This post follows on from this one. The purpose of posts in this category is for me to learn more about the research being done in quantum groups. This post looks at this paper of Schmidt.

# Preliminaries

## Compact Matrix Quantum Groups

The author gives the definition and gives the definition of a (left, quantum) group action.

### Definition 1.2

Let $G$ be a compact matrix quantum group and let $C(X)$ be a $\mathrm{C}^*-algebra$. An (left) action of $G$ on $X$ is a unital *-homomorphism $\alpha: C(X)\rightarrow C(X)\otimes C(G)$ that satisfies the analogue of $g_2(g_1x)=(g_2g_1)x$, and the Podlés density condition:

$\displaystyle \overline{\text{span}(\alpha(C(X)))(\mathbf{1}_X\otimes C(G))}=C(X)\otimes C(G)$.

## Quantum Automorphism Groups of Finite Graphs

Schmidt in this earlier paper gives a slightly different presentation of $\text{QAut }\Gamma$. The definition given here I understand:

### Definition 1.3

The quantum automorphism group of a finite graph $\Gamma=(V,E)$ with adjacency matrix $A$ is given by the universal $\mathrm{C}^*$-algebra $C(\text{QAut }\Gamma)$ generated by $u\in M_n(C(\text{QAut }\Gamma))$ such that the rows and columns of $u$ are partitions of unity and:

$uA=Au$.

_______________________________________

The difference between this definition and the one given in the subsequent paper is that in the subsequent paper the quantum automorphism group is given as a quotient of $C(S_n^+)$ by the ideal given by $\mathcal{I}=\langle Au=uA\rangle$… ah but this is more or less the definition of universal $\mathrm{C}^*$-algebras given by generators $E$ and relations $R$:

$\displaystyle \mathrm{C}^*(E,R)=\mathrm{C}^*( E)/\langle \mathcal{R}\rangle$

$\displaystyle \Rightarrow \mathrm{C}^*(E,R)/\langle I\rangle=\left(\mathrm{C}^*(E)/R\right)/\langle I\rangle=\mathrm{C}*(E)/(\langle R\rangle\cap\langle I\rangle)=\mathrm{C}^*(E,R\cap I)$

where presumably $\langle R\rangle \cap \langle I \rangle=\langle R\cap I\rangle$ all works out OK, and it can be shown that $I$ is a suitable ideal, a Hopf ideal. I don’t know how it took me so long to figure that out… Presumably the point of quotienting by (a presumably Hopf) ideal is so that the quotient gives a subgroup, in this case $\text{QAut }\Gamma\leq S_{|V|}^+$ via the surjective *-homomorphism:

$C(S_n^+)\rightarrow C(S_n^+)/\langle uA=Au\rangle=C(\text{QAut }\Gamma)$.

_______________________________________

## Compact Matrix Quantum Groups acting on Graphs

### Definition 1.6

Let $\Gamma$ be a finite graph and $G$ a compact matrix quantum group. An action of $G$ on $\Gamma$ is an action of $G$ on $V$ (coaction of $C(G)$ on $C(V)$) such that the associated magic unitary $v=(v_{ij})_{i,j=1,\dots,|V|}$, given by:

$\displaystyle \alpha(\delta_j)=\sum_{i=1}^{|V|} \delta_i\otimes v_{ij}$,

commutes with the adjacency matrix, $uA=Au$.

By the universal property, we have $G\leq \text{QAut }\Gamma$ via the surjective *-homomorphism:

$C(\text{QAut }\Gamma)\rightarrow C(G)$, $u\mapsto v$.

### Theorem 1.8 (Banica)

Let $X_n=\{1,\dots,n\}$, and $\alpha:F(X_n)\rightarrow F(X_n)\otimes C(G)$, $\alpha(\delta_j)=\sum_i\delta_i\otimes v_{ij}$ be an action, and let $F(K)$ be a linear subspace given by a subset $K\subset X_n$. The matrix $v$ commutes with the projection onto $F(K)$ if and only if $\alpha(F(K))\subseteq F(K)\otimes C(G)$

### Corollary 1.9

The action $\alpha: F(V)\rightarrow F(V)\otimes C(\text{QAut }\Gamma)$ preserves the eigenspaces of $A$:

$\alpha(E_\lambda)\subseteq E_\lambda\otimes C(\text{QAut }\Gamma)$

Proof: Spectral decomposition yields that each $E_\lambda$, or rather the projection $P_\lambda$ onto it, satisfies a polynomial in $A$:

$\displaystyle P_\lambda=\sum_{i}c_iA^i$

$\displaystyle \Rightarrow P_\lambda A=\left(\sum_i c_i A^i\right)A=A P_\lambda$,

as $A$ commutes with powers of $A$ $\qquad \bullet$

# A Criterion for a Graph to have Quantum Symmetry

### Definition 2.1

Let $V=\{1,\dots,|V|\}$. Permutations $\sigma,\,\tau: V\rightarrow V$ are disjoint if $\sigma(i)\neq i\Rightarrow \tau(i)=i$, and vice versa, for all $i\in V$.

In other words, we don’t have $\sigma$ and $\tau$ permuting any vertex.

### Theorem 2.2

Let $\Gamma$ be a finite graph. If there exists two non-trivial, disjoint automorphisms $\sigma,\tau\in\text{Aut }\Gamma$, such that $o(\sigma)=n$ and $o(\tau)=m$, then we get a surjective *-homomorphism $C(\text{QAut }\Gamma)\rightarrow C^*(\mathbb{Z}_n\ast \mathbb{Z}_m)$. In this case, we have the quantum group $\widehat{\mathbb{Z}_n\ast \mathbb{Z}_m}\leq \text{QAut }\Gamma$, and so $\Gamma$ has quantum symmetry.

Warning: This is written by a non-expert (I know only about finite quantum groups and am beginning to learn my compact quantum groups), and there is no attempt at rigour, or even consistency. Actually the post shows a wanton disregard for reason, and attempts to understand the incomprehensible and intuit the non-intuitive. Speculation would be too weak an adjective.

## Groups

A group is a well-established object in the study of mathematics, and for the purposes of this post we can think of a group $G$ as the set of symmetries on some kind of space, given by a set $X$ together with some additional structure $D(X)$. The elements of $G$  act on $X$ as bijections:

$G \ni g:X\rightarrow X$,

such that $D(X)=D(g(X))$, that is the structure of the space is invariant under $g$.

For example, consider the space $(X_n,|X_n|)$, where the set is $X_n=\{1,2,\dots,n\}$, and the structure is the cardinality. Then the set of all of the bijections $X_n\rightarrow X_n$ is a group called $S_n$.

A set of symmetries $G$, a group, comes with some structure of its own. The identity map $e:X\rightarrow X$, $x\mapsto x$ is a symmetry. By transitivity, symmetries $g,h\in G$ can be composed to form a new symmetry $gh:=g\circ h\in G$. Finally, as bijections, symmetries have inverses $g^{-1}$, $g(x)\mapsto x$.

Note that:

$gg^{-1}=g^{-1}g=e\Rightarrow (g^{-1})^{-1}=g$.

A group can carry additional structure, for example, compact groups carry a topology in which the composition $G\times G\rightarrow G$ and inverse ${}^{-1}:G\rightarrow G$ are continuous.

## Algebra of Functions

Given a group $G$ together with its structure, one can define an algebra $A(G)$ of complex valued functions on $G$, such that the multiplication $A(G)\times A(G)\rightarrow A(G)$ is given by a commutative pointwise multiplication, for $s\in G$:

$(f_1f_2)(s)=f_1(s)f_2(s)=(f_2f_1)(s)$.

Depending on the class of group (e.g. finite, matrix, compact, locally compact, etc.), there may be various choices and considerations for what algebra of functions to consider, but on the whole it is nice if given an algebra of functions $A(G)$ we can reconstruct $G$.

Usually the following transpose maps will be considered in the structure of $A(G)$, for some tensor product $\otimes_\alpha$ such that $A(G\times G)\cong A(G)\otimes_\alpha A(G)$, and $m:G\times G\rightarrow G$, $(g,h)\mapsto gh$ is the group multiplication:

\begin{aligned} \Delta: A(G)\rightarrow A(G)\otimes_{\alpha}A(G)&,\,f\mapsto f\circ m,\,\text{the comultiplication} \\ S: A(G)\rightarrow A(G)&,\, f\mapsto f\circ {}^{-1},\,\text{ the antipode} \\ \varepsilon: A(G)\rightarrow \mathbb{C}&,\, f\mapsto f\circ e,\,\text{ the counit} \end{aligned}

See Section 2.2 to learn more about these maps and the relations between them for the case of the complex valued functions on finite groups.

## Quantum Groups

Quantum groups, famously, do not have a single definition in the same way that groups do. All definitions I know about include a coassociative (see Section 2.2) comultiplication $\Delta: A(G)\rightarrow A(G)\otimes_\alpha A(G)$ for some tensor product $\otimes_\alpha$ (or perhaps only into a multiplier algebra $M(A(G)\otimes_\alpha A(G))$), but in general that structure alone can only give a quantum semigroup.

Here is a non-working (quickly broken?), meta-definition, inspired in the usual way by the famous Gelfand Theorem:

A quantum group $G$ is given by an algebra of functions $A(G)$ satisfying a set of axioms $\Theta$ such that:

• whenever $A(G)$ is noncommutative, $G$ is a virtual object,
• every commutative algebra of functions satisfying $\Theta$ is an algebra of functions on a set-of-points group, and
• whenever commutative algebras of functions $A(G_1)\cong_{\Theta} A(G_2)$, $G_1\cong G_2$ as set-of-points groups.

Keep in mind at all times:

“Any and all work, submitted at any time, will receive feedback.”

“Do not hesitate to contact me with questions at any time. My usual modus operandi is to answer all queries in the morning but sometimes I may respond sooner.”

## 25% Integration Test

60 minute, 25% Further Integration Test, 19:30 Tuesday 12 May 2020

This test will examine Chapter 4. A Summer 2020 paper was written, and this test will comprise of Q. 4 from that exam. The question comprises 25% of that exam, and that translates to 30 minutes of exam time. I will extend this to 45 minutes and allow 15 minutes to upload.

Here is video based on Q. 4, on p.226 of your manual.

The material for this test was covered in Week 10, Easter Week 1, and Easter Week 2 (lectures and exercises).

If you have any further questions on these topics, I am answering emails seven days a week. Email before midnight tonight to be guaranteed a response Monday 11 May. I cannot guarantee that I answer emails sent on Monday (although of course I will try).

## 10% Vectors Test

60 minute, 10% Vectors Test, 19:30 Tuesday 19 May 2020

This test will re-examine Chapter 1. A Summer 2020 paper was written, and this test will comprise of Q. 1 from that exam. The question comprises 25% of that exam, and that translates to 30 minutes of exam time. I will extend this to 45 minutes and allow 15 minutes to upload.

Here is a video based on Q. 1, on p.222 of your manual.

Chapter 1 Exercises may be found on:

• p.29
• p.39
• p.46

You can submit work for feedback by midnight Saturday 16 May to Vectors Exercises on Canvas. After this, email before midnight Monday 18 May to be guaranteed a response Tuesday 19 May. I cannot guarantee that I answer emails sent on Tuesday (although of course I will try).

Keep in mind at all times:

“Any and all work, submitted at any time, will receive feedback.”

“Do not hesitate to contact me with questions at any time. My usual modus operandi is to answer all queries in the morning but sometimes I may respond sooner.”

## 40% Test 1

90 minute, 40% Test, 11:00 Monday 11 May 2020

This test will examine Chapter 2, Section 3.6, and Chapter 4. A Summer 2020 paper was written, and this test will comprise of Q. 2, 3 (d), and 4 from that exam. The question comprises 40% of that exam, and that translates to 48 minutes of exam time. I will extend this to 75 minutes and allow 15 minutes to upload.

Here is tutorial video for this test.

Section 3.6 and Chapter 4 material can be found in Week 10, Easter Week 1, and Easter Week 2 (lectures and tutorial).

Chapter 2 Exercises that you should be looking at include:

• p.86, Q. 1-4
• p.91, Q. 1-7
• p.86, Q. 5-6 (harder)

I have developed the following for you to practise your differentiation which you need for Chapter 2 here:

If you have any further questions on these topics, I am answering emails seven days a week. Email before midnight tonight to be guaranteed a response Monday 11 May. I cannot guarantee that I answer emails sent on Monday (although of course I will try).

## 20% Linear Systems Test

90 minute, 20% Linear Systems Test, 11:00 Thursday 21 May 2020

This test will re-examine Chapter 1. A Summer 2020 paper was written, and this test will comprise of Q. 1 from that exam. The question comprises 35% of that exam, and that translates to 42 minutes of exam time. I will extend this to 75 minutes and allow 15 minutes to upload.

Here is video based on Q. 1 from the Summer 2019 paper on the back of your manual.

Chapter 1 Exercises may be found on:

• p.28
• p.38
• p.44
• p.51

You can submit work for feedback by midnight Monday 18 May to Linear Systems Revision Exercises on Canvas. After Monday, email before midnight Wednesday 20 May to be guaranteed a response Thursday 21 May. I cannot guarantee that I answer emails sent on Thursday (although of course I will try).

Keep in mind at all times:

“Any and all work, submitted at any time, will receive feedback.”

“Do not hesitate to contact me with questions at any time. My usual modus operandi is to answer all queries in the morning but sometimes I may respond sooner.”

## Week 13

### Catch Up/Revision of Lab 8 Material

The final assessment, based on Lab 8, takes place 11:00, Tuesday 12 May.

If you have not yet done so, you will undertake the learning described in Week 10. Perhaps you should also look at the theory exercises described in Week 10.

The final assessment will ask you to do the following:

Consider:

$\displaystyle k\cdot \frac{\partial^2 T}{\partial x^2}=\frac{\partial T}{\partial t}$      (1)

For one (i.e. $x_1$ or $x_2$ or $x_3$) or all (i.e. general $x_i$), use equations (2) and (3) to write (1) in the form:

$T^{\ell+1}_i=f(T_j^\ell),$

i.e. find the temperature at node $i$ at time $\ell+1\equiv(\ell+1)\cdot \Delta t$ in terms of the temperatures at the previous time $\ell\equiv \ell\cdot \Delta t$. You may take $\Delta t=0.5$.

\begin{aligned} \left.\frac{dy}{dx}\right|_{x_i}&\approx \frac{y(x_{i+1})-y(x_i)}{\Delta x}\qquad(2) \\ \left.\frac{d^2y}{dx^2}\right|_{x_i}&\approx \frac{y(x_{i+1})-2y(x_i)+y(x_{i-1})}{(\Delta x)^2}\qquad(3) \end{aligned}

so if you want to send on this work for feedback please do so.

If you submitted a Lab 8 either back before 6 April, or after, I will invite you, if necessary, to take on board the feedback I gave to that submission, and resubmit a corrected version for further feedback.

Submit work — VBA or Theory, catch-up or revision — based on Week 10 to the Lab 8 VBA/Theory Catch-up/Revision II assignment on Canvas by midnight 9 May. If you have any further questions, I am answering emails seven days a week. Email before midnight Monday 11 May to be guaranteed a response Tuesday 12 May. I cannot guarantee that I answer emails sent on Tuesday (although of course I will try).

It is my advice to try and find 7 hours per week for MATH6040, and spend that time on it, working your way down through the learning in this announcement.

Keep in mind at all times:

“Any and all work, submitted at any time, will receive feedback.”

“Do not hesitate to contact me with questions at any time. My usual modus operandi is to answer all queries in the morning but sometimes I may respond sooner.”

I recommend that you find (at least) 7 hours per week for MATH7021.

Keep in mind at all times:

“Any and all work, submitted at any time, will receive feedback.”

“Do not hesitate to contact me with questions at any time. My usual modus operandi is to answer all queries in the morning but sometimes I may respond sooner.”

## Week 13 to Sunday 10 May

### Catch Up

ASAP you need to catch up on Week 10, Easter Week 1, and Easter Week 2 (lectures and tutorial) for:

90 minute, 40% Test, 11:00 Monday 11 May 2020

This test will examine Chapter 2, Section 3.6, and Chapter 4. A Summer 2020 paper was written, and this test will comprise of Q. 2, 3 (d), and 4 from that exam. The question comprises 40% of that exam, and that translates to 48 minutes of exam time. I will extend this to 75 minutes and allow 15 minutes to upload.

Here is tutorial video for this test.

Chapter 2 Exercises that you should be looking at include:

• p.86, Q. 1-4
• p.91, Q. 1-7
• p.86, Q. 5-6 (harder)

I have developed the following for you to practise your differentiation which you need for Chapter 2 here:

Any exercises you do can be submitted to Week 13 Exercises by midnight Friday 8 May. If you have any further questions, I am answering emails seven days a week. Email before midnight Sunday 10 May to be guaranteed a response Monday 11 May. I cannot guarantee that I answer emails sent on Monday (although of course I will try).

If possible, submit the images as a single pdf file. To do this, select all the images in a folder, right-click and press print. It will say something like How do you want to print your pictures? Press (Microsoft?) Print to PDF. If possible choose an orientation that has all the images in portrait.

## Week 14 to 17 May

The 40% Test on Chapters 2, 4, and Section 3.6 will take place Monday 11 May.

In the form of the Test 1 trust pledge, instructions, and tables, practical information for Test 1 may be found here.

After this you will be invited to do revision on Linear Systems. Here is video based on Q. 1 from the Summer 2019 paper on the back of your manual.

## Week 15 to 24 May

The 20% Linear Systems Test will take place Thursday 21 May.

90 minute, 20% Linear Systems Test, 11:00 Thursday 21 May 2020

This test will re-examine Chapter 1. A Summer 2020 paper was written, and this test will comprise of Q. 1 from that exam. The question comprises 35% of that exam, and that translates to 42 minutes of exam time. I will extend this to 75 minutes and allow 15 minutes to upload.

The videos, here, comprise me going through a full Leaving Cert Higher Level Mathematics Paper, namely 2019, Paper 1.

They’re neither slick, perfect, nor as good as I would like them to be, but I am prepared to give some time every week to answering HL LC Maths student questions.

The videos are labelled in the descriptions, so if you are looking for, say, Q. 5 you can flick through the videos until you find the question you are looking for (e.g. Q. 5 starts at 17.05 here).

All students are looking for help: but perhaps the student I am best placed to help is a student (eventually) going for a H1 who needs something to be explained in more depth, or to give the thought process behind attacking a more challenging problem.

Students can ask me questions on whatever platform they want and I will try and address them. If I get no questions I will just tip away at these exam papers.

A colleague writes (extract):

I have an assessment with 4 sections in it A,B,C and D.

I have a question bank for each section. The number of questions in each bank is A-10, B-10, C-5, D-5.

In my assessment I will print out randomly a fixed number of questions from each bank. Section A will have 5 questions, B-5, C-2, D-2. 14 questions in total appear on the exam.

I can figure out how many different exam papers (order doesn’t matter) can be generated (I think!).

$\displaystyle \binom{10}{5}\cdot \binom{10}{5}\cdot \binom{5}{2}\cdot \binom{5}{2}=6350400$

But my question is: what is the uniqueness of each exam, or what overlap between exams can be expected.?

I am not trying to get unique exams for everyone (unique as in no identical questions) but would kinda like to know what is the overlap.

Following the same argument as here we can establish that:

## Fact 1

The expected number of students to share an exam is $\approx 0.00006$.

Let the number of exams $\alpha:=6350400$.

This is an approach that takes advantage of the fact that expectation is linear, and the probability of an event $E$ not happening is

$\displaystyle\mathbb{P}[\text{not-}E]=1-\mathbb{P}[E]$.

Label the 20 students by $i=1,\dots,20$ and define a random variable $S_i$ by

$\displaystyle S_i=\left\{\begin{array}{cc}1&\text{ if student i has the same exam as someone elese} \\ 0 & \text{ if student i has a unique exam}\end{array}\right.$

Then $X$, the number of students who share an exam, is given by:

$\displaystyle X=S_1+S_2+\cdots+S_{20}$,

and we can calculate, using the linearity of expectation.

$\mathbb{E}[X]=\mathbb{E}[S_1]+\cdots \mathbb{E}[S_{20}]$.

The $S_i$ are not independent but the linearity of expectation holds even when the addend random variables are not independent… and each of the $S_i$ has the same expectation. Let $p$ be the probability that student $i$ does not share an exam with anyone else; then

$\displaystyle\mathbb{E}[S_i]=0\times\mathbb{P}[S_i=0]+1\times \mathbb{P}[S_i=1]$,

but $\displaystyle\mathbb{P}[S_i=0]=\mathbb{P}[\text{ student i does not share an exam}]=p$, and

$\displaystyle \mathbb{P}[S_i=1]=\mathbb{P}[\text{not-}(S_i=0)]=1-\mathbb{P}[S_i=0]=1-p$,

and so

$\displaystyle\mathbb{E}[S_i]=1-p$.

All of the 20 $S_i$ have this same expectation and so

$\displaystyle\mathbb{E}[X]=20\cdot (1-p)$.

Now, what is the probability that nobody shares student $i$‘s exam?

We need students $1\rightarrow i-1$ and $i+1\rightarrow 20$ — 19 students — to have different exams to student $i$, and for each there is $\alpha-1$ ways of this happening, and we do have independence here (student 1 not sharing student $i$‘s exam doesn’t change the probability of student 2 not sharing student $i$‘s exam), and so $\mathbb{P}[\text{(student j not sharing) AND (student k not sharing)}]$ is the product of the probabilities.

So we have that

$\displaystyle p=\left(\frac{\alpha-1}{\alpha}\right)^{19}$,

and so the answer to the question is:

$\displaystyle\mathbb{E}[X]=20\cdot \left(1-\left(\frac{\alpha-1}{\alpha}\right)^{19}\right)\approx 0.00005985\approx 0.00006$.

We can get an estimate for the probability that two or more students share an exam using Markov’s Inequality:

$\displaystyle\mathbb{P}[X\geq 2]\leq \frac{\mathbb{E}[X]}{2}\approx 0.00003=0.003\%$

## Fact 2

This estimate is tight: the probability that two or more students (out of 20) share an exam is about 0.003%.

This tallies very well with the exact probability which can be found using a standard Birthday Problem argument (see the solution to Q. 7 here) to be:

$\mathbb{P}[X\geq 2]\approx 0.0000299191\approx 0.003\%$

The probability that two given students share an exam is $1/\alpha\approx 0.00001575\%$

## Fact 3

The expected number of shared questions between two students is 6.6

Take students 1 and 2. The questions are in four bins: two of ten, two of five. Let $B_i$ be the number of questions in bin $i$ that students 1 and 2 share. The expected number of shared questions, $Q$, is:

$\displaystyle \mathbb{E}[Q]=\sum_{i=1}^4\mathbb{E}[B_i]$,

and the numbers are small enough to calculate the probabilities exactly using the hypergeometric distribution.

The calculations for bins 1 and 2, and bins 3 and 4 are the same. The expectation

$\displaystyle\mathbb{E}[B_1]=\sum_{j=0}^5j\mathbb{P}[B_1=j]$.

Writing briefly $p_j=\mathbb{P}[B_1=j]$, looking at the referenced hypergeometric distribution we find:

$\displaystyle p_j=\frac{\binom{5}{j}\binom{5}{5-j}}{\binom{10}{5}}$

and we find:

$\displaystyle\mathbb{E}[B_1]=\mathbb{E}[B_2]=\frac52$

Similarly we see that

$\displaystyle\mathbb{E}[B_3]=\mathbb{E}[B_4]=\frac{4}{5}$

and so, using linearity:

$\displaystyle\mathbb{E}[Q]=\frac52+\frac52+\frac45+\frac45=6.6$

This suggests that on average students share about 50% of the question paper. Markov’s Inequality gives:

$\displaystyle\mathbb{P}[Q\geq 7]\underset{\approx}{\leq} 0.9429$,

but I do not believe this is tight.

Calculating this probability exactly is tricky because there are many different ways that students can share a certain number of questions. We would be looking at something like “multiple hypergeometric”, and I would calculate it as the event not-(0 or 1 or 2 or 3 or 4 or 5 or 6).

I think the $\mathbb{E}[Q]=6.6$ result is striking enough at this time!

In May 2017, shortly after completing my PhD and giving a talk on it at a conference in Seoul, I wrote a post describing the outlook for my research.

I can go through that post paragraph-by-paragraph and thankfully most of the issues have been ironed out. In May 2018 I visited Adam Skalski at IMPAN and on that visit I developed a new example (4.2) of a random walk (with trivial $n$-dependence) on the Sekine quantum groups $Y_n$ with upper and lower bounds sharp enough to prove the non-existence of the cutoff phenomenon. The question of developing a walk on $Y_n$ showing cutoff… I now think this is unlikely considering the study of Isabelle Baraquin and my intuitions about the ‘growth’ of $Y_n$ (perhaps if cutoff doesn’t arise in somewhat ‘natural’ examples best not try and force the issue?). With the help of Amaury Freslon, I was able to improve to presentation of the walk (Ex 4.1) on the dual quantum group $\widehat{S_n}$. With the help of others, it was seen that the quantum total variation distance is equal to the projection distance (Prop. 2.1). Thankfully I have recently proved the Ergodic Theorem for Random Walks on Finite Quantum Groups. This did involve a study of subgroups (and quasi-subgroups) of quantum groups but normal subgroups of quantum groups did not play so much of a role as I expected. Amaury Freslon extended the upper bound lemma to compact Kac algebras. Finally I put the PhD on the arXiv and also wrote a paper based on it.

Many of these questions, other questions in the PhD, as well as other questions that arose around the time I visited Seoul (e.g. what about random transpositions in $S_n^+$?) were answered by Amaury Freslon in this paper. Following an email conversation with Amaury, and some communication with Uwe Franz, I was able to write another post outlining the state of play.

This put some of the problems I had been considering into the categories of Solved, to be Improved, More Questions, and Further Work. Most of these have now been addressed. That February 2018 post gave some direction, led me to visit Adam, and I got my first paper published.

After that paper, my interest turned to the problem of the Ergodic Theorem, and in May I visited Uwe in Besancon, where I gave a talk outlining some problems that I wanted to solve. The main focus was on proving this Ergodic Theorem for Finite Quantum Groups, and thankfully that has been achieved.

What I am currently doing is learning my compact quantum groups. This work is progressing (albeit slowly), and the focus is on delivering a series of classes on the topic to the functional analysts in the UCC School of Mathematical Sciences. The best way to learn, of course, is to teach. This of course isn’t new, so here I list some problems I might look at in short to medium term. Some of the following require me to know my compact quantum groups, and even non-Kac quantum groups, so this study is not at all futile in terms of furthering my own study.

I don’t really know where to start. Perhaps I should focus on learning my compact quantum groups for a number of months before tackling these in this order?

1. 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.
2. Look at random walks on quantum homogeneous spaces, possibly using Gelfand Pair theory. Start in finite and move into Kac?
3. Following Urban, study convolution factorisations of the Haar state.
4. Examples of non-central random walks on compact groups.
5. 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)