I originally planned to record an introduction to universal algebra during the 2018-2019 academic year. At the end of the following academic year the COVID-19 pandemic began and forced me to start giving online lectures as part of my regular teaching duties. I started this series with basically the intention to record lectures by myself, but I did open it up as a seminar to other people in the University of Rochester math department. This was kind of a weird hybrid idea and the realities of the 2020 fall semester made me drop the seminar format and the regular schedule. Thanks to those of you who did attend a talk live!

I have taken an extended hiatus from this project since finishing graduate school but I very much would like to return to it when possible. Thanks to all of you who have watched and supported me in this over the years!

Universal algebra and lattice theory have many connections to other areas of math, so regardless of your primary interests you should find something in the special topics or connexions which suits your interests. I have done a number of the exercsies in the above-mentioned textbook and while I would like to avoid giving detailed answers during lectures I would be happy to talk about them with anyone who cares to email me. (Warning for undergrads: I'm not going to do your homework for you so please ask your TA or professor first if you're taking an actual course in universal algebra or lattice theory.)

If you're not sure whether this is all for you, just watch the first video in order to get an idea what's going on here.

Click on the name of a talk to see the slides from that talk.

Recordings of the talks are available on this YouTube playlist.

Number | Date | Topic | Suggested reading |
---|---|---|---|

1 | 1 September 2020 | Introduction and background | FST 1.1, An overview of modern universal algebra |

2 | 3 September 2020 | Examples of algebras | FST 1.2 |

3 | 8 September 2020 | Homomorphisms, subalgebras, and products | FST 1.3-1.4 |

4 | 10 September 2020 | Congruences and quotients | FST 1.5 |

5 | 15 September 2020 | Posets and lattices | FST 2.1, The Many Lives of Lattice Theory |

6 | 17 September 2020 | The distributive and modular laws | FST 2.2 |

7 | 23 December 2020 | Complete lattices | FST 2.3 |

8 | TBD | (Algebraic) closure operators | FST 2.4 |

9 | TBD | Galois connections | FST 2.5 |

10 | TBD | Ideals and filters | FST 2.6 |

11 | TBD | The isomorphism theorems | FST 3.1 |

12 | TBD | (Sub)direct products | FST 3.2-3.3 |

13 | TBD | A structure theorem for general Abelian groups | FST 3.4 |

14 | TBD | Subdirectly irreducible rings | FST 3.4 |

15 | TBD | p-algebras | FST 3.4 |

16 | TBD | Varieties | FST 3.5 |

17 | TBD | Clones | FST 4.1 |

18 | TBD | Invariant relations | FST 4.2 |

19 | TBD | Terms and free algebras | FST 4.3 |

20 | TBD | Identities | FST 4.4 |

21 | TBD | Birkhoff's theorem | FST 4.4 |

22 | TBD | Lattices of varieties | FST 4.5 |

23 | TBD | Equational theories and fully invariant congruences | FST 4.6 |

24 | TBD | Maltsev conditions | FST 4.7 |

25 | TBD | Interpretations | FST 4.8 |

26 | TBD | Topological algebras | |

27 | TBD | Ternary rings and projective planes | |

28 | TBD | Graph algebras | |

29 | TBD | RPS algebras | |

30 | TBD | What comes next |