Welcome to Math 202B at UCSD, Winter quarter 2026. Here is a New Year’s problem you can keep in the back of your mind over the course of the course. Of course, let me know if you solve it.
Problem 0: Prove that 26 is the only positive integer nestled between a square and a cube.
The basic parameters of Math 202B are the same as those in Math 202A: weekly problem sets due on Sundays at 23:59 via GradeScope together with a final exam, with a 70/30 split. The final exam is scheduled for 03/20 at 15:00, and if you cannot sit for the exam at the appointed time you should not enroll in the course.
Math 202A began in the classical computational category whose objects are finite sets with morphisms being functions. This is in contrast to the quantum computational category whose objects are finite-dimensional Hilbert spaces with linear transformations as morphisms. We considered the quantization functor from to which sends a finite set to the Hilbert space of complex-valued functions on with the pointwise operations and the -scalar product. We then found a miraculous tool, the Singular Value Decomposition, which completely describes all morphism in for any two finite sets and In the case the SVD gave us the Spectral Theorem for normal operators in
In retrospect, we now recognize that Math 202A was actually about two quantization functors departing from the classical computational category, the Schroedinger functor and the Heisenberg functor both of which land in In Math 202B, we recognize something new as well: both quantizations and come with vector products as well as scalar products. The product of two functions in is defined pointwise, and the product of two endomorphisms in is defined by composing them.
We now recognize that Schroedinger and Heisenberg are actually telling us to think about a subcategory of namely the category of finite-dimensional algebras. Math 202B is all about the category We will begin by defining algebras precisely and developing their basic theory axiomatically. Given an algebra in this category, two natural goals are to classify its subalgebras and measure its commutativity, and we will formulate these goals rigorously.
Our two basic examples, and , are two opposite extremes: the former is fully commutative and the latter is maximally noncommutative. The classification of subalgebras of is elementary, while a complete description of subalgebras of is more involved and requires the development of new linear algebraic concepts and methods you may not have seen before.
Once we have said everything there is to say about and , we will consider the question of what lies between these two extremes. In this range we find a beautiful class of algebras constructed from finite groups. Namely, if carries a group law, then it becomes possible to associate a third algebra to it, the convolution algebra As Hilbert spaces, but as algebras the two are very different: multiplication of functions in is given by convolution rather than pointwise product. The commutativity index of is the number of conjugacy classes contains, so can be as commutative as but cannot be as noncommutative as
When is an abelian group, is isomorphic to via an extremely useful map called the Discrete Fourier Transform (DFT), which is perhaps the most widely applied algebra isomorphism there is. When is nonabelian, is isomorphic to a subalgebra of via a noncommutative generalization of the DFT whose construction will occupy much of the course and lead us into a new realm where linear algebra and group theory interact in many remarkable ways.
There is no official textbook, but as we move through the course you should regularly consult the following texts:
Algebras of Linear Transformations by Farenick, for the structure theory of finite-dimensional operator algebras.
Linear Representations of Finite Groups by Serre, for the conceptual backbone of representation theory.
Representation Theory of the Symmetric Groups by Ceccherini-Silberstein et al, for a detailed treatment of symmetric groups as the fundamental nonabelian example.
The Symmetric Group by Sagan, for combinatorial viewpoints which make the algebra we encounter more concrete.
