Readings and Presentations

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]

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
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
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
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