Whoever is interested in writing a bachelor's thesis in algebra and geometry is encouraged to contact Mats Boij and Carel Faber as soon as possible to discuss possible projects and get advice on needed background material for their particular project. Email addresses are provided below.
Below are suggestions for projects
AUTOMORPHISMS OF PLANE CURVES
Given a plane curve, it is interesting to study the group of transformations of the plane that fix the curve. This is especially true over fields of positive characteristic, where large infinite and/or non-reduced automorphism groups occur.
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.
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.
APPLICATIONS OF PROJECTIVE GEOMETRY TO COMPUTER VISION
Projective geometry represents an important area in modern geometry, extending the basics of Euclidian geometry. Recently it has become an important tool in more applied disciplines like computer vision.
CONJUGACY OF MATRICES
A matrix encodes the information about a linear mapping, and it is important to understand conjugacy of matrices. 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?