UROP

My usual practise regarding UROP

UROP is only the formal framework for recognising research. For my projects, you ignore UROP and do research first. After you make enough progress, then we can think about how to fit the research into UROP. I will create new UROP project to fit you if necessary.

You can work on my projects anytime you like, within the semester or during the winter or summer break. There is absolutely no prerequisite such as your grade, your year of study, your major (even if you are a business student) etc.

You do not have to restrict yourself to the projects I listed. If you have any idea, please feel free to talk to me. If you idea makes sense, we can start a new project.

Tiling of the sphere by pentagon

The tilings of the sphere by congruent polygons can only have triangle, quadrilateral, or pentagon as the tiles. The classification (i.e., the complete list) of triangular tilings was started by Sommerville in 1923 and completely only recently in 2002 by Ueno and Agaoka.

The study of the pentagonal tilings was started by H.H. Gao and N. Shi (see paper no.4 below), for the classification of the minimal case. The follow up paper no.7 deals with the case of enough variation in arc length, and the paper no.8 deals with the case of no variation in arc length. The students are invited to complete the subsequent two papers. The third one is the remaining case of enough variation in arc length and should relatively easily follow the technique of paper no.7. The fourth is the most difficult, that I do not have the full confidence of completing.

A related research direction is the combinatorial aspects of pentagonal tiling. In this problem, we ignore the edges and angles, and concentrate on the topological structure only. Paper no.5 answers the case of one or two high degree vertices. Students are invited to study the case of three high degree vertices, and the case of many high degree vertices that are very evenly distributed.

1. J. Brooks, J. Strantzen: Spherical Triangles of Area π and Isosceles Tetrahedra.
2. Y. Ueno, Y. Agaoka: Classification of Tilings of the 2-Dimensional Sphere by Congruent Triangles.
3. Y. Ueno, Y. Agaoka: Examples of Spherical Tilings by Congruent Quadrangles.
4. H.H. Gao, N. Shi, M. Yan: Spherical Tiling by 12 Congruent Pentagons.
5. M. Yan: Combinatorial Tilings of the Sphere by Pentagons.
6. H.P. Luk, M. Yan: Angle Combinations in Spherical Tilings by Congruent Pentagons.
7. K.Y. Cheuk, H.M. Cheung, M. Yan: Tilings of the Sphere by Geometrically Congruent Pentagons I.
8. Y. Akama, M. Yan: Tilings of the Sphere by Geometrically Congruent Pentagons II.

Manipulation of series

In calculus, you learned that a conditionally convergent series can be rearranged to have any number as the sum. In this project, we impose some restrictions on the rearrangements and try to find out the possible limits. Since the answer will depend on the series we start with, the first series we wish to investigate is the alternating sum of 1/n.

Another manipulation of series is to consider subseries obtained by dropping some terms from the original series. The question is the range of the sum of all subseries. For example, what is the necessary and sufficient condition for the sums to form an interval?

1. R. Beigel: Rearranging Terms in Alternating Series.
2. F. Brown, L.O. Cannon, J. Elich, D.G. Wright: On Rearrangements of the Alternating Harmonic Series.
3. C.C. Cowen, K.R. Davidson, R.P. Kaufman: Rearranging the Alternating Harmonic Series.
4. L. Riddle: Rearranging the Alternating Harmonic Series.
5. H. Nover, A. Hajek: Vexing Expectations.
6. P. Schaefer: Sum-Preserving Rearrangements of Infinite Series.
7. R. Witula: On the Set of Limit Points of the Partial Sums of Series Rearranged by a Given Divergent Permutation.