Bachelor's Thesis in Algebra and Geometry
On this page you will find information specific for the course SA104X, with specialization towards Algebra and Geometry.
For writing a bachelor's thesis in Algebra and Geometry we strongly recommend to have studied basic algebraic structures such as groups, rings, and fields, at a level comparable to the course SF2729. The course SF2729 during 2015/16 is given in periods 2 and 3.
If you are interested in writing a bachelor's thesis in Algebra and Geometry you are encouraged to take contact as soon as possible to discuss possible projects and get advice on needed background material for their particular project.
SUGGESTIONS FOR PROJECTS
The Möbius transformations are automorphisms of the sphere. The sphere is identified with the plane and a point at infinity, and as such the Möbius transformations are the length preserving transformations of hyperbolic geometry. Consult Wikipedia for more information.
Conjugacy of matrices
A matrix encodes the information about a linear mapping, and conjugacy of matrices corresponds to choices of different bases and is therefore important to understand. In particular if two matrices are conjugate over a given field, or ring, and the coefficients of the matrices belong to a subfield, are the two matrices then conjugate over the smaller field? Another interesting, but harder problem is: given a pair of two (or more) matrices, when are they simultaneously conjugate, that is, when are A1 and B1 as well as A2 and B2 conjugate by the same matrix?
Thompson's group F
The group F, found by Richard Thompson, is an infinite group which nevertheless has a finite presentation, i. e. a finite set of generators subject to relations that are also generated by a finite set. One can describe it as a group of transformations between rooted binary trees. It does not fit into any of the standard classes of groups. This has made it a prime candidate as a counterexample for many conjectures. A famous open problem is whether F has a property called “amenability”.
Mapping class groups
Mapping class groups are groups of self-maps of compact oriented surfaces, for instance of a sphere, a torus, a figure-eight space, and so on. The number of “holes” of these surfaces is called their genus, and the genus determines such a surface. This means that there is a mapping class group Gamma_n for each natural number n. While these groups are well-understood for small n, they are notoriously difficult to describe in general. However, it is an interesting problem to relate them to each other for various n and study their common properties.
The braid group on n strands, Bn, has as elements braids with n incoming and n outgoing threads, and the group operation is given by tying two such braids together. Despite its simple geometric description, it has many applications in knot theory and also other areas of mathematics, but also in cryptography and even robotics.
Hyperbolic groups and the word problem
The word problem for a finitely presented group with generators g ∈ G and relations r ∈ R is the following algorithmic task: given a “word” g1 g2 … gn with gi ∈ G, does it represent the identity element in G or not? This initially simple question turns out to be undecidable in the sense of Turing.
However, one can ask whether the word problem is any easier if one restricts it to groups of a certain type. For instance, it is very easy for abelian groups. Another class of groups for which it is decidable are hyperbolic groups. These are groups that can be given a metric whose triangles have an angular sum of less than 180°, in a certain sense. This project is about studying the properties of hyperbolic groups and the algorithms one can apply to them to solve the word problem and other generally hard decision problems.
Invariant rings (cyclic invariants / Dickson-Mui invariants / divided power invariants)
Given a polynomial ring R=k[x1,…,xn], the symmetric group Sn acts on R by permuting the variables. The invariant ring S is the ring of polynomials that stay unchanged under all such permutations. For instance, for n=2, the polynomials σ1 = x1 + x2 and σ2 = x1x2 are in S, along with all their sums and products. In fact, the ring S is always again a polynomial ring in n variables σi.
For sub- or supergroups of the symmetric groups such as cyclic groups or general linear groups, or for variants of polynomial rings such as exterior algebras or divided power rings, this is no longer true, and the invariant rings can become quite complicated. Studying them is very interesting and important.
The Cayley-Dickson construction takes as input a k-algebra and outputs another k-algebra of the double dimension. If we feed it R, we get C. If we feed it C, we get the quaternions H, and if we feed it that, we get the octonions, or Cayley numbers, O. In the course of this process, we lose one nice property after the other: orderedness, commutativity, associativity. One can continue this further; in the next step one loses “power-associativity”, i.e. the value of the expression x•x•…•x depends on the bracketing even if all factors are equal. The Cayley-Dickson construction can be applied to other rings as well and it is an interesting problem to study the properties (or lack thereof) as one iterates it.
By combining the integers and the square root of -1, one obtains the Gaussian integers. This integer lattice of the complex plane is only one of many possible ones. Other lattices have unexpected behavior, which raises several natural questions such as when we have a Euclidean division algorithm in them.
Units in group rings
Given a group G and a commutative ring R, one can form a new ring RG whose elements are formal R-linear combinations of elements of G. Every element of G is a unit in this “group ring”, but the converse is usually not true. An interesting problem is whether one can reconstruct G from the group ring RG, and what kind of group extension of G the units in RG represent.
Your own idea
It is of course also possible to pursue a project not listed here. Maybe you have an idea of your own? The only requirements are that the project is of an adequate level of difficulty and relevance and that your KEX advisor feels competent to supervise it.