KTH Mathematics Bachelor Thesis in Algebra and Geometry   VT11   


Bachelor's Thesis in Algebra and Geometry

(Kandidatexamensarbete, SF140X)

Spring 2011

On this page you will find information specific for the course SA140X, with specialization towards Algebra and Geometry. General information about the course is found here, and general information about specialization in mathematics is found here.

COURSE STRUCTURE

The course SF2729, Groups and Rings, is given every spring. This course is required in order to write a bachelor's thesis in algebra and geometry but can be taken in parallel. 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.

EXAMPLES OF PROJECTS FOR BACHELOR THESIS IN ALGEBRA AND GEOMETRY

Below we give a brief description of possible projects. Students that will work with one of the projects will be given a much more detailed description concerning goals, reading material as well as working schedule.

  • 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.
  • MÖBIUS GROUP. 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. Click here for further reading.
  • GAUSSIAN INTEGERS. 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 behaviour, which raises several natural questions. More information is available here.
  • 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. More information, containing references to online literature can be found here.
  • 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?
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    PREVIOUS TOPICS

    In the spring of 2010 eight students wrote their bachelor's thesis in algebra and geometry. They formed two groups, one working on Rubik's cube and one working on p-groups.
  • RUBIK'S CUBE. This famous toy was originally designed to exemplify the complexity of finite groups. Which group is actually visualized with the Cube, and which normal subgroups does it have? are some questions to consider. Click here for much more information and generalizations.

  • Thesis on Rubik's cube

  • p-GROUPS. One of the results that we encounter in the course Groups and Rings states that every group consisting of p squared elements, for a prime number p, is an abelian group. Groups where the number of elements is a prime power, is a p-group, and such groups are well studied. It would be interesting to find elementary results characterizing the structure of p-groups of low prime powers, as e.g. three and four.

  • Thesis on p-groups

    • THE GROUP IN ALGEBRA AND GEOMETRY

    Here you will find a list of the present members of the research group in Algebra and Geometry. An idea of their research interests can be obtained by looking at the list of master's theses completed in Algebra and Geometry, and the list of examined Ph.D.'s. More specific details about the research interests can be found from their individual homepages, and by looking at their arXiv listings.

    CONTACT PERSONS

    During the spring of 2011 Mats Boij and Carel Faber are the contact persons for the course SA140X. If you are considering taking SA140X, with a specialization towards Algebra and Geometry, please do take contact at an early stage.
    Mats Boij, boij@kth.se
    Carel Faber, faber@kth.se



    KTH Matematik
    Sidansvarig Mats Boij