MATH6040: Winter 2024, Week 3

September 27, 2024 in MATH6040, MATH6040 - Day | Leave a comment

Test 1

The test will be in the first Tuesday lecture, 8 October, of Week 5. See Canvas (MATH6040 —> Assignments) for further details.

Week 3

We finished Chapter 1, and the material for Test 1, by talking at the applications of vectors to work and moments. We started Chapter 2, on Matrices.

Week 4

We will look at Matrix Inverses — “dividing” for Matrices. This will allow us to solve matrix equations. We might also look at determinants.

Some deeper discussion here: Why do we multiply matrices like we do?, Why can’t I divide by zero?

Study

Please feel free to ask me questions about the exercises via email.

Student Resources

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

Test 2

Test 2 is provisionally booked for the Wednesday of Week 11:

27/11/202418.00-19.00MATH6040Technological Maths 20140+A – C

MATH6055: Winter 2024, Week 2

September 26, 2024 in MATH6055 | Leave a comment

Weeks 2

This week we mostly spoke about the Laws of Sets, which form a Boolean Algebra. We just started talking about Cartesian Products.

In tutorial most students got to try applying the laws of sets in simplifying ‘complicated sets’. Students will get another chance to work on these in Week 3.

Week 3

In Week 3 we will talk more about Cartesian Products and then Relations.

In tutorial we will work on Laws of Sets and Cartesian Products. 

Assessment

See Canvas for the assessment plan and schedule.

Student Resources

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

MATH7019: Winter 2024, Week 3

September 26, 2024 in MATH7019 | Leave a comment

Week 3

We looked at models that are not linear but log-linear. We will have had two hours of tutorial time (the lecture tomorrow will be a tutorial on linear models).

Week 4

We will start the mathematical background for Chapter 2 and will possibly start looking properly at applications to beams: in particular at simply supported beams. We will probably have a normal lecture on Friday.

Assessment 1

Assessment 1 is now up on Canvas. It has a hand up date the Friday of Week 5, 11 October.

Academic Learning Centre

I would urge anyone having any problems with material that isn’t being addressed in the tutorial communication to use the Academic Learning Centre. If you are a little worried about your maths this semester you need to be aware of this resource. You will get best results if you come to the helpers there with specific questions.

Student Resources

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

MATH6040: Winter 2024, Week 2

September 23, 2024 in MATH6040, MATH6040 - Day | Leave a comment

The manuals are available in the Copy Centre (at a cost of €15.50). All students need a manual at this point.

Week 2

In Week 2 we spoke about vector products: the dot product and the vector product.

Week 3

In Week 3 we will finish Chapter 1, and the material for Test 1, by talking at the applications of vectors to work and moments.

Study

Please feel free to ask me questions about the exercises via email.

Student Resources

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

MATH7019: Winter 2024, Week 2

September 23, 2024 in MATH7019 | Leave a comment

Week 2

We continued talking about (linear) Least Squares curve fitting.

We had one and a half classes of tutorials.

Week 3

We will start the next section on models that are not linear but log-linear (after the very short section on correlation). We will aim for one and a half to two classes of tutorials.

Academic Learning Centre

I would urge anyone having any problems with material that isn’t being addressed in the tutorial communication to use the Academic Learning Centre. If you are a little worried about your maths this semester you need to be aware of this resource. You will get best results if you come to the helpers there ith specific questions.

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.

MATH6055: Winter 2024, Week 1

September 19, 2024 in MATH6055 | Leave a comment

Manuals

COMP1C are required to purchase an academic manual for MATH6055. This contains all the lecture notes and exercises for the module. The lecture notes contain gaps that we fill in during class. I will have notes for you for the first week but after that you must purchase the manual: the sooner the better.

The manuals can be purchased from Reprographics\Copy Centre beside the Student Centre for €14.10.

With all the materials (including worked examples, summaries, etc) it comes to 235 pages and provides a comprehensive resource for this module.

I will be writing in various colours, so maybe  a four colour pen would be useful.

Week 1

After listening to me go on about the importance of mathematics to your programme we started the first chapter on Sets and Relations by looking at some number sets. We saw something new with the concept of the power set of a set.

We started tutorials in Week 1.

Week 2

In Week 3 we will look at the Laws of Sets and move onto Cartesian Products and Relations.

Assessment

See Canvas for the assessment plan and schedule.

Student Resources

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

MATH6040: Winter 2024, Week 1

September 16, 2024 in MATH6040, MATH6040 - Day | Leave a comment

The manuals are available in the Copy Centre and should be purchased as soon as possible.

Week 1

We began our study of Chapter 1, Vector Algebra by studying the difference between a scalar (single number) and a vector (list of numbers). We looked at how to both visualise vectors and describe them algebraically. We learned how to find the magnitude  and direction of a vector, add them and scalar multiply them. We spoke about displacement vectors and started talking about vector products: the dot product and the vector product.

We had a tutorial, where students were actually ahead of the lectures.

Week 2

In Week 2 we continue to talk about vector products.

Test 1

The test will probably be the Wednesday evening, 9 October, of Week 5. Official notice will be given in Week 3. There is a sample test in the notes.

Study

Please feel free to ask me questions about the exercises via email.

Student Resources

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

MATH7019: Winter 2024, Week 1

September 16, 2024 in MATH7019 | Leave a comment

Manuals

The manuals are available in the Copy Centre. Please purchase ASAP. More information on Canvas

Week 1

We covered Lagrange interpolation and started the theory for Least Squares curve fitting.

Apologies about the cancelled lecture on Thursday. Unfortunately, unless we go to Friday afternoon, we have no chance of rescheduling. Last year’s class lost out on Bank Holiday Mondays (and we won’t) so I don’t see this being a huge issue. Just an unfortunate misstep in Week 1.

Week 2

We will continue talking about Least Squares curve fitting. We will have our first tutorial too.

Assessment 1

Assessment 1 has a provisional hand-in week of Week 5. Once the class list is established I can send on the student-number-personalised assessment.

Academic Learning Centre

I would urge anyone having any problems with material that isn’t being addressed in the tutorial communication to use the Academic Learning Centre. If you are a little worried about your maths this semester you need to be aware of this resource. You will get best results if you come to the helpers there ith specific questions.

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.

The classical permutation group is not normal in the quantum permutation group

September 10, 2024 in Quantum Groups, Research | Leave a comment

Unless I am being usually dumb, while the notion of a normal quantum subgroup is established in the theory of compact quantum groups, it’s something that rarely comes up. I am not entirely sure why this is.

I wondered if the classical permutation group S_N\subset S_N^+ is normal for N\geq 4. I am not sure why I didn’t just Google this: Shuzhou Wang covered this here. I mean it’s in the abstract yet this was my source for learning about normal quantum subgroups.

Tracial central states on C(S_N^+)

Let k=h_{C(S_N)}\circ \pi_{\text{ab}} be the Haar idempotent associated to S_N\subset S_N^+.

See Proposition 2.1 of Wang to see that if S_N is a normal subgroup then k is central. That k is central is to say that for all irreducible representation \alpha\in\mathrm{Irr}(S_N^+), there exists c_\alpha\in\mathbb{C} such that:

\displaystyle k(u_{ij}^\alpha)=c_\alpha\delta_{i,j}.

If a state \varphi is central in this way, and this holds for a general compact quantum group \mathbb{G}, then it is central in the sense that for all states on C(\mathbb{G}) we have:

\displaystyle \varphi\star \rho=\rho\star \varphi\qquad (\rho\in \mathcal{S}(C(\mathbb{G}))).

This is easy: both (\varphi\star \rho)(u_{ij}^\alpha) and (\rho\star \varphi)(u_{ij}^\alpha)=c_\alpha\,\rho(u_{ij}^{\alpha}), and those matrix elements form a basis for the norm-dense \mathcal{O}(\mathbb{G})\subset C(\mathbb{G}).

Recently, Freslon, Skalski & Wang have showed that the set of tracial central states is given by (Theorem 5.6):

\displaystyle \mathrm{TCS}(S_N^+)=\{t\varepsilon+(1-t)h\colon t\in [0,1]\}\qquad (N\geq 6).

The Haar idempotent k satisfies k\star k=k, and if S_N is normal, then k\in \mathrm{TCS}(S_N^+). If you solve \varphi\star \varphi=\varphi for \varphi\in \mathrm{TCS}(S_N^+), you find t=0,1, that is \varphi=\varepsilon\text{ or }h.

This also reproves that S_N^+ is simple for N\geq 6.

The Haar idempotent: it’s not central

So… there must exist a state \varphi on C(S_N^+) that does not commute with k. Can you find such a state and f\in C(S_N^+) such that:

(k\star \varphi)(f)\neq (\varphi\star k)(f)?

An approach to the maximality conjecture for quantum permutation groups

April 4, 2024 in C*-Algebras & Operator Theory, Quantum Groups, Research | Leave a comment

The following is an approach to the maximality conjecture for S_N\subset S_N^+ which asks what happens to a counterexample S_N\subsetneq \mathbb{G}_N\subsetneq S_N^+ when you quotient C(\mathbb{G}_N)\to C(\mathbb{G}_N)/\langle 1-u_{NN}\rangle. If C(\mathbb{G}_N)/\langle 1-u_{NN}\rangle is noncommutative, you generate another counterexample S_{N-1}\subsetneq \mathbb{G}_{N-1}\subsetneq S_{N-1}^+.}

Most of my attempts at using this approach were doomed to fail as I explain below.

Let \mathbb{G}\subseteq S_N^+ be a quantum permutation group with (universal) algebra of continuous functions C(\mathbb{G}) generated by a fundamental magic representation u\in M_N(C(\mathbb{G})). Say that \mathbb{G} is classical when C(\mathbb{G}) is commutative and genuinely quantum when C(\mathbb{G}) is noncommutative.


Definition 1 (Commutator and Isotropy Ideals)

Given \mathbb{G}\subseteq S_N^+, the commutator ideal J_N\subset C(\mathbb{G}) is given by:


\displaystyle J_N=\langle [u_{ij},u_{kl}]\,\mid\, 1\leq i,j,k,l\leq N\rangle.

The isotropy ideal I_N\subset C(\mathbb{G}) is given by:

\displaystyle I_N=\langle 1-u_{NN}\rangle.

Lemma 1

The commutator ideal J_N\subset C(\mathbb{G}) is equal to the ideal

\displaystyle K_N:=\langle u_{i_1j_1}u_{i_2j_2}u_{i_1j_3},u_{i_1j_1}u_{i_2j_2}u_{i_3j_1}\,\mid \, 1\leq i_k,j_k\leq N,\,j_1\neq j_3,i_1\neq i_3\rangle.

Proof:

\begin{aligned} [u_{i_2j_2},u_{i_3j_1}] & =u_{i_2j_2}u_{i_3j_1}-u_{i_3j_1}u_{i_2j_2} \\ \implies u_{i_1j_1}[u_{i_2j_2},u_{i_3j_1}] & = u_{i_1j_1}u_{i_2j_2}u_{i_3j_1}\qquad (i_1\neq i_3)\\ \implies u_{i_1j_1}u_{i_2j_2}u_{i_3j_1} & \in J_N, \end{aligned}
with a similar statement for u_{i_1j_1}u_{i_2j_2}u_{i_1j_3}.

On the other hand:
\begin{aligned} [u_{i_1j_1},u_{i_2j_2}] & =u_{i_1j_1}u_{i_2j_2}-u_{i_2j_2}u_{i_1j_1} \\ & =u_{i_1j_1}u_{i_2j_2}\sum_{k=1}^Nu_{i_1k}-\sum_{l=1}^{N}u_{i_1l}u_{i_2j_2}u_{i_1j_1} \\ & = u_{i_1j_1}u_{i_2j_2}u_{i_1j_1}+u_{i_1j_1}u_{i_2j_2}\sum_{k\neq j_1}u_{i_1j_1}u_{i_2j_2}u_{i_1k}-\\&u_{i_1j_1}u_{i_2j_2}u_{i_1j_1}\sum_{l\neq j_1}u_{i_1l}u_{i_2j_2}u_{i_1j_1}\\ \implies [u_{i_1j_1},u_{i_2j_2}]& \in K_N \end{aligned}

Proposition 1

The commutator and isotropy ideals are Hopf*-ideals. The quotient C(\mathbb{G})\to C(\mathbb{G})/J_N gives a classical permutation group G\subset S_N, the classical version G\subseteq \mathbb{G}, and the quotient C(\mathbb{G})\to C(\mathbb{G})/I_N giving an isotropy quantum subgroup. If \mathbb{G} is classical, this quotient is the isotropy subgroup of N for the action \mathbb{G}\curvearrowright {1,2,\dots,N}.

Via C(S_N^+)\to C(S_N^+)/J_N, the classical permutation group is a quantum subgroup S_N\subset S_N^+. It is conjectured that for all N it is a maximal subgroup.


Theorem 1 (Wang/Banica/Bichon)

S_N\subseteq S_N^+ is maximal for 1\leq N\leq 5.

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