David Bressoud Macalester College, St. Paul, MN

Transcription

1 MAA David Bressoud Macalester College, St. Paul, MN PowerPoint available at Pacific Northwest Section Juneau, AK June 24, 2011

2 Imre Lakatos, Hungarian. Born Imre Lipschitz Changed his name to Imre Lakatos (Locksmith) because he had shirts monogrammed IL. 1956: Hungarian uprising, flees to Vienna, then on to England 1961: PhD in Philosophy at Cambridge, with help from George Pólya.

3 Sir Karl Popper, The Logic of Scientific Discovery Science advances by 1. Observing nature 2. Creating a theory to explain what is happening 3. Looking for consequences of this theory 4. Testing the predicted consequences 5. Adjusting the theory when predictions do not pan out In science, nothing can be proven to be true. Real progress in science comes from establishing that something is false.

4 Lakatos: Is this relevant to Mathematics?

5 Lakatos: Is this relevant to Mathematics? Icosahedron 20 faces 30 edges 12 vertices = 2 Theorem (Euler): For all polyhedra, V E + F = 2. Definition: A polyhedron is a solid whose surface consists of polygonal faces.

6 V = 16 E = 24 F = = 4 V = 7 E = 12 F = = 3

7 For any polygon, V = E. Can even be a self-intersecting polygon:

8 Small Stellated Dodecahedron 12 faces (pentagrams) 30 edges 12 vertices = 6

9 Great Stellated Dodecahedron V = 20 E = = 30 F = = = 2 What s different? Every closed curve made up of edges is the boundary of a chain of contiguous faces (faces that share an edge).

10 Here we have a closed circuit of faces that is not the boundary of the solid. Here we have a closed circuit of edges that is not the boundary of a chain of contiguous faces.

11 The appendix from Lakatos s Proof and Refutations would be the inspiration for my own A Radical Approach to Real Analysis

12 Cauchy, Cours d analyse, 1821 explanations drawn from algebraic technique cannot be considered, in my opinion, except as heuristics that will sometimes suggest the truth, but which accord little with the accuracy that is so praised in the mathematical sciences

13 Niels Henrik Abel (1826): Cauchy is crazy, and there is no way of getting along with him, even though right now he is the only one who knows how mathematics should be done. What he is doing is excellent, but very confusing

14 Cauchy, Cours d analyse, 1821, p. 120 Theorem 1. When the terms of a series are functions of a single variable x and are continuous with respect to this variable in the neighborhood of a particular value where the series converges, the sum S(x) of the series is also, in the neighborhood of this particular value, a continuous function of x. ( ) = f k ( x) S x, f k continuous S continuous k =1

15 S n n ( x) = f k ( x), R n x k =1 ( ) = S( x) S n ( x) ( x) as small as we wish Convergence can make R n by taking n sufficiently large. S n is continuous for n <.

16 S n n ( x) = f k ( x), R n x k =1 ( ) = S( x) S n ( x) ( x) as small as we wish Convergence can make R n by taking n sufficiently large. S n is continuous for n <. S continuous at a if can force S(x) - S(a) as small as we wish by restricting x a.

17 S n n ( x) = f k ( x), R n x k =1 ( ) = S( x) S n ( x) ( x) as small as we wish Convergence can make R n by taking n sufficiently large. S n is continuous for n <. S continuous at a if can force S(x) - S(a) as small as we wish by restricting x a. S( x) S( a) = S ( n x) + R ( n x) S ( n a) R ( n a) S n ( x) S ( n a) + R ( n x) + R ( n a)

18 Abel, 1826: It appears to me that this theorem suffers exceptions. sin x 1 2 sin 2x sin 3x 1 sin 4x + 4

19 S n n ( x) = f k ( x), R n x k =1 ( ) = S( x) S n ( x) ( x) as small as we wish Convergence can make R n by taking n sufficiently large. S n is continuous for n <. S continuous at a if can force S(x) - S(a) as small as we wish by restricting x a. S( x) S( a) = S ( n x) + R ( n x) S ( n a) R ( n a) S n x depends on n ( x) S ( n a) + R ( n x) + R ( n a) n depends on x

20 S n n ( x) = f k ( x), R n x k =1 ( ) = S( x) S n ( x) ( x) as small as we wish Convergence can make R n by taking n sufficiently large. S n is continuous for n <. Uniform convergence eliminates the possibility that n depends on x. S( x) S( a) = S ( n x) + R ( n x) S ( n a) R ( n a) S n x depends on n ( x) S ( n a) + R ( n x) + R ( n a) n depends on x

21 A mathematician [is] an observer, a man who gazes at a distant range of mountains and notes down his observations There are some peaks which he can distinguish easily, while others are less clear. He see A sharply, while of B he can obtain only transitory glimpses. At last he makes out a ridge which leads from A, and following it to its end he discovers that it culminates in B. B is now fixed in his vision, and from this point he can proceed to further discoveries. G. H. Hardy, Rouse Ball Lecture, 1928.

22 I see the mathematician as someone who has parachuted into dense woods and needs to find her (or his) way back to familiar territory.

23 Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture

24 Alternating sign matrix: Square matrix of 1 s, 1 s, and 0 s Each row and column adds to 1 Nonzero entries in any row or column alternate in sign

25 n A n How many n n alternating sign matrices? = = = = = =

26 There is exactly one 1 in the first row n 1 A n

27 There is exactly one 1 in the first row n 1 A n

28

29

30 ???? 0???? 0???? 0????

31 1 1 2/ /3 3 3/ / / / / / /2 429

32 1 1 2/ /3 3 3/ /4 14 5/5 14 4/ / / / / / / / / /2 429

33 2/2 2/3 3/2 2/4 5/5 4/2 2/5 7/9 9/7 5/2 2/6 9/14 16/16 14/9 6/2

34

35 Numerators:

36 Numerators: Conjecture 1: A n,k A n,k +1 = n 2 k 1 + n 1 k 1 n 2 n k 1 + n 1 n k 1

37 Conjecture 1: A n,k A n,k +1 = n 2 k 1 + n 1 k 1 n 2 n k 1 + n 1 n k 1 Conjecture 2 (corollary of Conjecture 1): A n = n 1 ( 3j + 1)! = ( n + j)! j =0 1! 4! 7! ( 3n 2 )! n! ( n + 1)! ( 2n 1)!

38 Exactly the formula found by George Andrews for counting descending plane partitions. Conjecture 2 (corollary of Conjecture 1): George Andrews Penn State A n = n 1 ( 3j + 1)! = ( n + j)! j =0 1! 4! 7! ( 3n 2 )! n! ( n + 1)! ( 2n 1)!

39 Connections to partitions, determinant evaluations, orthogonal polynomials, counting lattice paths, tiling problems. Connection to representation theory led to Proof of the Macdonald Conjecture, (Mills, Robbins, Rumsey, Inv. Math., 1982). Previously described by Richard Stanley as the most interesting open problem in all of enumerative combinatorics. Ian Macdonald University College, London

40 1996 Kuperberg announces a simple proof of A n = n 1 j =0 ( 3j + 1)! ( n + j)! Physicists had been studying ASM s for decades, only they call them the six-vertex model Greg Kuperberg UC Davis

41 Horizontal = 1 Vertical = 1

42 Rodney J. Baxter Australian National University Anatoli Izergin Vladimir Korepin, SUNY Stony Brook

43 1996 Doron Zeilberger uses the connection to statistical mechanics to prove the original conjecture Proof of the refined alternating sign matrix conjecture, New York Journal of Mathematics

44 These slides are available at