Meeting Notes 2024-04-02

This Week’s Finds

We had quite a few interesting finds this week.

Locations of Prime Numbers

Kokkimidis opened with an interesting observation that primes seem to be “near” multiples of six, conjecturing that numbers close to multiples of 6 are neither divisible by 2 or 3. Examples: 11 and 13 are near 12, 17 and 19 are near 18, etc. The relation to Mersenne primes was discussed, as well as an interesting fact about digit sums and divisibility. The digit sum \(F_b(n)\) of a number \(n\) is the sum of the digits in the number (forgive the circular definition), and the digital root is the result of applying the digit sum repeatedly until a single digit number is achieved. The digital root \(dr_b\) has a relation to the base of the number system \(b\), as \(dr_b(n) = n mod (b – 1)\). If the digital root of a number is divisible by 3, then the number is divisible by 3. [1] [2]

 

Uncertainty in Ground States and Chern Numbers

Amogh described recent applications of topology in determining uncertainty in ground states of 2-dimensional band structure systems. Specifically, he discussed how non-zero Chern numbers act as a signature for nontrivial topological band structure. More on this soon. [3]

 

Projective Groups and Quotients

Jim introduced an example of quotients often encountered in gauge theory, in particular Yang Mills \(SU(n)\) theories, where symmetry groups are modded by their centers to produce the “interesting” symmetry groups. For example, given \(U(n)\), the space of unitary \(n \times n\) matrices, the center is \(Z(U(n)) = U(1) = \{e^{i\theta} \mathbb{I}_n\}\) is the set of constant phase scalars applied to the identity matrix. To remove symmetries that vary only by constant phase, the project group is created using a quotient \(PU(n) \equiv U(n) / U(1)\). The exercise can be repeated in the case of special unitary matrices to produce the group \(PSU(n) \equiv SU(n) / Z(SU(n))\). Also, the name of the group is “PSU”, which is neat. [4]

 

Subfactorial Numbers and Derangements

Michael ran into an interesting integer sequence that he was able to identify as the “subfactorial” numbers, which count derangements, or permutations that have no fixed points. For example, given a set \(X = \{1, 2, 3\}\), the cycle \((12)\) would not be a derangement because it leaves 3 unchanged (or 3 is a fixed point of the permutation); however, the cycle \((123)\) would be a derangement, as each number is permuted. The number of derangements for a particular number \(n\) of objects to permute is denoted \(!n\), can be given by a recursive relation \(!n = (n – 1)\cdot (!(n-1) + !(n-2))\) for \(n \geq 2\) where \(!0 = 1\) and \(!1 = 0\). An explicit non recursive form is also given by \(!n = n! \sum_{I=0}^n \frac{(-1)^i}{i!}\) for \(n \geq 0\). These numbers apply to physics by helping to count the number of possible linearly independent tensors of rank \(n\) with a given gauge group.

 

Focus Session:

Reading review for Dummit & Foote chapter 3.

 

Isomorphism Theorems

We reviewed the isomorphism theorems (while noting that the first is the most useful by a wide margin).

Theorem 1: Let \(\varphi G \to H\) be a group hom., then (i) \(\mathrm{ker}\ \varphi \underline{\triangleleft} G\), and (ii) \(G / \mathrm{ker}\ \varphi \cong \varphi(G)\)

Theorem 2 (Diamond): Let \(A, B \leq G\) and \(A \leq N_G(B)\), then (i) \(AB \leq G\), (ii) \(B \underline{\triangleleft} AB\), (iii) \(A\cap B \underline{\triangleleft} AB\), and (iv) \(AB/B \cong A / A\cap B\)

Theorem 3: Let \(H, K \underline{\triangleleft} G\) and \(H\leq K\), then (i) \(K/H \underline{\triangleleft} G/H\) and (ii) \(\left(G/H\right) / \left(K/H\right) \cong G / K\)

Theorem 4 (Lattice): Let \(N\underline{\triangleleft} G\) and \(\mathcal{A} = \{A \leq G\ : \ N \leq A\}\) and \(\overline{\mathcal{A}} = \{A/N \leq G/N\}\), then (i) \(\exists \phi: \mathcal{A} \to \overline{\mathcal{A}}\) bijective.

 

 

Up Next:

Interesting or Relevant Talks:

• Math Colloquium

References:

1. Mersenne Primes: https://en.wikipedia.org/wiki/Mersenne_prime
2. Digital roots: https://en.wikipedia.org/wiki/Digital_root
3. A. Anakru and Z. Bi, Topological Phases and Phase Transitions with Dipolar Symmetry Breaking, arXiv:2403.19601.
4. Projective Special Unitary (PSU) Group: https://en.wikipedia.org/wiki/Projective_unitary_group#Projective_special_unitary_group
5. Subfactorial Numbers (Derangements): https://en.wikipedia.org/wiki/Derangement#Counting_derangements