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Prerequisites: MAT220.

Description: This project involves studying a particular correlation between (ultimately) groups and commutative rings.

Every finite set comes with a natural number, namely the number of elements in the set. Conversely, the natural numbers are constructed from finite sets. By collecting all the sets with the same number, we get the natural numbers. Note that two finite sets have the same number of elements if and only if there is a bijective mapping between them. Multiplication and addition of natural numbers corresponds to the Cartesian product and union of sets, respectively. Given a finite group G, we can look at the finite G-sets, that is to say, the finite sets that the group G operates on. We can construct a set of numbers by setting two finite G-sets equal if there is a bijection from one to the other that is compatible with the G-operations. If G is the trivial group with one element, we have only repeated the construction of the natural numbers. In the same way that we construct the ring of the integers from the natural numbers, we can construct a ring from this new set of numbers. This ring is called the Burnside ring. We shall generalise the Burnside ring to a construction called the Burnside-Witt ring. Once we have the construction in place, we shall see that the characteristic polynomials from linear algebra can be regarded as an element in a particular Burnside-Witt ring. In fact, we shall see that the Burnside-Witt ring gives information about linear algebra over arbitrary commutative rings. There are several ways to complete the project. A possibility is to derive formulas from classical combinatorics from the properties of the Burnside-Witt ring. Another possibility is to create explicit calculations of Burnside-Witt rings. Finally, it is possible to finish by going deeper into the construction of the Burnside-Witt ring and generalising it.

Although the Burnside-Witt ring should not be used in further studies, the methods in this project are important in both algebraic geometry and topology. If desired, the project can lead on to active research. Within algebraic geometry the Burnside-Witt ring is used to study phenomena in positive characteristics. In algebraic topology, the Burnside-Witt ring is closely related to algebraic K-theory.

What you will learn:

1. group theory (Burnside's formula, etc. See Fraleigh's book, which is used in MAT220),
2. linear algebra over commutative rings (K-theory of endomorphisms),
3. combinatorics (Möbius inversion formula).

References:

[1] , .
[2] A.Dress, C. Siebeneicher, The Burnside ring of profinite groups and the Witt vector construction. Adv. in Math. 70 (1988), no. 1, 87—132. ().