Universal algebra and lattice theory


Description

I'll give lectures primarily based on Clifford Bergman's «Universal Algebra: Fundamentals and Selected Topics» at roughly the rate of one section per lecture. The tentative idea is to cover the first part of this text (the «Fundamentals») after which I will give talks on a few of my own chosen topics of interest in the area, concluding with the subject of my own recent paper on tournament algebras.
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've since become preoccupied with applying to jobs and have taken an extended break from recording new lectures in this series. I may start again in the spring of 2022 once I have solid plans for the fall of 2022.
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.

Lectures

In the suggested reading the abbreviation FST stands for Clifford Bergman's «Universal Algebra: Fundamentals and Selected Topics». For example, «FST 1.2-1.4» refers to sections 1.2, 1.3, and 1.4 of that book. More suggested readings will be added as we move forward.
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 Introduction and background FST 1.1, An overview of modern universal algebra
2 3 September Examples of algebras FST 1.2
3 8 September Homomorphisms, subalgebras, and products FST 1.3-1.4
4 10 September Congruences and quotients FST 1.5
5 15 September Posets and lattices FST 2.1, The Many Lives of Lattice Theory
6 17 September The distributive and modular laws FST 2.2
7 23 December 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