Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

This summer I will have a chance to work on a 16-week summer research project under a professor in convex/discrete geometry. I'm a first-year student with a fairly good background for my age and I've been working through a graduate text in convex geometry, "Introduction to Convex Polytopes" by Brondsted. I need to decide with my supervisor what topic to focus on for a research proposal.

In general I am interested in any topics related to properties of polytopes, tesselations of spaces, illumination problems, packing of convex bodies, etc.

Which of these areas would be most accessible for a first-year student to work on (around 8-weeks to study and learn the background of the problem, and around 8-weeks to work on the problem and continue to learn background material). I'm interested in what sub-field of convex geometry/discrete geometry offers the opportunity to work on a problem that would fit within this time schedule with a potential tangible result towards the solution.

  • Sphere Packings (kissing numbers, density, voronoi cells of sphere packings, etc.)
  • Finite Packings by Translates of Convex Bodies (Hadwiger numbers, touching numbers)
  • Illumination Conjecture and Related Problems
  • Coverings by Planks and Cylinders (Covering with hyperplanes, etc.)
  • Volumes of Sphere Arrangements (Kneser-Poulsen Conjecture, etc.)
  • Ball-Polyhedra (rigidity, disk-polygons, spindle convex sets)
  • Convex Polytopes (antipodality, supporting hyperplanes, etc.)

Any suggestions as to what sort of area of topics (please be at least as specific as the list I've just mentioned) I might be able to access would be very helpful! So far I have a basic understanding of some topics to do with affine spaces, basic topology, graph theory, knot theory, hyperplanes, and convex sets.

Thanks!

share|improve this question

closed as too localized by J. M., Potato, Danny Cheuk, Micah, Quixotic Jun 16 '13 at 5:34

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

5  
What your supervisor suggests, after some discussion. –  André Nicolas Jan 28 '12 at 2:32
1  
I'm sure that it is not an unfair question to ask what areas of discrete geometry are more accessible than others? –  Samuel Reid Jan 28 '12 at 2:46
4  
It is certainly a fair question. But most areas of mathematics have accessible problems and not so accessible problems. Your supervisor will know intimately about accessible problems in her/his general research area, and will be able to provide assistance when you have difficulties. Congratulations on having earned the opportunity to do supervised research so early! –  André Nicolas Jan 28 '12 at 2:53
1  
Günter Ziegler's Lectures on Polytopes brings you closer to the forefront of research than does Brøndsted, in my opinion. –  Joseph O'Rourke Jan 30 '12 at 0:38
4  
@Samuel: I don't think your question is answerable, as it depends upon personal preferences. They are all accessible; they are all inaccessible. Choose whichever topic appeals to you most aesthetically, and the attraction will fuel your research. –  Joseph O'Rourke Jan 30 '12 at 15:27