SENIOR SEMINAR

OFFICE HOURS: Tuesday 1-3:30 and Thursday 1-3:30; other times by discovery.

TEXT: Proofs from THE BOOK, Third Edition, by Martin Aigner and Gunter M. Ziegler.

Number Theory:
      **1. Six proofs of the infinity of primes: Jeshon
      **2. Bertrand's postulate: Nick
      3. Binomial coefficients are (almost) never powers
      **4. Representing numbers as sums of two squares: Jennifer P.
      5. Every finite division ring is a field
      **6. Some irrational numbers: Beth
      **7. Three times pi squared over 6: Pye
Geometry:
      8. Hilbert's third problem: decomposing polyhedra
      **9. Lines in the plane and decompositions of graphs: Jennifer R.
      **10. The slope problem: Jennifer P.
      **11. Three applications of Euler's formula: Beth
      12. Cauchy's rigidity theorem
      13. Touching simplices
      **14. Every large point set has an obtuse angle: Sam
      15. Borsuk's conjecture
Analysis:
      16. Sets, functions, and the continuum hypothesis
      **17. In praise of inequalities: Pye
      18. A theorem of Polya on polynomials
      19. On a lemma of Littlewood and Offord
      **20. Cotangent and the Herglotz trick: Dusty
      **21. Buffon's needle problem: Aaron
Combinatorics:
      **22. Pigeon-hole and double counting: Nick
      **23. Three famous theorems on finite sets: Sam
      24. Shuffling cards
      25. Lattice paths and determinants
      **26. Cayley's formula for the number of trees: Dusty
      **27. Completing Latin squares: Jeshon
      28. The Dinitz problem
      29. Identities versus bijections
Graph Theory:
      30. Five-coloring plane graphs
      **31. How to guard a museum: Jennifer R.
      32. Turan's graph theorem
      33. Communicating without errors
      **34. Of friends and politicians: Aaron
      35. Probability makes counting (sometimes) easy

COURSE OBJECTIVES: Mathematics 402 is required of all senior mathematics majors. Each student chooses two chapters from the above list (no two from the same category), then delivers a 30-45 minute oral presentation to the class on each. The student will experience reading, learning, and presenting interesting mathematical results, indeed results considered beautiful for their elegance or insightfulness, that may not normally be seen in our typical course sequence.

GRADING:
	Preparation	15%
	Presentations	80%
	Attendance and Participation	5%

COMMENTS:
1. The topics you research and the resultant talks are on mathematical results that belong in The Book, a collection of perfect proofs envisioned by Paul Erdos (1913-1996) to be maintained by God. We hope that you enjoy these gems from our fellow mathematicians.
2. The talks are to be 30-45 minutes in length, depending on the material; this will take planning on your part. You might want to give the talk to friends or a fictitious audience before your actual presentation. You should feel rather well prepared a week ahead of your scheduled talk. You will find that during the last week many unexpected questions will come to mind; it is then that you are really coming to understand the depth of your presentation.
3. The talks are to be well-planned, using teaching aids such as overhead transparencies, class handouts, power point, colored chalk, et cetera as appropriate. We have some supplies available in the mathematics and computer science office.
4. During and/or after each talk members of the class may ask questions or seek clarifications.
5. Absolutely feel free to stop by to discuss problem areas that you encounter.
6. Chapters and presentation dates will be reserved on a first come, first served basis.
7. Timeliness on the choice of topics and dates is expected.