University of Edinburgh, September 22 - December 1, 2022

John Baez

Seminar on This Week's Finds

I wrote 300 issues of a colum called This
Week's Finds, where I explained math and physics. This fall
I'm running a seminar on selected topics from this column. For
information on how to attend, go here.

Here you'll find lecture notes and videos. I'll keep
updating and improving these notes.

Young diagrams and classical groups

Young diagrams are combinatorial structures that show up in a myriad
of applications. Among other things, they can be used to classify
conjugacy classes in the symmetric groups \(S_n\), irreducible
representations of \(S_n\), and irreducible representations of the
classical groups \(\mathrm{GL}(n)\), \(\mathrm{SL}(n)\), \(\mathrm{U}(n)\)
and \(\mathrm{SU}(n)\).

Coxeter and Dynkin diagrams classify a wide
variety of structures, most notably Coxeter groups, lattices having
such groups as symmetries, and simple Lie algebras. The simply laced
Dynkin diagrams also classify the Platonic solids and quivers with
finitely many indecomposable representations. This tour of Coxeter and
Dynkin diagrams will focus on the connections between these
structures.

q-mathematics

A surprisingly large portion of mathematics generalizes to something
called \(q\)-mathematics, involving a parameter \(q\). For example,
there is a subject called \(q\)-calculus that reduces to ordinary
calculus at \(q = 1\). There are important applications of
\(q\)-mathematics to the theory of quantum groups and also to
algebraic geometry over \(\mathbb{F}_q\), the finite field with \(q\)
elements. These seminars will give an overview of \(q\)-mathematics
and its applications.

The three-strand braid group

The three-strand braid group has striking connections to the trefoil
knot, rational tangles, the modular group \(\mathrm{PSL}(2,
\mathbb{Z})\), and modular forms. This group is also the simplest of
the Artin–Brieskorn groups, a class of groups which map surjectively
to the Coxeter groups. The three-strand braid group will be used as
the starting point for a tour of these topics.

Clifford algebras and Bott periodicity

The Clifford algebra \(\mathrm{Cl}_n\) is the associative real algebra
freely generated by \(n\) anticommuting elements that square to -1. I
will explain their role in geometry and superstring theory, and the
origin of Bott periodicity in topology in facts about Clifford
algebras.

The threefold and tenfold way

Irreducible real group representations come in three kinds, a fact
arising from the three associative normed real division algebras: the
real numbers, complex numbers and quaternions. Dyson called this the
threefold way. When we generalize to superalgebras this becomes part
of a larger classification, the tenfold way. We will examine these
topics and their applications to representation theory, geometry and
physics.

Exceptional algebras

Besides the three associative normed division algebras over the real
numbers, there is a fourth one that is nonassociative: the
octonions. They arise naturally from the fact that Spin(8) has three
irreducible 8-dimensional representations. We will explain the
octonions and sketch how the exceptional Lie algebras and the
exceptional Jordan algebra can be constructed using octonions.