MATH 489 – Section  900

Topics in Geometry and Topology


Instructor: Dr. Eric Rowell. email: rowell@math.tamu.edu, webpage: www.math.tamu.edu/~rowell. Office Blocker 510B

Place and Time: BLOC 605AX MWF 10:20-11:10am.

 Office hours:  Wednesdays 1-2pm or by appointment or drop in (if I am not too busy...)


Course Objectives: In this course you will develop geometric and topological intuition through a hands-on approach to major elementary results in geometry and topology. Although proofs of many theorems will be presented, the emphasis will be on understanding the concepts intuitively. You will learn to think geometrically and understand topological arguments. The format of the class will be approximately 50% lecture and 50% group activities.


The following areas will be covered, with demonstrations and applications included.
1. Euler characteristic, curvature, genus, Gauss-Bonnet Theorem
2. 6-color theorem
3. Classification of Archimedean solids, platonic solids in higher dimensions.
4. Wallpaper groups and symmetries.
5. Knots and braids, basic invariants, Reidemeister’s, Alexander’s and Markov’s theorems
6. Conway’s rational tangles
7. Fold and One Cut Theorem
8. Other topics as interest and time permits.

Text: Jeffrey Weeks: The Shape of Space (2nd edition): ISBN-13: 978-0824707095.

Grading: The grading will be based on journals (40%) that the students will turn in periodically (approximately every 3 weeks), class participation (10%), a final project that will be written (40%) and presented (10%). The journals will be on specific prompts, much like ordinary homework problems, and are graded equally on effort and correctness. For example a correct, 1 word answer will get at most 1/2 credit, whereas a well-thought-out but inconclusive explanation of an attempted solution will not be given a 0. The writing project will be turned in twice: once as a rough draft (25%) and again as a final draft (15%).  You must pass the writing component of the course to pass the course.


Exams: There are no  exams. There is NO in-class final exam.

Research Project: The writing project will be a research paper of 5-10 pages on a topic in geometry or topology: you should choose.

You may include images, figures and indented/block quotes, but these do not count towards the page constrants (so an 11 page paper with lots of images. might be ok, but a 5 page paper with images might not be).  Moreover, the works cited/bibliography does not count towards the page constraints.

Homework/Journal Entries: Prompts for journal entries will be given in class, which will be collected and graded as described above.  

Course Policies: Late homework and make-ups for missed exams will only be guaranteed for a university approved excuse in writing, other situtations (interviews etc.) will be considered on a case-by-case basis, but must always be supported with documentation. Wherever possible, students should inform the instructor before an exam or major assignment is missed. Consistent with University Student Rules, students are required to notify an instructor by the end of the next working day after missing an exam. Otherwise, they forfeit the right to a make-up.

An Aggie does not lie, cheat, or steal or tolerate those who do. Copying work that was done by others is an act of scholastic dishonesty and any instance of it will be prosecuted according to University Student Rules.

The Americans with Disabilities Act (ADA) is a federal anti-discrimination statute that provides comprehensive civil rights protection for persons with disabilities. Among other things, this legislation requires that all students with disabilities be guaranteed a learning environment that provides for reasonable accommodation of their disabilities. If you believe you have a disability requiring an accommodation, please contact Disability Services, currently located in the Disability Services building at the Student Services at White Creek complex on west campus or call 979-845-1637. For additional information visit http://disability.tamu.edu.

Writing Project/Plagiarism: Your Writing Project should be the product of your own research.  This means you should cite references properly (use any standard format you choose).  See this page by Prof. Gregory of Washington and Lee University for a description of plagiarism (which should be avoided, as it is heavily penalized).  Long quotes (4+ standard lines) should usually be indented/blocked, but remember these do not count towards your page constraints.  I reserve 5% of the writing project grades to reward good and interesting writing.  At least 3 sources must be cited, and at least 2 must be static (i.e. unchanging, unlike Wikepedia).  Help is available for general writing questions at The University Writing Center

 

Copyright Policy: All printed materials disseminated in class or on the web are protected by Copyright laws. One photocopy (or printout from the web) is allowed for personal use. Multiple copies or sale of any of these materials is strictly prohibited.


Important Announcements: 

    Key Dates (tentative):



Journal Entry Assignments:

  1. Due Sept. 14. Classify the capital letters A-Z (in the font provided in class) by a) symmetry (reflections, rotations, order) b) up to homeomorphism and c) up to homotopy.
  2. Due Sept. 14. Consider the following frequently encountered topological spaces:  an interval I=[0,1], a circle S1={(x,y): x2+y2=1} and a disk D=B2={(x,y):x2+y2≤1}.  a) Describe and name the following product spaces:  IxI, IxS1, S1xS1. DxS1, DxI. b)  Notice that S1 is 1 dimensional (a curve) while D=B2 is 2 dimensional.  What should S0 and S2 denote (describe them)? What about B1 and B3? c) Visual the following, and give a common name for the result (up to homeomorphism): i) glue the endpoints of I together ii) remove a point from a sphere iii) remove any point on D that is not on the boundary.  (feel free to draw pictures!)
  3. Due Sept. 14. Determine how many Frieze patterns there are, and draw a sample of each one.  Remember you must have translations, but you could also have vertical/horizontal reflections, 180 rotations and glide reflections.  (to guide your eye, you should box in a fundamental domain: a smallest piece that generates the whole pattern).
  4. Due Sept. 28  Notice that the vertex figure (5,4,3) is impossible for an Archimedean solid.  Explain why, and see if you can see this as part of a more general phenomenon.
  5. Due Sept. 28 Work out at least one example of a NS+1=EW Frieze pattern.  What kinds of patterns are possible?
  6. Due Sept. 28 Write a 1 page outline of your paper.
  7. Due Sept. 28 Try to use the constraints from class to show that there cannot by infinitely many Archimedean solids.  For example, if (n1,n2,...,nk) is a vertex figure, what is the largest value of k?  What about ni? You may use the fact that the total angle defect times the number of vertices must be 4*Pi, as well as the symmetry constraint from 4.
  8. Due Sept. 28 Write your thoughts on the lecture by Steve Simon on Friday Sept. 21.
  9. Due Oct. 19 Compute the Euler characteristic of: a disk, a torus with 2 holes, a sphere with 6 punctures/holes, a mobius strip, a cylinder, a Klein bottle, a torus with 3 punctures/holes.
  10. Due Oct. 19 Imagine the following 3-dimensional manifolds: a solid torus, a solid ball, the 3-sphere or the product of 3 circles (a 3-torus!): If you cut it up into cells (things that are homeomorhic to a 2-sphere) and compute V-E+F-C you get an invariant that generalizes the Euler characteristic.  Compute this for each of the above 3-manifolds.
  11. Due Oct. 19 Draw a picture (or make a model and photograph it! Mashed potatoes work well, or modeling clay) of an island with a single shoreline and 4 peaks, 2 pits and 5 passes.
  12. Due Oct. 19 Suppose a surface S has an irregular polygonalization with 15 pentagons, 9 squares, 12 triangles and 4 decagons.  How many edges does it have?  Is the surface closed?
  13. Due Oct. 19 Draw a graph on a torus that is not 6-colorable (but will be 7-colorable).
  14. Due Oct. 19 From the knot diagram given in class on 10/8, give a sequence of drawings that reduces the number of crossings to as few as possible. Draw a braid whose closure is the knot 5-2 from class. 
  15. Due Nov. 9 From the unicursal curve coming from the 5-crossing knot 5-2, consider all (32) of the possible knot projections.  Identify which knots can be obtained in this way.  Can both of the trefoil knots appear?  How often does the unknot appear?
  16. Due Nov. 9 We played some tic-tac-toe on the torus in class, and saw that the first player could always win. Is the same true for tic-tac-toe on a Klein bottle (see the text: Shape of Space)?
  17. Due Nov. 9 What surface do you get if you glue two hemispheres (disks) together using 3 mobius bands?  How many boundary components does it have?  What is its euler characteristic?  Is it orientable?  Reading Chapters 8,11,12 of the text could be useful.
  18. Due Nov. 9 Determine which rational knot has value 7/3.  You should try both the "square dance" way and the rational tangle way to make sure you understand both, and how to translate between them.
  19. Due Nov. 9 Do exercise 8.1 in the text (Shape of Space, 2nd edition), carefully explaining your answer.
  20. Due Nov. 30 Show that mirror symmetry in any wallpaper pattern can only have 1,2,3,4 or 6 mirrors at a point.
  21. Due Nov. 30 Take a photo of something with symmetry, and analyze the kinds of symmetry.
  22. Due Nov. 30 Draw a sample of each of the 17 kinds of wallpaper patterns.
  23. Due Nov. 30 Make a fold-and-one-cut pattern for some letter of the alphabet.