Cool theorems proved by undergraduates Ken Ono Emory University - PowerPoint PPT Presentation

Cool theorems proved by undergraduates Recurrence sequences Lucas Sequences A Lucas sequence ( b , c ) has characteristic polynomial and roots √ b 2 + 4 c α, β = b ± g ( z ) = z 2 − bz − c , . 2 Fact 1. u n = α n − β n v n = α n + β n α − β 2. u 2 k = u k v k 3. ( b 2 + 4 c ) u 2 n = v 2 n − 4( − c ) n

Cool theorems proved by undergraduates Recurrence sequences Specific Examples Theorem (Silliman, Vogt) For the following values of b and c: ( b , c ) = (3 , − 2) , (5 , − 6) , (7 , − 12) , (17 , − 72) , (9 , − 20) the Lucas sequence u n has no nontrivial pth powers, except u 2 = 3 2 in (9 , − 20) .

Cool theorems proved by undergraduates Recurrence sequences General Bound Conjecture (Frey-Mazur) Let E , E ′ be two elliptic curves defined over Q . If E [ p ] ∼ = E ′ [ p ] for some p > 17 , then E and E ′ are isogenous.

Cool theorems proved by undergraduates Recurrence sequences General Bound Conjecture (Frey-Mazur) Let E , E ′ be two elliptic curves defined over Q . If E [ p ] ∼ = E ′ [ p ] for some p > 17 , then E and E ′ are isogenous. Theorem (Silliman, Vogt) Assume the Frey Mazur conjecture. Then a pth power u n = y p satisfies p ≤ Ψ( b , c ) . where Ψ( b , c ) is an explicit constant.

Cool theorems proved by undergraduates Recurrence sequences More Examples Example The sequence (3 , 1): 0 , 1 , 3 , 10 , 33 , 109 , 360 , 1189 , 3927 , 12970 , 42837 , 141481 , · · · We haven’t found any nontrivial perfect powers . . .

Cool theorems proved by undergraduates Recurrence sequences More Examples Example The sequence (3 , 1): 0 , 1 , 3 , 10 , 33 , 109 , 360 , 1189 , 3927 , 12970 , 42837 , 141481 , · · · We haven’t found any nontrivial perfect powers . . . Theorem (Silliman, Vogt) Assume the Frey-Mazur Conjecture. There are no nontrivial powers in the sequences ( b , c ) = (3 , 1) , (5 , 1) , and (7 , 1) .

Cool theorems proved by undergraduates Recurrence sequences

Cool theorems proved by undergraduates Partitions Beautiful identities

Cool theorems proved by undergraduates Partitions Beautiful identities Euler proved that ∞ (1 − q 24 n ) = q − q 25 − q 49 + q 121 + q 169 − q 289 · · · . � q n =1

Cool theorems proved by undergraduates Partitions Beautiful identities Euler proved that ∞ (1 − q 24 n ) = q − q 25 − q 49 + q 121 + q 169 − q 289 · · · . � q n =1 Jacobi proved that ∞ (1 − q 8 n ) 3 = q − 3 q 9 + 5 q 25 − 7 q 49 + 9 q 81 − 11 q 121 + · · · . � q n =1

Cool theorems proved by undergraduates Partitions Beautiful identities Euler proved that ∞ (1 − q 24 n ) = q − q 25 − q 49 + q 121 + q 169 − q 289 · · · . � q n =1 Jacobi proved that ∞ (1 − q 8 n ) 3 = q − 3 q 9 + 5 q 25 − 7 q 49 + 9 q 81 − 11 q 121 + · · · . � q n =1 Gauss proved that ∞ (1 − q 16 n ) 2 � (1 − q 8 n ) = q + q 9 + q 25 + q 49 + q 81 + q 121 + q 169 + q 225 + · · · q n =1

Cool theorems proved by undergraduates Partitions Such rare identities have been discovered by:

Cool theorems proved by undergraduates Partitions Such rare identities have been discovered by: Crazy combinatorial manipulation of power series. Higher identities such as Jacobi’s ∞ z 2 m q m 2 . � � (1 − q 2 n )(1 + z 2 q 2 n − 1 )(1 + z − 2 q 2 n − 1 ) = n =1 m ∈ Z Modular forms.

Cool theorems proved by undergraduates Partitions Nekrasov-Okounkov Theory

Cool theorems proved by undergraduates Partitions Nekrasov-Okounkov Theory Deeper structure for such identities.

Cool theorems proved by undergraduates Partitions Nekrasov-Okounkov Theory Deeper structure for such identities. One doesn’t have to multiply out and combine terms.

Cool theorems proved by undergraduates Nekrasov-Okounkov Partitions Definition A nonincreasing sequence of positive integers summing to n is a partition of n .

Cool theorems proved by undergraduates Nekrasov-Okounkov Partitions Definition A nonincreasing sequence of positive integers summing to n is a partition of n . p ( n ) := # { partitions of n } .

Cool theorems proved by undergraduates Nekrasov-Okounkov Partitions Definition A nonincreasing sequence of positive integers summing to n is a partition of n . p ( n ) := # { partitions of n } . Example { Partitions of 4 } = { 4 , 3 + 1 , 2 + 2 , 2 + 1 + 1 , 1 + 1 + 1 + 1 }

Cool theorems proved by undergraduates Nekrasov-Okounkov Partitions Definition A nonincreasing sequence of positive integers summing to n is a partition of n . p ( n ) := # { partitions of n } . Example { Partitions of 4 } = { 4 , 3 + 1 , 2 + 2 , 2 + 1 + 1 , 1 + 1 + 1 + 1 } = ⇒ p (4) = 5 .

Cool theorems proved by undergraduates Nekrasov-Okounkov Generating function for p ( n )

Cool theorems proved by undergraduates Nekrasov-Okounkov Generating function for p ( n ) Lemma We have that ∞ ∞ 1 p ( n ) q n = � � 1 − q n . n =0 n =1

Cool theorems proved by undergraduates Nekrasov-Okounkov Wishful thinking

Cool theorems proved by undergraduates Nekrasov-Okounkov Wishful thinking Question Is there a “combinatorial theory” of infinite products where coefficient of q n = “formula in partitions of n ” ?

Cool theorems proved by undergraduates Nekrasov-Okounkov Hooklengths of partitions

Cool theorems proved by undergraduates Nekrasov-Okounkov Hooklengths of partitions Definition Hooks are the maximal ∗ ∗ ∗···∗ ∗ ∗ . . . ∗ in the Ferrers board of λ .

Cool theorems proved by undergraduates Nekrasov-Okounkov Hooklengths of partitions Definition Hooks are the maximal ∗ ∗ ∗···∗ ∗ ∗ . . . ∗ in the Ferrers board of λ . Hooklengths are their lengths, and H ( λ ) = { multiset of hooklengths of λ }

Cool theorems proved by undergraduates Nekrasov-Okounkov An example Example ( λ = 5 + 3 + 2) 7 6 4 2 1 4 3 1 2 1

Cool theorems proved by undergraduates Nekrasov-Okounkov An example Example ( λ = 5 + 3 + 2) 7 6 4 2 1 4 3 1 2 1 An so H ( λ ) = { 1 , 1 , 1 , 2 , 2 , 3 , 4 , 4 , 6 , 7 } .

Cool theorems proved by undergraduates Nekrasov-Okounkov Nekrasov-Okounkov q -series Definition For z ∈ C , let 1 − z q | λ | · � � � � O z ( q ) := . h 2 λ h ∈H ( λ )

Cool theorems proved by undergraduates Nekrasov-Okounkov Example ( λ = 5 + 3 + 2) 7 6 4 2 1 4 3 1 2 1

Cool theorems proved by undergraduates Nekrasov-Okounkov Example ( λ = 5 + 3 + 2) 7 6 4 2 1 4 3 1 2 1 The λ - contribution to O z ( q ) is 1 − z q 10 · � � � h 2 h ∈H ( λ ) 1 − z 1 − z 1 − z 1 − z 1 − z � 2 � � 2 � = q 10 (1 − z ) 3 � � � � � � 4 9 16 36 49

Cool theorems proved by undergraduates Nekrasov-Okounkov A famous identity revisited

Cool theorems proved by undergraduates Nekrasov-Okounkov A famous identity revisited Letting z = 4, we consider � 1 − 4 � q | λ | · � � O 4 ( q ) = . h 2 λ h ∈H ( λ )

Cool theorems proved by undergraduates Nekrasov-Okounkov A famous identity revisited Letting z = 4, we consider � 1 − 4 � q | λ | · � � O 4 ( q ) = . h 2 λ h ∈H ( λ ) We only need λ where every hook h � = 2.

Cool theorems proved by undergraduates Nekrasov-Okounkov A famous identity revisited Letting z = 4, we consider � 1 − 4 � q | λ | · � � O 4 ( q ) = . h 2 λ h ∈H ( λ ) We only need λ where every hook h � = 2. = ⇒ { Triangular partitions 1 + 2 + · · · + k }

Cool theorems proved by undergraduates Nekrasov-Okounkov A famous identity revisited.

Cool theorems proved by undergraduates Nekrasov-Okounkov A famous identity revisited. � (1 − 4 / h 2 ) λ 8 | λ | + 1 H ( λ ) 0 1 φ 1 1 9 { 1 } − 3 2 + 1 25 { 1 , 1 , 3 } 5 3 + 2 + 1 49 { 1 , . . . , 5 } − 7

Cool theorems proved by undergraduates Nekrasov-Okounkov A famous identity revisited. � (1 − 4 / h 2 ) λ 8 | λ | + 1 H ( λ ) 0 1 φ 1 1 9 { 1 } − 3 2 + 1 25 { 1 , 1 , 3 } 5 3 + 2 + 1 49 { 1 , . . . , 5 } − 7 And so we have ⇒ q O 4 ( q 8 ) = q − 3 q 9 + 5 q 25 − 7 q 49 + · · · =

Cool theorems proved by undergraduates Nekrasov-Okounkov A famous identity revisited. � (1 − 4 / h 2 ) λ 8 | λ | + 1 H ( λ ) 0 1 φ 1 1 9 { 1 } − 3 2 + 1 25 { 1 , 1 , 3 } 5 3 + 2 + 1 49 { 1 , . . . , 5 } − 7 And so we have ⇒ q O 4 ( q 8 ) = q − 3 q 9 + 5 q 25 − 7 q 49 + · · · = ∞ Jacobi ? � (1 − q 8 n ) 3 = q n =1

Cool theorems proved by undergraduates Nekrasov-Okounkov Big Identity

Cool theorems proved by undergraduates Nekrasov-Okounkov Big Identity Theorem (Nekrasov-Okounkov) If z is complex, then ∞ 1 − z q | λ | · � � � � � (1 − q n ) z − 1 . O z ( q ) := = h 2 n =1 λ h ∈H ( λ )

Cool theorems proved by undergraduates Nekrasov-Okounkov Big Identity Theorem (Nekrasov-Okounkov) If z is complex, then ∞ 1 − z q | λ | · � � � � � (1 − q n ) z − 1 . O z ( q ) := = h 2 n =1 λ h ∈H ( λ ) Remark Letting z = 0 gives the generating function for p ( n ) .

Cool theorems proved by undergraduates Nekrasov-Okounkov Euler, Gauss, and Jacobi-type identities

Cool theorems proved by undergraduates Nekrasov-Okounkov Euler, Gauss, and Jacobi-type identities Question 1 When do the sums below vanish? � 1 − ab � � � A ( a , b ; n ) := . h 2 | λ | = n h ∈H a ( λ )

Cool theorems proved by undergraduates Nekrasov-Okounkov Euler, Gauss, and Jacobi-type identities Question 1 When do the sums below vanish? � 1 − ab � � � A ( a , b ; n ) := . h 2 | λ | = n h ∈H a ( λ ) 2 Are there more identities of Euler, Gauss and Jacobi-type?

Cool theorems proved by undergraduates Nekrasov-Okounkov Euler, Gauss, and Jacobi-type identities Question 1 When do the sums below vanish? � 1 − ab � � � A ( a , b ; n ) := . h 2 | λ | = n h ∈H a ( λ ) 2 Are there more identities of Euler, Gauss and Jacobi-type? 3 If so, find them all .

Cool theorems proved by undergraduates Nekrasov-Okounkov A cool theorem

Cool theorems proved by undergraduates Nekrasov-Okounkov A cool theorem Theorem (Clader, Kemper, Wage) The list of all pairs ( a , b ) for which “almost all” the A ( a , b ; n ) vanish are (1 , 2) , (1 , 3) , (1 , 4) , (1 , 5) , (1 , 7) , (1 , 9) , (1 , 11) , (1 , 15) , (1 , 27) , (2 , 2) , (2 , 3) , (2 , 5) , (2 , 7) , (3 , 3) , (3 , 5) , (3 , 9) , (4 , 5) , (4 , 7) , (7 , 9) , (7 , 15) .

Cool theorems proved by undergraduates Nekrasov-Okounkov A cool theorem Theorem (Clader, Kemper, Wage) The list of all pairs ( a , b ) for which “almost all” the A ( a , b ; n ) vanish are (1 , 2) , (1 , 3) , (1 , 4) , (1 , 5) , (1 , 7) , (1 , 9) , (1 , 11) , (1 , 15) , (1 , 27) , (2 , 2) , (2 , 3) , (2 , 5) , (2 , 7) , (3 , 3) , (3 , 5) , (3 , 9) , (4 , 5) , (4 , 7) , (7 , 9) , (7 , 15) . Remark These pairs are the Euler, Gauss and Jacobi identities for ∞ (1 − q an ) b A ( a , b ; n ) q n := � � (1 − q n ) . n =1

Cool theorems proved by undergraduates Nekrasov-Okounkov

Cool theorems proved by undergraduates Number Fields Number fields Wrong Definition This is a number field .

Cool theorems proved by undergraduates Number Fields Number fields Wrong Definition This is a number field . .

Cool theorems proved by undergraduates Number Fields Number fields Wrong Definition This is a number field . . Definition A finite dimensional field extension of Q is called a number field .

Cool theorems proved by undergraduates Number Fields An invariant Remark The discriminant ∆ K ∈ Z \ { 0 } of a number field K does: