# Math4991 Capstone Project in Pure Mathematics

Week 1: Tiling, by Min Yan

The history of tiling is as old as human civilisation. We review the rich history of the mathematics of tiling, and then concentrate on the tiling of sphere by congruent pentagons.

Homework: Exercise 1, 2, 7 in Note and exercise. Due on Friday, September 15.

Research papers.

Classification of tilings of the 2-dimensional sphere by congruent triangles by Ueno and Agaoka.

Combinatorial Tilings of the Sphere by Pentagons by Min Yan. Earth map tiling.

Spherical tiling by 12 congruent pentagons by Hong-Hao Gao, Nan Shi and Min Yan. Classification for the minimal case.

Tilings of Sphere by Congruent Pentagons I by Ka-Yue Cheuk, Ho-Man Cheung and Min Yan. Classification for variable edge length, and there is a 35-tile.

Tilings of Sphere by Congruent Pentagons II by Er-Xiao Wang and Min Yan. Classification for variable edge length, and there is no 35-tile.

Tilings of Sphere by Congruent Pentagons III by Yohji Akama and Min Yan. Classification of equilateral pentagonal tilings.

Week 2: Perspective Drawing and Projective Geometry, by Weiping Li

The perspective technique was developed in Western oil paintings. This gave rise to the new concepts of vanishing points and horizon, and led to the discovery of projective geometry. We will introduce the concept of projective spaces, and relate the projective geometry with Euclid geometry that you studied in high school.

Note.

Homework: Exercise 1 and second part of Exercise 2 in Note. Due on Friday, September 22.

Week 3: Poincaré-Bendixson Theorem, by Frederick Tszho Fong

Poincaré-Bendixson Theorem is a powerful and elegant theorem in both pure and applied mathematics. Its proof uses some very elegant and smart arguments in elementary analysis and topology. Its can be applied to study how chemical reactions are stable by predicting whether the ODE system has periodic solutions. We will go over the ingenious proof and its wonderful applications.

Exercises for week 3, due on Friday, September 29.

Week 4: Regular Polyhedra and Finite Symmetry Groups, by Jingsong Huang

This is a fascinating topic. There is a wealth of literature out there both in academic journals and popular websites and books. I list a couple of them here.

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A very nice mathematical introduction to symmetry and groups

Exercises for week 4, due on Friday, October 6.

Problem 1: Classify the regular polyhedra in R^3 (there are five of them).

Problem 2: Prove that the isometry group of Euclidean space Rn is the semi direct product of O(n) and Rn.

A potential project: Find all finite symmetry groups of R3. From now on, your homework should be typeset in LaTeX.

Week 5: Kepler Problem and Lorentz Transformations, by Guowu Meng

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Exercises for week 5, due on Friday, October 20.

Week 6: Hardamad Matrices and Reed-Mueller Code, by Kinyin Li

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Lecture and Exercises for week 6, due on Friday, October 27.

Week 7: Elliptic Curves and Theta Functions, by Yongchang Zhu

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Week 8: An introduction to Arithmetic and Riemann Surfaceâ€‹, by Huailiang Chang

Lecture. The homework is Exercises 3.1-3.5, due on Friday, November 10.

Week 9: Flows on Signed Graph, by Beifang Chen

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Week 10: Congruence Number Problem, by Maosheng Xiong

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Exercises for week 10, due on Wednesday, November 22.

Week 11: Picard's Theorem and Ramanujan, by Yikman Chiang

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Week 12: Random Sorting Networks, by Eric Marberg