Class readings included both a textbook and approved papers from recent mathematical literature. In each class, two students were each assigned / asked to give a 35-minute presentation based on a chapter in the textbook or on one of the papers. With 16 students in the class, each student gave three presentations throughout the term.
The textbook chapter titles are listed below, since most of the student presentations were based upon the text:
Aigner, Martin, Günter M. Ziegler, and Karl Heinrich Hofmann. Proofs from THE BOOK. Springer, 2014. ISBN: 9783662442043. [Preview with Google books]
| CHAPTER # | PRESENTATION TOPICS / CHAPTER TITLES |
|---|---|
| Number Theory | |
| 1 | Six proofs of the infinity of primes |
| 2 | Bertrand’s postulate |
| 3 | Binomial coefficients are (almost) never powers |
| 4 | Representing numbers as sums of two squares |
| 5 | The law of quadratic reciprocity |
| 6 | Every finite division ring is a field |
| 7 | The spectral theorem and Hadamard’s determinant problem |
| 8 | Some irrational numbers |
| 9 | Three times π2/6 |
| Geometry | |
| 10 | Hilbert’s third problem: decomposing polyhedra |
| 11 | Lines in the plane and decompositions of graphs |
| 12 | The slope problem |
| 13 | Three applications of Euler’s formula |
| 14 | Cauchy’s rigidity theorem |
| 15 | The Borromean rings don’t exist |
| 16 | Touching simplices |
| 17 | Every large point set has an obtuse angle |
| 18 | Borsuk’s conjecture |
| Analysis | |
| 19 | Sets, functions, and the continuum hypothesis |
| 20 | In praise of inequalities |
| 21 | The fundamental theorem of algebra |
| 22 | One square and an odd number of triangles |
| 23 | A theorem of Pólya on polynomials |
| 24 | On a lemma of Littlewood and Offord |
| 25 | Cotangent and the Herglotz trick |
| 26 | Buffon’s needle problem |
| Combinatorics | |
| 27 | Pigeon-hole and double counting |
| 28 | Tiling rectangles |
| 29 | Three famous theorems on finite sets |
| 30 | Shuffling cards |
| 31 | Lattice paths and determinants |
| 32 | Cayley’s formula for the number of trees |
| 33 | Identities versus bijections |
| 34 | The finite Kakeya problem |
| 35 | Completing Latin squares |
| Graph Theory | |
| 36 | The Dinitz problem |
| 37 | Permanents and the power of entropy |
| 38 | Five-coloring plane graphs |
| 39 | How to guard a museum |
| 40 | Turán’s graph theorem |
| 41 | Communicating without errors |
| 42 | The chromatic number of Kneser graphs |
| 43 | Of friends and politicians |
| 44 | Probability makes counting (sometimes) easy |