Here are some potential topics for your presentations, which will be 30% of your grade. They are loosely organized by area. You can also propose your own topic. Please email me by 10 January 2022 with your proposal.

You will give a 30 minute presentation on the topic we agree upon. You should prepare slides for this presentation.

There will be two presentation sessions:

- Tuesday, 22 February 14:00 - 16:30, P3 Pavilhão Matemática: Anita, Guilherme, Luís D, Rui.
- Monday, 28 February 14:00 - 16:30, 3.10 Pavilhão Matemática: João, Marco, Luís L (by Zoom), Simão.

Here is how you will be evaluated:

- quality of your presentation;
- quality of your slides;
- amount of research done;
- mastery of the material in your presentation;
- attendance of other presentations in your session.

**Mathematics**

- The classification of finite simple groups.
- Crystallographic groups.
- The Erlangen program and homogeneous spaces.
- Finite geometry and the card game Dobble.
- The monster group, an example of a sporadic group.
- Rubik's cube group.
- Wallpaper groups.

**Physics**

- Gauge theories.
- Molecular symmetries.
- Noether's theorem.
- Representations of SU(2) and spin.
- Representations of SU(3) and quarks.
- Spherical harmonics and atomic orbitals.
- The standard model of particle physics and grand unified theories.

**Engineering**