J.P. McCarthy: Math Page

MATH7021: Spring 2024, Week 5

February 22, 2024 in MATH7021 | Leave a comment

Week 5

We completed Chapter 2.

Week 6

On Monday we will have some tutorial time on Chapter 2.

On Wednesday we will look at The Engineer’s Transform” — the Laplace Transform.

Things are more serious now as we start Chapter 3. We must work hard on this material:

because without doing so you could be very, very lost on 15% Assignment 2 (and 35% exam question), and
because if you are going into Level 8 Structural Engineering it will be assumed that you are competent with the Chapter 3 material

Week 7

On Monday, we may or may not have a final Chapter 2 tutorial. It depends on how much Chapter 3 we get done.

Then the rest of the class will be given over to partial fractions and then the inverse Laplace transform.

Week 8

On Monday, we are due to have a double tutorial.

The rest of the week will be focused on finishing Chapter 3.

Week 9

No Monday class with the bank holiday

This should be a bumper week of Chapter 3 tutorials if we have finished Chapter 3. It will be in this week that we catch up as we will skip Section 3.6.

The below is very much provisional

Week 10

We will do systems of differential equations.

Week 11

We will work on double integration and when we finish that section we will have tutorials.

Week 12

We will work on triple integration. When finished we will have tutorial time on that topic.

Week 13, Review Week

We will go through an exam paper, answer questions, and have tutorial time as appropriate.

Study

Please feel free to ask me questions about the exercises via email or on this webpage.

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..

Study

Please feel free to ask me questions about the exercises via email or even better on this webpage.

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..

MATH7016: Spring 2024, Week 5

February 22, 2024 in MATH7016 | Leave a comment

20% VBA 1 Assessment, Week of 26 February to 1 March

Please see Canvas for more information.

20% Written Assessment 1, 6 March

Please see Canvas for more information.

Week 5

Lots of case studies of second order differential equations. We will also introduce methods even more accurate than Heun’s Method: the general Runge–Kutta methods.

In VBA we worked towards Heun’s Method for second order differential equations.

Week 6

We will look at the Felix Baumgartner situation before moving onto Boundary Value Problems.

In VBA you will have your assessment.

Week 7

We will continue looking at boundary value problems (in particular the Shooting Method and Goal Seek) before you have your written assessment on Wednesday.

In VBA we will finish what we are doing on Lab 4 and then look at Lab 5.

Week 8

Finish Section 1.10 and start Chapter 2.

Lab 6

Week 9

Section 2.1

Lab 7

Because every lab counts, the Bank Holiday labs of DME2A are going to be rescheduled to Tuesday: on Tuesday March 19 we will have the following:

DME2A, 14:40-16:20, DME2E 16:20-18:00, A285

Here on is provisional:

Week 10

Section 2.2

Lab 8 (material examinable in final assessment)

Week 11

Finish Section 2.2 and then tutorial time

Assessment

See Canvas for the assessment schedule.

Study

Study should consist of

doing exercises from the notes
completing VBA exercises

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..

A strange property of quantum permutations

February 13, 2024 in Probability Theory, Quantum Groups, Research | Leave a comment

Classical Warm-up

Let $S_N$ be the classical permutation group on $N$ symbols. The algebra of continuous functions $C(S_N)$ on $S_N$ is generated by indicator functions:

$\displaystyle \mathbf{1}_{j\to i}(\sigma)=\delta_{i,\sigma(j)}$ .

This algebra is commutative, but we will use some non-commutative algebraic analogues. Let $\widetilde{\pi}\in M_p(S_N)$ be the uniform probability distribution on $S_N$ and by $\pi$ the associated state on $C(S_N)$ :

$\displaystyle\pi(f)=\sum_{t\in G}f(t)\,\widetilde{\pi}(\{t\})=\frac{1}{|S_N|}\sum_{t\in G} f(t)\qquad (f\in C(S_N))$

When choosing a permutation at random what is the probability that it sends $1\to 1$ ? Well, this will equal, in some notation we won’t fully explain here:

$\displaystyle \mathbb{P}[\pi(1)=1]=\pi(\mathbf{1}_{1\to 1})=\frac{1}{N}$ .

Once this has been observed, there is a conditioning of the state $\pi\mapsto \pi_{1\to 1}$ :

$\displaystyle \pi_{1\to 1}(f)=\dfrac{\pi(\mathbf{1}_{1\to 1}f\mathbf{1}_{1\to 1} )}{\pi(\mathbf{1}_{1\to 1})}=N \pi(\mathbf{1}_{1\to 1}f\mathbf{1}_{1\to 1} )) \qquad (f\in C(S_N))$ .

The state was given by $\pi$ but is now given by $\pi_{1\to 1}$ . Now, after having observed this random permutation mapping $1\to 1$ , we can ask it another question.

Consider a subset $S\subseteq \{3,\dots,N\}$ and define:

$\displaystyle\mathbf{1}_{3\to S}:=\sum_{s\in S}\mathbf{1}_{3\to s}$ .

This is an observable, which basically asks of a permutation, do you map three into $S$ ? One for yes, zero for no. So let us ask this of $\pi_{1\to 1}$ :

What is the probability that a random permutation maps three into $S$ after having been observed mapping one to one?

We find, with a fairly elementary calculation that the answer to this question is:

$\displaystyle \mathbb{P}[\pi_{1\to 1}(3)\in S]=\frac{|S|}{N-1}$ .

If $\pi_{1\to 1}(3)\in S$ , we get further conditioning, to let us say $\nu$ :

$\displaystyle \nu(f)=\frac{N(N-1)}{|S|}\pi(\mathbf{1}_{1\to 1}\mathbf{1}_{3\to S}f\mathbf{1}_{3\to S}\mathbf{1}_{1\to 1})\qquad (f\in C(S_N))$

What is the probability that after seeing $\pi_{1\to 1}(3)\in S$ after $\pi(1)=1$ we see $\nu(2)=2$ ? We find it is:

$\displaystyle \mathbb{P}[\nu(2)=2]=\frac{1}{N-2}$ .

Therefore, where $\succ$ means after, and the state-conditioning implicit

$\displaystyle \mathbb{P}[(\pi(2)=2)\succ (\pi(3)\in S)\succ(\pi(1)=1)]$

$\displaystyle =\frac{1}{N}\cdot \frac{|S|}{N-1}\cdot\frac{1}{N-2}=\frac{|S|}{N(N-1)(N-2)}$

Now let us ask a ridiculous question. What was the probability that $\nu(1)=2$ ? But $\nu$ is just a conditioning of $\pi_{1\to 1}$ , which mapped one to one with probability one:

$\displaystyle \mathbb{P}[\pi_{1\to 1}(1)=1]=\dfrac{\pi(\mathbf{1}_{1\to 1}\mathbf{1}_{1\to 1}\mathbf{1}_{1\to 1})}{\pi(\mathbf{1}_{1\to 1})}=1$ .

Of course the answer is zero. Can we actually see it in the above framework: well, yes, because the algebra of functions is commutative. You end up with evaluating a state at

$\displaystyle \mathbf{1}_{1\to 1}\mathbf{1}_{3\to S}\mathbf{1}_{1\to 2}\mathbf{1}_{3\to S}\mathbf{1}_{1\to 1}$ ,

but commutativity sees $\mathbf{1}_{1\to 1}\mathbf{1}_{1\to 2}$ … and no permutation maps one to one and one to two (you’d need a relation to do that), and so $\mathbf{1}_{1\to 1}\mathbf{1}_{1\to 2}=0$ , so is the length five monomial above, and so is the probability we spoke about.

We don’t have to do state conditioning and multiplication to calculate the probability that a random permutation maps one to one, then maps three into $S$ , then maps two to two though. Where $|f|^2=f^*f$ :

$\displaystyle \mathbb{P}[(\pi(2)=2)\succ (\pi(3)\in S)\succ(\pi(1)=1)]$

$=\pi(|\mathbf{1}_{2\to 2}\mathbf{1}_{3\to S}\mathbf{1}_{1\to 1}|^2)=\dfrac{|S|}{N(N-1)(N-2)}$ .

Let us remark that this probability is increasing in $|S|\in\{0,1,\dots,N-2\}$ : the less we specify about the event, in this case represented by the projection $\mathbf{1}_{3\to S}$ , the more general it is, the larger the probability.

Let’s Go Quantum

In the case of quantum permutations, so talking $S_N^+$ the ridiculous question of asking what is the probability that a quantum permutation maps one to two after it had previously mapped one to one… is no longer zero. Where $q(N)=N(N-1)(N^2-4N+2)$ , the probability of the analogue of a “random” quantum permutation, the Haar state $h$ doing this is:

$\displaystyle \mathbb{P}[(h(1)=2)\succ(h(3)=3)\succ (h(1=1))]=\frac{N-3}{q(N)}$

Anyway, the quantum versions of the $\mathbf{1}_{j\to i}$ are entries $u_{ij}\in M_N(C(S_N^+))$ of a universal magic unitary. We say:

$\mathbb{P}[(h(2)=2)\succ (h(3)\in S)\succ (h(1)=1)]:=h(|u_{22}u_{S,3}u_{11}|^2)$ ,

where $u_{S,3}=\sum_{s\in S}u_{s3}$ . We find this probability is:

$\displaystyle \mathbb{P}[(h(2)=2)\succ (h(3)\in S)\succ (h(1)=1)]=\frac{|S|(N^2-5N+5+|S|)}{(N-2)q(N)}$ .

This is also increasing in $|S|$ . Again, the less specificity about the event, the greater the probability.

This quantity is related to the classical probability above, and there are asymptotic similarities. Writing $|S|=N-2-|S^c|$ , where $S^c$ is the complement of $S$ in $\{3,4,\dots,N\}$ :

$\displaystyle \mathbb{P}[(h(2)=2)\succ (h(3)\in S)\succ (h(1)=1)]$

$\displaystyle=\mathbb{P}[(\pi(2)=2)\succ (\pi(3)\in S)\succ(\pi(1)=1)]\cdot\left[1-\frac{|S^c|+1}{q(N)}\right]$

I guess this also says, the larger $|S|$ , the more similar the classical and quantum probabilities.

A twist

What if instead of looking at $h(2)=2$ after $h(3)\in S$ after $h(1)=1$ but instead we asked about $h(1)=2$ ? This should be non-zero, but what is the dependence on $|S|$ ?

From our classical intuition, we would probably just expect, well, the same really. The larger the value of $|S|$ , the less we are saying about the event. The larger the probability. But in fact this is not the case. The probability is largest when $|S|$ is close to $(N-2)/2$ . We will write down the probability, and then explain, mathematically, why the symmetry with respect to $|S|\leftrightarrow N-2-|S|$ , with respect to $S\leftrightarrow S^c$ :

$\displaystyle \mathbb{P}[(h(1)=2)\succ (h(3)\in S)\succ (h(1)=1)]=\dfrac{|S|(N-2-|S|)}{q(N)}$ .

Proof: Consider $u_{21}u_{11}=0$ . Insert between them $1_{C(S_N^+)}=\sum_{k=1}^Nu_{k3}$ :

$\displaystyle u_{21}\sum_{k=1}^Nu_{k3} u_{11}=\sum_{k=1}u_{21}u_{k3}u_{11}=0$ .

Like in the classical case, by definition here, $u_{21}u_{23}=0=u_{13}u_{11}=0$ . So split into two sums:

$\displaystyle \sum_{s\in S}u_{21}u_{i3}u_{11}+\sum_{j\in S^c}u_{21}u_{j3}u_{11}=0$ .

Now, classically, this is just a sum of zero terms equal to zero, but not in the quantum world where we have these $u_{21}u_{j3}u_{11}\neq 0$ . They are not positive though. We can now split:

$\displaystyle \sum_{s\in S}u_{21}u_{i3}u_{11}=-\sum_{j\in S^c}u_{21}u_{j3}u_{11}$ .

Now take the $|\cdot|^2$ of both sides to show that:

$\displaystyle |u_{21}u_{S,3}u_{11}|^2=|u_{21}u_{S^c,3}u_{11}|^2$

This yields the strange probability above. I think any intuition I have for this would be very much post-hoc. I think I will just let it hang there as something weird, cool and mysterious about quantum permutations…

MATH7016: Spring 2024, Week 4

February 12, 2024 in MATH7016 | Leave a comment

20% VBA 1 Assessment, Week of 26 February to 1 March

Please see Canvas for more information.

Week 4

We did more on Heun’s Method, and also looked at “Taylor Methods”.

We also introduced second order differential equations and how to attack them numerically.

In VBA we look at Lab 3, on Heun’s Method. If you have not completed this lab you are strongly urged to do so as the 20% VBA 1 Assessment will be based on this lab. There are lab videos on Canvas to help you outside the lab.

Read the rest of this entry »

MATH7021: Spring 2024, Week 5

February 12, 2024 in MATH7021 | Leave a comment

15% Assignment 1

This should be started ASAP. This has a hand in time and date of 10:45 Thursday 22 February. Please see Canvas.

Gaussian Elimination Tutor

If you download Maple (see Student Resources), there is a Maple Tutor that is easy to use and will help you with Gaussian Elimination. Open up Maple and go to Tools -> Tutors -> Linear Algebra -> Gaussian Elimination.

Week 5

We are now going to push to complete Chapter 2 as soon as reasonable.

Study

Please feel free to ask me questions about the exercises via email or even better on this webpage.

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..

An interesting addend to Tracing the orbitals of the quantum permutation group

February 6, 2024 in C*-Algebras & Operator Theory, Quantum Groups, Research | Leave a comment

This paper referenced in the title (preprint here) tries to establish some very basic properties of a so-called exotic quantum permutation group, one intermediate to the classical and quantum permutation groups:

$\displaystyle S_N\subsetneq \mathbb{G}\subsetneq S_N^+$ .

If a compact matrix quantum subgroup $\mathbb{H}\subseteq \mathbb{G}$ (here $\mathbb{G}$ is generic, not exotic) given by a quotient $\pi:C(\mathbb{G})\to C(\mathbb{H})$ , is such that for all monomials $f$ in the generators $u_{ij}\in C(\mathbb{G})$ :

$\displaystyle h_{C(\mathbb{H})}(\pi(f))=h_{C(\mathbb{G})}(f)$ ,

then in fact $\mathbb{H}=\mathbb{G}$ . In the paper above I explored calculating the Haar state on $C(S_N^+)$ using invariances related to the inclusion $S_N\subseteq S_N^+$ … the point being that the Haar measure on exotic $\mathbb{G}$ shares these invariances too… and perhaps, very speculatively, we could show that these Haar states coincide on such monomials… and a famous open problem in the theory would be solved.

The paper above fails at generators of length four though. The invariances do not determine the Haar measure on $S_N^+$ and we have to include a representation theory result around the law of the main character to give the explicit formulae. If we leave out the representation theory input, we have the Haar state up to a parameter.

However, a very basic algebraic result, that a product of three generators in exotic $C(\mathbb{G})$ can be zero for trivial reasons only, allows us to put bounds on the Haar state. If these bounds could force the fourth moment of the main character (with respect to the Haar state) to equal 14, then the Haar state on length four monomials would be equal for both. Alas… they only bound them less than or equal to 15. At this point the paper gives up… but this moment is a whole number, either 14 or 15… so I added the relation that the fourth moment is 15… and what comes out is that the parameter above is zero… but when this parameter is zero we find, in exotic $\mathbb{G}$ :

$\displaystyle h(|u_{22}u_{11}u_{23}|^2)=0$ ,

but this is not the case, as $u_{22}u_{11}u_{23}$ is not zero for trivial reasons, and the Haar state is faithful… the fourth moment equal to 15 gives classical $S_N\subseteq S_N^+$ . Therefore the fourth moment for exotic $\mathbb{G}\subset S_N^+$ is forced equal to 14, giving the same explicit formula for length four generators as for $S_N^+$ . Further work needed here.

A note on universal C*-algebras

February 2, 2024 in C*-Algebras & Operator Theory, Quantum Groups, Research | Leave a comment

Idea and Intuition

Let $\mathcal{G}=\{g_i\}_{i\geq 1}$ be a (usually finite) set of generators and $\mathcal{R}$ a (usually finite) set of relations between the generators. The generators at this point are indeterminates, and we will be momentarily vague about what is and isn’t a relation. We write (if it exists!) $\mathrm{C}^*(\mathcal{G}\mid \mathcal{R})$ for the universal $\mathrm{C}^*$ -algebra generated by generators $\mathcal{G}$ and relations $\mathcal{R}$ .

It has the following universal property. Suppose $|\mathcal{G}|=n$ . Let $A$ be a $\mathrm{C}^*$ -algebra with elements $f_1,\dots,f_n$ that satisfy the relations $\mathcal{R}$ , then there is a (unique) *-homomorphism $\mathrm{C}^*(\mathcal{G}\mid\mathcal{R})\to A$ mapping $g_i\mapsto f_i$ . This map will be a surjective *-homomorphism $\mathrm{C}^*(\mathcal{G}\mid\mathcal{R})\to \mathrm{C}^*(f_1,\dots,f_n)$ (aka a quotient map and $\mathrm{C}^*(f_1,\dots,f_n)$ a quotient of $\mathrm{C}^*(\mathcal{G}\mid\mathcal{R}))$ .

If the $f_i\in A$ generate $A$ , $\mathrm{C}^*(f_1,\dots,f_n)$ , then $\mathrm{C}^*(\mathcal{G}\mid\mathcal{R})\to A$ is a surjective *-homomorphism.

My (highly non-rigourous, the tilde reminding of hand-waving) intuition for this object is that you collect all (really all, not just isomorphism classes, we want below $C([0,1])$ and e.g. $C([0,2]$ ) of $\mathrm{C}^*$ -algebras $A_\lambda$ generated by $n$ generators satisfying the relations $\mathcal{R}$ and forming a “big direct sum/product thingy” out of all of them:

$\displaystyle \mathrm{C}^*(\mathcal{G}\mid\mathcal{R})\sim \widetilde{\prod_{\lambda\in \Lambda}}A_\lambda$ .

Then the *-homomorphism $\pi_\mu:A\to A_\mu$ given by the universal property is given by projection onto that factor (which is a surjective *-homomorphism, a quotient):

$\displaystyle\pi_\mu:\widetilde{\prod_{\lambda\in \Lambda}}A_\lambda\to A_\mu$ .

This intuition works well but we should give a brief account of things are done properly.

First off, not every relation will give a universal $\mathrm{C}^*$ -algebra. For example, consider $\mathcal{G}=\{g\}$ and $\mathcal{R}=\{g=g^*\}$ . The problem here is one of norm. Recall our rough intuition from above. What is the norm on $a\in \mathrm{C}^*(\mathcal{G}\mid\mathcal{R})$ , this big thing $\widetilde{\prod_{\lambda\in\Lambda}}A_\lambda$ ? It should be something like, where the norm of $A_\lambda$ is $\|\cdot\|_\lambda$ the supremum over the factors:

$\displaystyle \|f\|\sim\sup_{\lambda\in \Lambda}\|\pi_{\lambda}(f)\|_{\lambda}\qquad (f\in \mathrm{C}^*(\mathcal{G}\mid\mathcal{R})).$

The first approach to show that $\mathrm{C}^*(g\mid g=g^*)$ does not exist is to consider for each $\alpha\in [0,\infty)$ the $\mathrm{C}^*$ -algebra $A_\alpha=C([0,\alpha])$ which is singly-generated by the self-adjoint $f_\alpha:=t\mapsto t$ . But the norm $\|\cdot \|_\alpha$ in $A_\alpha$ is the supremum norm, so we find $\|f_\alpha\|_{\alpha}=\alpha$ . From here:

$\displaystyle \|g\|\sim\sup_{\lambda\in \Lambda}\|\pi_{\lambda}(g)\|_{\lambda}\geq \sup_{\alpha\in [0,\infty)}\|f_{\alpha}\|_{\alpha}=\sup_{\alpha\in [0,\infty)}\alpha,$

which is unbounded. The relations must give a bounded norm to the generators.

As an example of relations that do bound the generators, consider, $n$ self-adjoint generators such that the sum of their squares is the identity:

$\displaystyle \mathrm{C}^*\left(g_1,\dots,g_n\mid g_i=g_i^*,\,\sum_i g_i^2=1\right)$ .

These relations bounds the norm of the generators $\|g_i\|\leq 1$ ¹, and this gives existence to this algebra, using the Gelfand philosophy giving the “algebra of continuous functions on the free sphere”, $C(\mathcal{S}^{n-1}_+)$ (I think first considered by Banica and Goswami).

Note here the relations are given by polynomial relations. If the polynomial relations are suitably admissible (i.e. give a bound to the generators), in this setting there is a real construction (real vs our ridiculous $\widetilde{\prod}$ ) of $\mathrm{C}^*(\mathcal{G}\mid\mathcal{R})$ . See p.885 (link to *.pdf quantum group lecture notes of Moritz Weber).

In fact, this is only a small class of the possible relations. I suggest there are at least two more types:

$\mathrm{C}^*$ relations that would be (admissible) norm relations (for example, in one generator, adding $\|g\|\leq 1$ , a non-polynomial relation, to the polynomial relation $g=g^*$ gives an admissible set of relations, and $\mathrm{C}^*(g\mid g=g^*,\|g\|\leq 1)\cong C([-1,1])$ . For $\mathrm{C}^*$ /norm relations see here and maybe here.
(admissible) strong relations (see here for a use of this, with reference)

The constructions in one or both of cases might be constructive, as in the case (admissible) polynomial relations, but there is also an approach using category theory. But the main feature in all such definitions is the universal property, whose use could be summarised as follows:

Let $\mathrm{C}^*(\mathcal{G}\mid\mathcal{R})$ be a universal $\mathrm{C}^*$ -algebra. The universal property can be used to answer questions about $\mathrm{C}^*(\mathcal{G}\mid\mathcal{R})$ such as:

is some polynomial $p(g_1,\dots,g_n)$ in the generators non-zero,

is $\mathrm{C}^*(\mathcal{G}\mid\mathcal{R})$ infinite dimensional,

is $\mathrm{C}^*(\mathcal{G}\mid\mathcal{R})$ non-commutative;

because, if $A$ is a $\mathrm{C}^*$ -algebra with elements $f_1,\dots,f_n$ that satisfy the relations then there is a unique *-homomorphism $\pi:\mathrm{C}^*(\mathcal{G}\mid\mathcal{R})\to \mathrm{C}^*(f_1,\dots,f_n)$ . So, for example, where $B$ is some such $\mathrm{C}^*(f_1,\dots,f_n)$ then

if $p(f_1,\dots,f_n)\neq 0$ , then $p(g_1,\dots,g_n)\neq 0$ as $\pi(p(g_1,\dots,g_n))=p(f_1,\dots,f_n)$ ,

if $B$ is infinite dimensional then $\pi$ is a surjective *-homomorphism onto an infinite dimensional algebra, and so the domain $\mathrm{C}^*(\mathcal{G}\mid\mathcal{R})$ is infinite dimensional too.

if the commutator $[f_i,f_j]\neq 0$ , so that $B$ is non-commutative, then so is $\mathrm{C}^*(\mathcal{G}\mid\mathcal{R})$ as $\pi([g_i,g_j])=[f_i,f_j]$

These quotients $B$ can be considered models of $\mathrm{C}^*(\mathcal{G}\mid\mathcal{R})$ .

Two Examples

A projection $p$ in a $\mathrm{C}^*$ -algebra is such that $p=p^2=p^*$ . Consider

$A:=\mathrm{C}^*(p,q,1\mid p\text{ and }q\text{ are projections})$ .

Existence is easy, because the norm of a non-zero projection is one. To answer questions about this algebra consider the infinite dihedral group $D_\infty=\langle a,b\mid a^2=b^2=e\rangle$ . This has group algebra $\mathbb{C}D_\infty$ and group $\mathrm{C}^*$ -algebra $\mathrm{C}^*(D_\infty)$ . Note that $a$ and $b$ in $\mathrm{C}^*(D_\infty)$ satisfy the relations of $A$ , and so we have a *-homomorphism (in fact a *-isomorphism) $\pi:A\to \mathrm{C}^*(D_\infty)$ . This tells us that any monomial in the generators of $A$ is non-zero, $A$ is infinite dimensional, and $A$ is non-commutative.

A partition of unity is a finite set of projections that sum to the identity, $\sum_{i}p_i=1$ . A magic unitary in a $\mathrm{C}^*$ -algebra $A$ is a matrix $u\in M_N(A)$ such that the entries along any one row or column form a partition of unity. Consider (notation to be kept mysterious):

$\displaystyle C(S_N^+)=\mathrm{C}^*\left(u_{ij}\mid u\text{ an }N\times N\text{ magic unitary}\right)$ .

Consider the following magic unitary:

$\displaystyle v=\begin{bmatrix}p & 1-p & 0 & 0 \\ 1-p & p & 0 & 0 \\ 0 & 0 & q & 1-q \\ 0 & 0 & 1-q & q\end{bmatrix}$ .

Note that the $v_{ij}$ satisfy the relations of $C(S_N^+)$ and in fact generate $A=\mathrm{C}^*(p,q)$ from above. Thus we have a quotient $\pi:C(S_4^+)\to A$ which shows that $C(S_4^+)$ is infinite dimensional and noncommutative.

It is possible to show using a magic unitary with entries in $M_N(\mathbb{C})$ that for $N>3$ , a monomial with entries in $u\in M_N(C(S_N^+))$ is zero for trivial reasons only (link to *.pdf, from Theorem 1 on).

In addition it can be shown that for $u$ (and similarly $v$ ) the matrix in $M_N(C(S_N^+)\otimes_{\min} C(S_N^+))$ with $(i,j)$ – entries

$\displaystyle \sum_{k=1}^Nu_{ik}\otimes u_{kj}$

is a magic unitary, and thus by the universal property $u_{ij}\mapsto \sum_{k=1}^Nu_{ik}\otimes u_{kj}$ is a *-homomorphism… the comultiplication giving $C(S_N^+)$ the structure of a compact quantum group.

Commutative Examples

If a universal $\mathrm{C}^*$ algebra is commutative (as in commutativity, $[g_i,g_j]=0$ , is one of the relations, vs the relations imply commutativity, as in $C(S_3^+)$ (nice exercise)), we write $\mathrm{C}^*_{\text{comm}}(\mathrm{G}\mid \mathrm{R})$ . In this case Gelfand’s Theorem, that $\mathrm{C}^*_{\text{comm}}(\mathrm{G}\mid \mathrm{R})\cong C(\text{characters})$ , often allows us to easily identity the algebra (vs the noncommutative case where the universal algebra is mostly studied via models, quotients).

Theorem

If $A$ is a (polynomial) universal commutative $\mathrm{C}^*$ -algebra, then it of the form $C(X)$ , and $X$ is given by the tuples $(z_1,\dots,z_n)\subset\mathbb{C}^n$ that satisfy the relations of of $A$ .

Proof: Characters are *-homomorphisms $A\to \mathbb{C}$ .

Suppose that $z_1,\dots,z_n$ satisfy the relations. Then by the universal property, $\varphi(g_i)=z_i$ is a *-homomorphism.

On the contrary, suppose that $\varphi$ is a character. Then the relations are preserved under a *-homomorphism.

Examples

For $A_0:=\mathrm{C}_{\text{comm}}^*(p,q,1\mid p\text{ and }q\text{ are projections})$ , projections in $\mathbb{C}$ are just the scalars zero and one. Thus the spectrum is $\{(0,0),(1,0),(0,1),(1,1)\}$ and $A_0$ is the algebra of continuous functions on four points.

For $\mathrm{C}_{\text{comm}}^*\left(u_{ij}\mid u\text{ an }N\times N\text{ magic unitary}\right)$ collect the tuple of $N^2$ generators in a matrix. The relations imply that each such tuple is in fact a permutation matrix, and so the universal algebra above is the algebra of continuous functions on $S_N$ .

For

$\displaystyle \mathrm{C}_{\text{comm}}^*\left(g_1,\dots,g_n\mid g_i=g_i^*,\,\sum_i g_i^2=1\right)$

you end up with tuples of $n$ real numbers in $[-1,1]$ whose sum of squares is one… otherwise known as the sphere $\mathcal{S}^{n-1}$ .

Liberations

An interesting business here is to start with a universal commutative $\mathrm{C}^*$ algebra, say one of the three examples above… and see do you get something strictly bigger, necessarily non-commutative, if you drop commutativity. In the above, yes you do. Gelfand’s theorem says that a commutative unital $\mathrm{C}^*$ -algebra is the algebra of continuous functions on a compact space $X$ (which we call a classical space). The Gelfand Philosophy says therefore that a noncommutative unital $\mathrm{C}^*$ -algebra $A$ can be thought of as the algebra of continuous functions on a compact quantum space $\mathbb{X}$ . Note here $\mathbb{X}$ is not a set-of-points, but a virtual object, and strictly $A=C(\mathbb{X})$ is just notation (but see here).

Liberating the second example above from commutativity is the passage from the permutation group $S_N$ to the quantum permutation group $S_N^+$ . Liberating the third example above gives the passage from the real sphere $\mathcal{S}^{n-1}$ to a quantum sphere called the free sphere $\mathcal{S}^{n-1}_+$ .

We can also, of course, work in the other direction, imposing commutativity on not-necessarily universal $\mathcal{A}$ . If we write $\mathcal{A}=C(\mathbb{X})$ , then imposing commutativity (qotienting by commutator ideal) gives us the classical version $X$ of $\mathbb{X}$ .

Imposing commutativity is not so scary: using the above you just get $\mathrm{C}^*(\mathcal{G}\mid\mathcal{R})\to C(\text{characters})$ … and everything we said above about identifying characters on $\mathrm{C}_{\text{comm}}^*(\mathcal{G}\mid\mathcal{R})$ holds for $\mathrm{C}^*(\mathcal{G}\mid\mathcal{R})$ too.

This can be used: for example if a quantum group acts on a structure $S$ , then its classical version acts on $S$ . This idea was used by Banica and I to show that not every quantum permutation group is the quantum automorphism group of a finite graph (link to *.pdf).

If you have positive elements $f_i$ in a $\mathrm{C}^*$ -algebra with bounded sum, say $\|\sum_i f_i\|\leq C$ then we can bound the summands $\|f_i\|\leq C$ . Write $\sum_if_i=f$ . Note $f$ is positive with norm $C$ . Let $\varphi$ be a state such that $\varphi(f_j)=\|f_j\|$ and apply this to $f$ , $\varphi(\sum_i f_i)=\varphi(f)$ which yields, with $\varphi$ bounded of norm one, $\varphi(f_j)+\sum_{i\neq j}\varphi(f_i)\leq C$ . The result follows by positivity of the sum and the state. To apply to the above play with the $\mathrm{C}^*$ identity. Incidentally, I cannot remember how Murphy proves the existence of $\varphi$ but I like for positive $f$ in a $\mathrm{C}^*$ -algebra, $\mathrm{C}^*(f)\cong C(\sigma(f))$ . Then let $x:=\max_{\lambda\in\sigma(f)}\lambda$ and define a state $\mathrm{ev}_x$ on $\mathrm{C}^*(f)\cong C(\sigma(f))$ . It is the case that $\mathrm{ev}_x(f)=f(x)=\|f\|_{C(\sigma(f))}=x$ , also equal to the norm of $f$ in the ambient algebra (by a spectral radius theorem). Extend the evaluation functional to the whole algebra by Hahn-Banach. ↩︎

MATH7016: Spring 2024, Week 3

February 1, 2024 in MATH7016 | Leave a comment

DME2A and DME2E in Bank Holiday Weeks

Because every lab counts, the two Bank Holiday labs of DME2A are going to be rescheduled to Tuesday and on those two dates Tuesdays February 6 and March 19 we will have the following:

DME2A, 14:40-16:20, DME2E 16:20-18:00, A285

Week 3

We did some further study on the Euler Method. The global error with the Euler Method is $\mathcal{O}(h)$ and we need to reduce this by coming up with a better method or adjusting the Euler Method.

We started this by looking at Huen’s Method,.

In VBA we finished off the Euler Method Lab 2. People who have not complete this lab in the two labs of Weeks 2 and 4 should really complete this in their own time. Help here.

Week 4

We will do more on Heun’s Method, and also look at “Taylor Methods”.

We might also introduce second order differential equations and how to attack them numerically.

In VBA we look at Lab 3, on Heun’s Method.

Assessment

See Canvas for the assessment schedule.

MCQ League

See how you are getting on vs your peers on Canvas.

Study

Study should consist of

doing exercises from the notes
completing VBA exercises

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..

MATH7021: Spring 2024, Week 3

February 1, 2024 in MATH7021 | Leave a comment

15% Assignment 1

This can be started. This has a provisional hand in time and date of 10:45 Thursday 22 February. Please see Canvas.

Week 3

We finished Chapter 1 on Wednesday by looking at applications to temperature distribution, where the Jacobi Method is used to find approximate solutions to a diagonally dominant linear system.

If you want a deep dive into the temperature of a plate stuff there is a post here which explains — in great detail — how the ‘iterations’ are actually physically meaningful.

We had four classes of tutorials — Monday, the BH pre-catch up, and the Thursday double. We should be well and truly prepped for Assignment 1.

Gaussian Elimination Tutor

Week 4

We are now going to push to complete Chapter 2 over the next two weeks. Any remaining classes will be given over to tutorial time on Chapter 2.

If there is full attendance on Wednesday we might do a Chapter 1 “concept MCQ”.

Study

Please feel free to ask me questions about the exercises via email or even better on this webpage.

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc..

MATH7016: Spring 2024, Week 2

January 25, 2024 in MATH7016 | Leave a comment

Here is a graph showing the relationship between Concept MCQ Performance and Final Grade:

Week 2

We looked at the Euler Method, and then started looking at big $\mathcal{O}$ notation, and Taylor Series. We started doing error analysis for Euler’s Method.

In VBA we started programming the Euler Method.

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Week 13, Review Week

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20% VBA 1 Assessment, Week of 26 February to 1 March

20% Written Assessment 1, 6 March

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Classical Warm-up

Let’s Go Quantum

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20% VBA 1 Assessment, Week of 26 February to 1 March

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15% Assignment 1

Gaussian Elimination Tutor

Week 5

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Idea and Intuition

Two Examples

Commutative Examples

Theorem

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DME2A and DME2E in Bank Holiday Weeks

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15% Assignment 1

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Gaussian Elimination Tutor

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