Summer 2003 `B' Term-- Proofs from the Book -- MAT 5932-24

Meets MTWRF 9:30-10:50 112 MCH
Instructor Steve Bellenot (bellenot at math.fsu.edu) 850.644.7189
OFFICE HOURS MTWR 11:00-11:30am
Text: Proofs From the Book, by Martin Aigner and Gunter Ziegler
Amazon link to the book.

Chapters spoken for include ch 5 -- Goce; and ch20 -- Fazhe;

Tentative order of topics for the five week. (Jun 9-13, 2003)

1. chap 4 Representing numbers as the sums of two squares
2. chap 8 Lines in the plane and decompositions of graphs
3. chap 18 On a lemma of Littlewood and Offord
4. chap 29 Turan's graph theorem

Tentative order of topics for the fourth week. (Jun 2-6, 2003)

1. chap 13 Every large set has an obtuse angle
2. chap 32 Probability makes counting (sometimes) easy.
3. chap 24 Caylay's formula for the number of trees
4. chap 17 A Theorem of Polya on polynomials

Tentative order of topics for the third week. (May 27-30, 2003)

1. chap 22 Three famous theorems on finite sets.
2. chap 19 Cotangent and the Herglotz trick
3. chap 2 Bertrands postulate (there is always a prime between n and 2n).
4. chap 13 Every large set has an obtuse angle -- but it is a short week so perhaps we will not get this far.

Tentative order of topics for the second week. (May 19-23, 2003)

1. chap 15 Sets, functions, and the continuum hypothesis
2. chap 6 Some irrational numbers
3. chap 7 Hilbert's third problem: decomposing polyhedra
4. chap 27 Five-coloring plane graphs

Tentative order of topics for the first week. (May 12-16, 2003)

1. 6 proofs of the infinity of primes (chap 1 part of Number Theory)
2. How to guard a museum (chap 28 from Graph theory)
3. Sperners's Lemma (sec 6 of chap 21 (Pigeon-hole and double counting) part of Combinatorics) and an application, the Brouwer fixed point theorem of topology.
4. In praise of inequalites (chap 16 part of Analysis) --Omitted
5. Three applications of Euler's formula (chap 10 part of Geometry)

• F 20 Jun:
  • Peano axioms are listed here
  • Characterization of |R as a complete ordered field. For example look here
  • The rest of chapter 29.
• R 19 Jun:
  • Chang -- Chapter 20 Buffon's needle
  • Undecidable, for many mathematical statements, undecidable => there do not exist counterexamples => true.
  • Non-standard analysis, how to have uncountable sets in a countable model,
• W 18 Jun:
  • Penny -- Book:Proofs and Refutations
  • James -- Math:Origami
• T 17 Jun:
  • Goce -- Chapter 5 finite division rings are fields
• M 16 Jun:
  • Erhan -- Book:Gamma
  • Edwin -- Math:The hat root
• F 13 Jun:
  • Guney -- Book:The man who loved only numbers
  • Bill -- Book:An Imaginary Tale: The story of sqrt(-1)
• R 12 Jun:
  • Chapter 29, the first 3 proofs.
• W 11 Jun:
  • Chapter 18, almost exactly as written.
  • Haar basis, Rademacher and Schauder systems.
• T 10 Jun:
  • Chapter 8, almost exactly as written. Also part of chap 10 we skipped
• M 9 Jun:
  • Chapter 4, almost exactly as written.
• F 6 Jun:
  • Handed out tentative schedule for student talks.
  • Finished chapter 17
  • discussion about chapter 4's topic. Every integer can be written as the sum of four squares, there are infinitely many like `7' that require more than 3 squares.
  • Waring's problem on cubes, fourth powers and so on.
• R 5 Jun:
  • Finished Cayley chapter 24 proof 4
  • Started chapter 17
  • Chebyshev polynomials.
• W 4 Jun:
  • Ramsey numbers, generalizations to unorder k-subsets, to the graphs on the infinte |N, and to large cardinals.
  • finished 32.
  • Started Cayley chapter 24 proof 1
• T 3 Jun:
  • Intro to prob.
  • Finished 13
  • Started chapter 32
• M 2 Jun:
  • Convex sets, extreme points (C - C)/2, functionals.
  • Chapter 13
• F 30 May:
  • How to quess a formula for sum i^2.
  • Allowing calculators in calculus and other math courses
  • The 400 year battle (1100-1500) between the abacists and the algorists.
  • a Mathematican is a person willing to spend 8 hours trying to figure out how to aviod 20 minutes of calculation.
  • Finished chap 2.
• R 29 May:
  • Bernoulli numbers, generating function definition, recurrence relation, some properties, what Jakob was looking them for.
  • Finished chap 19, got formula for zeta(2n) in terms of B_2n.
  • Started chap 2, stuff on estimating n choose k and the first two steps of the theorem.
• W 28 May:
  • Fixed the missing step in yesterday's thm2. If 1 in A, and cyclic rotation has 1 at the top, there are k choices for where in the arc the 1 goes. Then (k-1)! ways to fill the other A and (n-k)! to fill the non-A's.
  • Bernoulli's numbers introduction
  • Continuous f:|R->|R s.t. f(x+y)=f(x)+f(y) is f(x)=ax where a = f(1).
  • Normalized alternating mulitlinear is the determinate
  • Chap 19, proof of cot = partial fraction series
• T 27 May:
  • This is the week to decide what your big project will be on.
  • Three famous theorems on finite sets.
  • Missing step on thm2
  • A regular bipartite graph has a complete matching.
• F 23 May:
  • Books: Prime Obesession, Four colors suffice, Edwards book on the zeta function. [Topic "not `pretty proofs'"]
  • The four color theorem (here is a link to the new version [1])
  • finished the topics in chap 27 on five coloring.
• R 22 May:
  • Rest of chap 7 on Hilbert's 3rd problem.
  • Started 27, showed every planar graph can be 6-colored.
  • This used a lemma from ch 10 that a plane simple graph has a vertex of degree 5 or less.
  • Defined list coloring (actually from ch 26).
• W 21 May:
  • the rest of chap 6. Euler's original proof.
  • Start of chap 7. How to divide a cube into 3 pyramids. See this old calc 2 project actually this animation folding pyramids
• T 20 May:
  • chap 6, some irrational numbers. straight from book after some mention of square 2 and the pythagorians.
• M 19 May:
  • chap 15, sets, functions and the continuum hypothesis. Included a side track into 128 equivalents to the Axiom of Choice. A recent discussion why CH should be considered false http://www.u.arizona.edu/~chalmers/notes/continuum.html. Godel and Cohen results on the independence of CH and Axiom of Choice.
• F 16 May:
  • Millennium problems, Hilberts problems, more on the fix point theorem. Chap 10 three applications of Euler's formula. Picks theorem.
• R 15 May:
  • chap 21 (Pigeon-hole and double counting) Sperner lemma, Brouwer's fixed point theorem (with side track into n=1, IVT)
• W 14 May:
  • Riemann Hypothesis side track
  • How to guard a museum (chap 28 from Graph theory)
  • chap 21 (Pigeon-hole and double counting) introduction
  • Fixed Points.
• T 13 May:
  • How to guard a museum (chap 28 from Graph theory)
  • 6 proofs of the infinity of primes (chap 1 part of Number Theory)
  • Riemann Hypothesis side track
• M 12 May:
  • Erdos and Erdos numbers.
  • 6 proofs of the infinity of primes (chap 1 part of Number Theory) -- well we made it through 4.

The class "ad"

The great mathematican Paul Erdos said God maintains perfect mathematical proofs in ``The Book''. Aigner and Ziegler (with many suggestions from Erdos) have collected a number of candidates for such ``perfect proofs'', those which contain brilliant ideas, clever connections, and wonderful observations, bringing new insight and surprising perspectives to problems in number theory, geometry, analysis, combinatorics and graph theory. The book was to be a tribute to Erdos on his 85th birthday.

While there are results in all fields, most are inspired by the wide ranging interests of Paul Erdos. Proofs are chosen because of their simplicity and elegance.