## Symmetry

## Pure Mathematics

Pure Mathematics代写 This project is all about understanding the various manifestations of symmetry in mathematics, with a particular focus…

This project is all about understanding the various manifestations of symmetry in mathematics, with a particular focus on algebra and geometry. For example, this includes symmetric groups, rotational and reflectional symmetry groups in three dimensions, wallpaper symmetry groups, and matrix groups. Their actions are fundamental to our understanding in every mathematical and physical subject.

**Tentative Calendar: Pure Mathematics代写**

**Wednesday 25 May: **Lecture.

**25–29 May: **Please select a tentative topic for your project and discuss it on the EdTech forum and/or with me or the GTA, Haiping Yang ([email protected]).

**Monday 30 May: **Workshop 1, 10–12, Huxley 139. The purpose of the workshops is to discuss what you’d like to do for the poster, to go over any questions, and to interact with others doing related projects.

**30–31 May: **If you haven’t already, please do select your topic and discuss it with us.

**Wednesday 1 June: **Workshop 2, 10–12, Huxley 139

**Monday 6 June: **Workshop 3, 10–12, Huxley 139

**Wednesday 8 June: **Workshop 4, 10–12, Huxley 139

**Monday 13 June: **Online Q+A, tentatively 2–4 PM, on Teams.

**Wednesday 15 June: **Final Q+A, 10–11:30, Huxley 139

We will also discuss over the EdTech forum between meetings.

**1 Symmetry Groups Pure Mathematics代写**

The following is a non-exhaustive list of possible projects related to symmetry groups.

A general reference on examples of symmetry groups: https://math.mit.edu/~etingof/groups.pdf. A textbook including this material is Artin’s *Algebra*.

**Project: Matrix groups. **You can understand the groups GL(*n, *F)*, *SL(*n, *F)*, *SO(*n, *F)*, *O(*n, *F), and their relationships, for F = R*, *C, or more general fields. Additionally you can understand the symplectic groups, Sp(2*n, *F), the quaternions H, the octonions O, and *G*2 = AutR(O) (here the subscript means we look at real-linear automorphisms of O, which are compatible with the multiplication).

**References: **http://math.uchicago.edu/~may/REU2014/REUPapers/Collins.pdf and references therein (Tapp, Baker, and Fulton and Harris).

**Project: The spin group. **The Dirac Belt Trick shows that in SO(3*, *R), if we take a closed path which goes from a 0*◦ *to a 720*◦ *rotation about an axis, we can contract this closed path to a trivial one: see https://www.gregegan.net/APPLETS/21/21.html. On the other hand, this is not true for half of this path, going only to 360*◦ *. This is a manifestation of the fact that there is a nontrivial two-to-one covering *S *3 *→ *SO(3*, *R). You can understand this and generalise it to two-to-one coverings Spin(*n, *R) *→ *SO(*n, *R). These coverings are in fact *simply-connected*, which means that there is no connected *m*-to-1 covering of Spin(*n, *R), other than by itself for *m *= 1.

**References: **For the spin homomorphism, see. e.g., Artin, and for the general construction of the Spin group, see, e.g., Fulton and Harris, *Representation Theory*, Lecture 20. For more on covering spaces (of which this is a very special example), see e.g., http: //math.uchicago.edu/~may/REU2017/REUPapers/Bhatnagar.pdf and references therein, or Allen Hatcher’s book, *Algebraic Topology *(more advanced).

**Platonic symmetry groups: **This could include one or more of: Pure Mathematics代写

- Classify the finite subgroups of SO(3
*,*R) - Study the connections between the classification of finite subgroups and the classification of Platonic solids (given the Platonic solids you get a rotational symmetry group, and given rotational symmetry groups you can take its action on a point to get a possible collection of vertices, etc.)
- Classify the finite subgroups of
*S*3*∼*= SU(2*,*C), which is the spin group from lectures (there is two-to-one homomorphism*S*3*→*SO(3*,*R), see also the next project).

Possible references: Artin’s *Algebra*, end of chapter 5; Etingof’s notes; https://math. uchicago.edu/~may/REU2020/REUPapers/Bui,An.pdf; many more!

**Wallpaper groups and the Alhambra: **The purpose of this project could include one or more of:

- Understanding the classification of symmetry groups of wallpaper patterns—there are 17 such groups;
- Finding examples of these symmetry groups at the Alhambra or elsewhere, and explaining how to identify the symmetry.
- Making your own wallpaper patterns demonstrating the various groups;
- Generalisations of this classification, such as to three dimensions, to other two-dimensional surfaces, etc.

#### Possible references: Artin, Chapter 5; Jos´e Mar´ıa Montesinos-Amilibia, *Classical Tes**sellations and Three-Manifolds*, pages 94–95, available online; https://www.youtube.com/ watch?v=KWAEM4_QDN0 (if you understand Spanish)

**Rubik’s Cube: **This project is a bit open-ended, but the idea is to study to structure of the group of symmetries of the Rubik’s cube, defined as the group of symmetries of the cube generated by face twists, considering two symmetries the same if all they do is rotate the center squares of faces (we consider two configurations the same if all they do is this).

For instance, it’s an interesting fact that every solvable configuration can be solved in at most 20 moves (and there are some where it takes at least 20).

Possible References: There is something on this in Etingof’s notes; you can probably find many more online, such as https://people.math.harvard.edu/~jjchen/docs/ Group%20Theory%20and%20the%20Rubik%27s%20Cube.pdf, or https://web.mit.edu/sp. 268/www/rubik.pdf.

**2 Representation Theory Pure Mathematics代写**

General references: My course notes for representation theory (there are videos to go along with that, in case you need any), and the references contained Etingof’s notes.

**Project: Symmetry in the hydrogen atom and/or Kepler’s problem: **This project studies how rotational symmetry can be applied to understanding the electron states of the hydrogen atom on one hand and Kepler’s problem (planetary orbits) on the other. To simplify, we model these systems as having a stationary centre (the proton on the one hand, or the Sun on the other) and one particle orbiting it (the electron on the one hand, and a single planet on the other). The system is described by the electromagnetic or gravitational force, whose strength is inversely proportional to the square of the distance. Alternatively, this can be stated as a potential energy which is inversely proportional to the distance itself. There is a 3D rotational symmetry SO(3). This means that rotating any solution about the origin produces another. This structure is seen in the *spherical harmonics*, closely related to orbital configurations, as shown in lecture. In fact, there is an additional “hidden” 4D rotational symmetry, an action of SO(4) on bound states, which is much more powerful—this is captured by the Laplace–Runge–Lenz vector.

#### Some possible references: Pure Mathematics代写

**Noether’s theorem: **This is a more general result about the relationship between symmetry and conserved quantities which is in the background of all of the above: see, e.g., https://math.ucr.edu/home/baez/noether.html, https://www.digital-science. com/blog/2018/03/noethers-theorem-symmetry-runs-game-internationalwomensday/ and the references therein.

**On the hydrogen atom: **https://math.mit.edu/~etingof/lnlg.pdf (using some basics about differential equations), https://arxiv.org/pdf/quant-ph/0212010.pdf (on Pauli’s original approach using SO(4) symmetry), https://arxiv.org/pdf/quant-ph/0611287. pdf (on further SO(4*, *2) symmetry)

**On Kepler’s planetary motion: **See for instance https://math.ucr.edu/home/baez/ gravitational.html and the references therein.

**Representation theory of ***S**n ***or **GL*n***: **Note that you won’t be able to delve into this in great detail, but you could explain one aspect, such as the definition of Young diagrams 3and the construction of a representation of *S**n *or of GL*n *from such a diagram. For *S**n*, see, e.g., Gordon James, *The representation theory of the symmetric groups*; for GL*n *see, for instance, Fulton and Harris, *Representation Theory: A First Course*.

**Classification of simple Lie groups/algebras: **You won’t be able to delve into this in great detail, but you could learn about what Dynkin diagrams are and a bit about their classification. See for instance https://uu.diva-portal.org/smash/get/diva2:1231357/ FULLTEXT01.pdf; Edrmann and Wildon, *Introduction to Lie algebras *(available online); Fulton and Harris, *Representation Theory: A First Course*; and Humphreys, *Introduction to **Lie algebras and Representation Theory Pure Mathematics代写*

**Variation on this: McKay quivers and extended Dynkin diagrams. **See for instance https://www.math.miami.edu/~armstrong/686sp13/McKay_Yi_Sun.pdf and https: //ncatlab.org/nlab/show/McKay+quiver. You need to learn what the tensor product is; I can give you access to a video lecture with Dr. Lawn on the subject if you like.