecture is that there is no intermediate quantum subgroup S_N<\mathbb{G}<S_N^+ at any N.

Let \mathfrak{G}_0 be the Kac–Paljutkin quantum group of order eight and consider the quantum permutation \varsigma_0:= (\varepsilon+ \varsigma_{e_5})/2\in \mathfrak{G}_0, the state space of the algebra of functions on \mathfrak{G}_0, F(\mathfrak{G}_0). Where u^{\mathfrak{G}_0<S_N^+}:=\text{diag}(u^{\mathfrak{G}_0},\mathbf{1}_{\mathfrak{G}_0},\dots, \mathbf{1}_{\mathfrak{G}_0}) and \pi_{\mathfrak{G}_0}:C(S_N^+)\twoheadrightarrow F(\mathfrak{G}_0), \varsigma\circ \pi_{\mathfrak{G}_0}\in S_N^+. Where \pi_C:C(S_N^+)\rightarrow F(S_N) and (by abuse of notation) h_{S_N}:=h_{S_N}\circ\pi_C\in S_N^+ the Haar state of S_N in S_N^{+}. Where \varsigma_1:=h_{S_N}\star \varsigma_0, consider the idempotent state on C(S_N^+):

\displaystyle \varsigma:=w^{*}-\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{k=1}^{n}\varsigma_1^{\star k}.

There are three possibilities and all three are interesting:

\varsigma=h_{S_N^+} — this is what I expect to be true. If this could be proven, the approach would be to hope that it might be possible to construct a state like \varsigma_{e_5} on any compact quantum group. The state \varsigma_{e_5} has a nice “constraint” property: I can only map 1 or 2 to 3 or 4 and vice versa. A starting idea in the construction might be to take the unital \mathrm{C}^* algebra generated by a non-commuting pair u^{\mathbb{G}}_{i_1j_1},u^{\mathbb{G}}_{i_2j_2}. Using Theorem 4.6, there is a *-representation \pi_2:\mathrm{C}^*(u^{\mathbb{G}}_{i_1j_1},u^{\mathbb{G}}_{i_2j_2})\rightarrow B(\mathbb{C}^2) such that for some t\in (0,1) we have a representation and from this representation we have a nice vector state that is something like \varsigma_{e_5}. Well it doesn’t have the “constraint” property… one idea which I haven’t thought through is to condition this lovely quantum permutation in a clever enough way… we could condition it to only map j_1 to i_1 or some i_3… but that seems to only be the start of it.

\varphi=h_{\mathbb{G}} for S_N<\mathbb{G}<S_N^{+} — this would obviously be a counterexample to the maximality conjecture.

\varphi is a non-Haar idempotent — this is a possibility that I don’t think many have thought of. It wouldn’t disprove the conjecture but would be an interesting example. This might be something non-zero on e.g. |u_{31}u_{22}u_{11}|^2 but zero on some strictly positive |f|^2\in C(S_N^+).

Doing something with abelian quantum permutation groups

I can’t really describe this so instead I quote from the paper:

It could be speculated that the dual of a discrete group \Gamma=\langle\gamma_1,\dots,\gamma_k\rangle could model a k particle “entangled” quantum system, where the p-th particle, corresponding to the block B_p, has |\gamma_p| states, labelled 1,\dots,|\gamma_p|. Full information about the state of all particles is in general impossible, but measurement with x(B_p) will see collapse of the pth particle to a definite state. Only the deterministic permutations in \widehat{\Gamma} would correspond to classical states.

Quantum Automorphism Groups of Graphs

So I want to read Schmidt’s PhD thesis and maybe answer the question of whether or not there is a graph whose quantum automorphism group is the Kac–Paljutkin quantum group. Also see can anything be done in the intersection of random walks and quantum automorphism groups.

Write locally compact to compact dictionary

I want to able to extract results on locally compact quantum groups to compact quantum groups.

Use of Stopping Times and Classical Probabilistic Methods for Random Walks on Quantum Permutation Groups

This is a bit mad… bottom of p.35 to p.36.

Maybe take x_0:=1\in\{1,\dots,N\} and define a Markov chain on \{1,\dots,N\} using a quantum permutation. So for example you measure \varsigma with x(1) to get x_1. Then measure x(x_1) and iterate. The time taken to reach N or some other k\in\{1,\dots,N\} or maybe hit all of \{1,\dots,N\}… this is a random time T. Is there any relationship between the expectation of T and the distance of the convolution powers of \varsigma to the Haar state. Lots of things to think about here.

Alternating Measurement

More mad stuff. See Section 8.5.

Infinite Discrete Duals with no Finite Order Generators

If \Gamma has a finite set of finite-order generators it is a quantum permutation group. What about if there are only infinite order generators? I guess this isn’t a quantum permutation group (subgroup of some S_N^+) but maybe using Goswami and Skalski such a dual can be given the structure of a quantum permutation group on infinite many symbols. At the other end of the scale… are the Sekine quantum groups quantum permutation groups?

Other random walk questions

From April 2020:

  1. Look at random walks on quantum homogeneous spaces, possibly using Gelfand Pair theory. Start in finite and move into Kac? — No interest in this by me at the moment
  2. Following Urban, study convolution factorisations of the Haar state. — ditto.
  3. Examples of non-central random walks on compact quantum groups — Freslon and coauthors have cornered the market on interesting examples of random walks on compact quantum groups… I don’t think I will be spending time on this.

Note that Simeng Wang has sorted: extending the Upper Bound Lemma to the non-Kac case. There are a handful of other problems here, here, and here that I am no longer interested in.

A state-space approach to quantum permutations

April 6, 2021 in C*-Algebras & Operator Theory, Quantum Groups, Research, Research Updates | Leave a comment

An expository piece, the watered down version of the madness here… should be on the Arxiv Thursday.

Click here for pdf.

Limit Profiles for Quantum Random Transpositions

April 2, 2021 in arXiv, Quantum Groups, Random Walks on Finite Quantum Groups, Research | Leave a comment

In this post here, I outlined some things that I might want to prove. Very bottom of that list was:

Extending the Upper Bound Lemma to the non-Kac case. As I speak, this is beyond what I am capable of. This also requires work on the projection and quantum total variation distances (i.e. show they are equal in this larger category).

After that post, Simeng Wang wrote with the, for me, exciting news that he had proven the Upper Bound Lemma in the non-Kac case, and had an upcoming paper with Amaury Freslon and Lucas Teyssier. At first glance the paper is an intersection of the work of Amaury in random walks on compact quantum groups, and Lucas’ work on limit profiles, a refinement in the understanding of how the random transposition random walk converges to uniform. The paper also works with continuous-time random walks but I am going to restrict attention to what it does with random walks on S_N^+.

Introduction

For the case of a family of Markov chains (X_N)_{N\geq 1} exhibiting the cut-off phenomenon, it will do so in a window of width w_N about a cut-off time t_N, in such a way that w_N/t_N\rightarrow 0, and, where d_N(t) is the ‘distance to random’ at time t, d_N(t_N+cw_N) will be close to one for c<0 and close to zero for c>0. The cutoff profile of the family of random walks is a continuous function f:\mathbb{R}\rightarrow[0,1] such that as N\rightarrow \infty,

\displaystyle d_N(t_N+cw_N)\rightarrow f(c).

I had not previously heard about such a concept, but the paper gives a number of examples in which the analysis had been carried out. Lucas however improved the Diaconis–Shahshahani Upper Bound Lemma and this allowed him to show that the limit profile for the random transpositions random walk is given by:

\displaystyle d_{\text{TV}}(\text{Poi}[1+e^{-c}],\text{Poi}[1])

Without looking back on Lucas’ paper, I am not sure exactly how this d_{\text{TV}} works… I will guess it is, where \xi_1\sim \text{Poi}[1+e^{-c}] and \xi_2\sim \text{Poi}[2], and so:

\displaystyle f_{\text{RT}}(c)=\frac{1}{2}\sum_{k=0}^{\infty}\left|\mathbb{P}[\xi_1=k]-\mathbb{P}[\xi_2=k]\right|,

and I get f(0)\approx 0.330, f(2)\approx 0.0497 and f(-2)\approx 0.949 on CAS. Looking at Lucas’ paper thankfully this is correct.

The article confirms that Lucas’ work is the inspiration, but the study will take place with infinite compact quantum groups. The representation theory carries over so well from the classical to quantum case, and it is representation theory that is used to prove so many random-walk results, that it might have been and was possible to study limit profiles for random walks on quantum groups.

More importantly, technical issues which arise as soon as N>4 disappear if the pure quantum transposition random walk is considered. This is a purely quantum phenomenon because the random walk driven only by transpositions in the classical case is periodic and does not converge to uniform. I hope to show in an upcoming work how something which might be considered a quantum transposition behaves very differently to a classical/deterministic transposition. My understanding at this point, in a certain sense (see here)) is that a quantum transposition has N-2 fixed points in the sense that it is an eigenstate (with eigenvalue N-2) of the character \text{fix}=\sum_i u_{ii}. I am hoping to find a dual with a quantum transposition that for example does not square to the identity (but this is a whole other story). This would imply in a sense that there is no quantum alternating group.

The paper will show that the quantum version of the ordinary random transposition random walk and of this pure random transposition walk asymptotically coincide. They will detect the cutoff at time \frac12 N\ln N, and find an explicit limit profile (which I might not be too interested in).

I will skip the stuff on O_N^+ but as there are some similarities between the representation theory of quantum orthogonal and quantum permutation groups I may have to come back to these bits.

Quantum Permutations

Character Theory

At this point I will move away and look at this Banica tome on quantum permutations for some character theory.

Read the rest of this entry »

MATH6040: Spring 2021, Week 9

March 27, 2021 in MATH6040, MATH6040 - Night | Leave a comment

Easter might provide extra time to put into Chapter 3.

Matrices Test — Corrections

I have a bit of a backlog of corrections but hopefully I can get these to ye before Easter Sunday.

Week 9

Lectures

Final two sections of Chapter 3:

Here are last year’s lectures of the same material:

Exercises

How much time you put into homework is up to you: of course the more time you put in the better but we all have competing interests. Please feel free to ask me questions about the exercises. 

Try:

  • p. 143, Q. 1-5 [Note these exercises are interleaved – there are questions here from earlier sections in Chapter 3]
  • p. 150, Q. 1-6 [Note these exercises are interleaved – there are questions here from earlier sections in Chapter 3]

Additional Exercises: p. 143, Q. 6, p. 151, Q.7-8, 9-10

Submit work for Canvas feedback by Sunday 28 March for video feedback after Monday 29 March.

Outlook

I will be providing learning support over Easter.

Looking further ahead, to after Easter, a good revision of integration/antidifferentiation may be found here. Here is some video of revision of antidifferentiation.

Week 10 

Perhaps of the order of 1.5 hours of lectures on starting Chapter 4 on (Further) Integration with a revision of antidifferentiation, and a look at Integration by Parts. We will use implicit differentiation to differentiate inverse sine.

Week 11

Perhaps of the order of 1.5 hours of lectures on completing the square, and work.

Week 12

Perhaps of the order of 1.5 hours of lectures on centroids of laminas and centres of gravity of solids of revolution.

Assessment Schedule

Week 11  – 25% Differentiation Test (Zoom Tutorial in Week 10, after Easter)

Week 14 – 25% Integration Test (Zoom Tutorial in Week 13)

Study

Please feel free to ask me questions about the exercises via email. I answer emails every morning seven days a week.

Student Resources

Please see Student Resources for information on the Academic Learning Centre, etc.

MATH7016: Spring 2021, Week 9

March 27, 2021 in MATH7016 | Leave a comment

Assessment Corrections

I cannot promise at this point when these will be completed. It is my intention to have these corrected before the end of Week 9. Watch this space.

Assessments 3 and 4

I will give proper notice for VBA Assessment 2 by the end of Week 9 and for Written Assessment 2 during Easter.

Week 9

Lectures

There is a longer, more in-person (old, last year) version of the above material: 

Lab 7

See VBA Lab 7

Theory Exercises and Q & A

Q&A on Tuesday as normal:

p.97, Q. 1,2 and p. 98 exercises

Week 10

I have linked here to last year’s videos but will be recording fresh videos. How different they will be I am not too sure.

Outlook

Week 11 and 12

Lectures

None. Perhaps additional tutorial time in Week 11 for Written Assessment in Week 12.

Labs

We will do our last lab, Lab 8 in Week 10. VBA Assessment 2 in Week 11.

Assessment

This is provisional and subject to change.

  1. Week 6, 25% First VBA Assessment, Based (roughly) on Weeks 1-4
  2. Week 7, 25 % In-Class Written Test, Based (roughly) on Weeks 1-5
  3. Week 11, 25% Second VBA Assessment, Based (roughly) on Weeks 6-9
  4. Week 12, 25% Written Assessment(s), Based on Weeks 6-10

Study

Study should consist of

  • doing exercises from the notes
  • completing VBA exercises

Student Resources

Please see Student Resources  for information on the Academic Learning Centre, etc..

MATH7021: Spring 2021, Week 9

March 27, 2021 in MATH7021 | Leave a comment

You have to put regular time to work on Chapter 3:

  1. because without doing so you could be very, very lost on 35% Assignment 3, and
  2. because if you are going into Level 8 Structural Engineering it will be assumed that you are competent with the Chapter 3 material

You need to get cracking on Chapter 3.

Assignment 1 & 2 Corrections

I am experiencing a hardware issue at the moment and it is making it difficult to get corrections done.. My promise is to complete corrections of both Assignment 1 and Assignment 2 before the end of Week 9. Sorry for the delay.

Week 9

Lectures

No lectures in Week 9. If you are behind please put big time into Chapter 3, catching up.

Exercises

If you are up to date on the Week 7 exercises, put some time into:

  • p.127, Q.1-11

Additional exercises p.130, Q.12-19.

Submit work for Canvas feedback by Monday 29 March for video feedback after Tuesday 30 March.

Outlook

In Week 10 you will also spend all your time doing Chapter 3 Exercises/Assignment 3. It is my intention to continue providing learning support throughout the Easter break.

Section 3.5 Systems of Differential Equations. is important for next year but I have messed up and so it will not be examined. I will record lectures and fill in the notes for ye though during Easter once I am caught up on corrections.

Weeks 11 and 12 are given over to Chapter 4. Perhaps of the order of 2 hours of lectures on Double Integrals in Week 11 and of the order of 2 hours of lectures on Triple Integrals. If I see that students have submitted Assignment 3 early I may record these earlier than Weeks 11 and 12 (or student could look at the old lectures, the ones that say “integrals” here)

Assignments

35% Assignment 1 on Chapter 1 — due end of Week 5, 28 February.

15% Assignment 2 on Chapter 2 — due end of Week 7, Sunday 14 March

35% Assignment 3 on Chapter 3 — due end of Week 11, Sunday 25 April will be released soon

15% Assignment 4 on Chapter 4 — due end of Week 13/14/15, (7/14/21 May) tbd

Student Resources

Please see Student Resources for information on the Academic Learning Centre, etc.

Intuition for Quantum Permutations

March 19, 2021 in C*-Algebras & Operator Theory, Quantum Groups, Research, Research Updates | Leave a comment

What started about ten months ago as a technical question to an expert, led to a talk, and led to me producing this weird production here.

Now that it is complete, although I really like all its contents (well except for the note to reader and introduction I spilled out very hastily), I can see on reflection it represents rather than a cogent piece of mathematics, almost a log of all the things I have learnt in the process of writing it. It also includes far too much speculation and conjecture. So I am going to post it here and get to work on editing it down to something a little more useful and cogent.

EDIT: Edited down version here.

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