David M. Bressoud Macalester College, St. Paul, MN Talk given MAA KY section March 28, 2008

Transcription

1 David M. Bressoud Macalester College, St. Paul, MN Talk given MAA KY section March 28, 2008

2 We mathematicians often delude ourselves into thinking that we create proofs in order to establish truth. In fact, that which is "proven" is often not true, and mathematical results are often known with certainty to be true long before a proof is found.

3 Imre Lakatos, Born Imre Lipschitz, Jewish, Hungarian Changed name to Imre Molnar during war so as not to be identified as Jewish Mother and grandmother died in Auschwitz 1944, graduated college with degrees in mathematics, physics, and philosophy After war changed his name to Imre Lakatos (Locksmith) because he had shirts monogrammed IL.

4 Imre Lakatos, Studied in Budapest and at Moscow State University 1947: position in Ministry of Education. 1950: arrested, three years in prison 1956: Hungarian uprising, flees to Vienna, then on to England 1961: PhD in Philosophy at Cambridge

5 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.

6 Lakatos: Is this relevant to Mathematics?

7 Lakatos: Is this relevant to Mathematics? Icosahedron 20 faces 30 edges 12 vertices = 2 Theorem (Euler): For all polyhedra, V E + F = 2.

8 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.

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

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

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

12 Great Stellated Dodecahedron

13 Great Stellated Dodecahedron

14 Great Stellated Dodecahedron V = 20 E = = 30 F = = = 2

15 Great Stellated Dodecahedron V = 20 E = 30 F = = 2 What s different?

16 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).

17 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.

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

19 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

20 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

21 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

22 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 < $.

23 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.

24 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)

25 Abel, 1826: It appears to me that this theorem suffers exceptions. sin x! 1 2 sin2x sin 3x! 1 sin 4x +! 4

26 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) x depends on n n depends on x

27 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)! S ( n a) + R ( n x) + R ( n a) x depends on n n depends on x

28 The existence of a proof does not (necessarily) imply the truth of a mathematical statement. But also, the absence of a proof does not (necessarily) mean that we are in any doubt about the truth of a mathematical statement. Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture

29 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

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

31 n A n = = = = = =

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

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

34

35

36 ! $ # 0????& # & # 0????& # 0???? & " # 0????% &

37 1 1 2/ /3 3 3/ / / / / / /2 429

38 1 1 2/ /3 3 3/ /4 14 5/5 14 4/ / / / / / / / / /2 429

39 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

40

41 Numerators:

42 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& '

43 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)!

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

45 Exactly the formula found by George Andrews for counting descending plane partitions. The attempt to find the relationship between descending plane partitions and alternating sign matrices would lead to proofs of important open problems in the study of plane partitions, with connections to representation theory. George Andrews Penn State

46 December, 1992 Zeilberger announces a proof that # of ASM s equals n!1 " j =0 ( 3j + 1)! ( n + j)! 1995 all gaps removed, published as Proof of the Alternating Sign Matrix Conjecture, Elect. J. of Combinatorics, 1996.

47 Zeilberger s proof is an 84-page tour de force, but it still left open the original conjecture: 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& '

48 1996 Kuperberg announces a simple proof Another proof of the alternating sign matrix conjecture, International Mathematics Research Notices Greg Kuperberg UC Davis

49 1996 Kuperberg announces a simple proof Another proof of the alternating sign matrix conjecture, International Mathematics Research Notices Greg Kuperberg UC Davis Physicists have been studying ASM s for decades, only they call them square ice (aka the six-vertex model ).

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

51 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

52 Soichi Okada, Nagoya University 2004, Okada shows that the formulas for counting ASM s, including those subject to symmetry conditions, are simply the dimensions of certain irreducible representations, i.e. specializations of Weyl Character formulas.

53 This PowerPoint will be available at