1. Happy 60 th B’day Noga

2. Elementary problems encoding computational hardness Avi Wigderson IAS, Princeton or Some problems Noga never solved

3. Explicit object (graph, number, set,…) n = -Explainable - Reasonable - Efficiently constructible - Not random - Not generic

4. Linear transformations + + + + + + + + X1X1 X2X2 …. XnXn + + M: F n  F n y = Mx + + + + + + + + + + + + + + + + Y1Y1 Y2Y2 …. YnYn For most M, size(M) ≈ n 2 Challenge: Find M with size(M) ≠ O(n) c c’

5. Gauss complexity F any field M an n×n non-singular matrix over F. gc(M) = the smallest number of Gauss elimination steps to make M diagonal = min {s : M = E 1 E 2 … E s, E i elementary} For most M, gc(M) ≈ n 2 Challenge: Find explicit M with gc(M) ≠ O(n)

6. Matrix rigidity [Valiant] F any field M an n×n matrix over F. M “rigid” if a all matrices in Hamming ball around it has high rank rig(M) = the smallest number of nonzeros in a matrix S such that rank(M-S) < n/10 For most M, rig(M) ≈ n 2 Challenge: Find explicit M with rig(M) ≠ O(n)

7. Formula size ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ XnXn -X j XjXj -X 5 ∧ ∧ f: {0,1} n  {0,1} X1X1 X5X5 Fact: For most f, Fsize(f) ≥ exp(n) Thm Explicit f, Fsize(f) ≥ n 3 Challenge: Find f with Fsize(f) ≠ poly(n) [Andreev, Hastad,Tal]

8. Composition [Karchmer-Raz-W] f : {0,1} n  {0,1} g : {0,1} k  {0,1} f ° g : {0,1} nk  {0,1} Fact: For all f,g Fsize(f ° g) ≤ Fsize(f)  Fsize(g) Challenge: Prove Fsize(f ° g) ≥ α Fsize(f)  Fsize(g) for some α >0 f g gg  f g

9. Communication Complexity [Karchmer-W] Alice and Bob communicate to solve: Task: Find (i,j) which is an edge in one of H or G Task*: Find (i,j) which is an edge in H but not G Fact: comm(Task) ≤ O(n) bits Challenge: Prove comm(Task) ≠ O(log n) bits Thm[KW]: comm(Task) ≤ log Fsize(Hamilton) Thm[Raz-W]: comm(Task*) ≥ Ω (n) bits Alice non-Hamiltonian Hamiltonian Bob 1 2 3 4 0 5 6 7 8 9 1 2 3 4 0 5 6 7 8 9 GH 0 1 1

10. Permanent & Determinant Det n (X) =   Sn sgn(  )  i  [n] X i  (i) Per n (X) =   Sn  i  [n] X i  (i) Easy! Hard? × × × × + + + + × × XiXi XjXj XiXi c + + f Arithmetic circuits

11. Determinantal complexity [Valiant] Affine map L: M n (F)  M k (F) is good if Per n = Det k  L dc(n): the smallest k for which there is a good map Thm[Polya]: dc(2) =2 Per 2 = Det 2 Thm[Valiant]: dc(n) < exp(n) Thm[Mignon-Ressayre]: dc(n) > n 2 Challenge: Prove dc(n)  poly(n) Thm[V]: Implies exponential lower bounds for Permanent a b -c d a b c d

12. Polynomial identities [Heintz-Schnorr,Agrawal,Kabanets-Impagliazzo] Symbolic matrix M (0,1,X 1 …,X n ) Is det(M) = 0 identically ? Small Hitting Set (of small integer substitutions). Set k=n 10 (say). H = {v 1, v 2, … v k }, a set of vectors in [k] n det(M) ≠ 0  det(M(v i )) ≠ 0 for some i Most H are small hitting sets Challenge: Find an explicit H Thm[A,KI]: Implies exponential lower bounds for Permanent X1X1 X1X1 X2X2 XnXn X5X5 XnXn 1 0 1 0X1X1 n n

13. Elusive curves [Raz] Everything over C (works for other fields) Fact: The moment curve avoids every hyperplane! f(x) = (x,x 2,x 3,…, x n ) of degree n, avoids deg 1 maps: for every G: C n -1  C n of degree 1, Im(f) Im(G) Challenge: Find an explicit curve of degree poly(n) which avoids all degree 2 maps. Thm[Raz]: Implies exponential lower bounds for Permanent  /

14. Sum-of-Squares [Hrubes-W- Yehudayoff] (X 1 2 +X 2 2 + … +X k 2 )(Y 1 2 +Y 2 2 + … +Y k 2 ) = B 1 2 +B 2 2 + … +B n 2 B i =B i (X,Y) Bilinear functions. n = n(k) ≤ k 2 Do better? n=1 (X 1 2 )(Y 1 2 ) = (X 1 Y 1 ) 2 n=2 (X 1 2 +X 2 2 )(Y 1 2 +Y 2 2 ) = (X 1 Y 2 +X 2 Y 1 ) 2 + (X 1 Y 1 -X 2 Y 2 ) 2 n=4 (X 1 2 +…+X 4 2 )(Y 1 2 +…+Y 4 2 ) = B 1 2 +B 2 2 + … +B 4 2 Euler n=8 (X 1 2 +…+X 8 2 )(Y 1 2 +…+Y 8 2 ) = B 1 2 +B 2 2 + … +B 8 2 Hamilton Thm[Hurwitz] 1,2,4,8 are the only ones with n=k Thm[Radon] n Z (k) < k 2 /logk Thm[James] n R (k) > 2k Thm[HWY] n Z (k) > k 6/5 Challenge: Prove n C (k) > k 1+ ε Thm[HWY]  exponential non-commutative circuit lower bound

15. The Fusion method Everything over GF(2) (works for other fields) M 1, M 2, … M k, a list of n×n invertible matrices S a subset of [k]. M S = ∑ i  S M i Task: cover the universe {S: S odd, M S singular} P=(P 1,P 2,P 3 ) is a 3-partition of [k]. P covers S if all three |S  P j | is odd. cov(n) = min size of covering family Challenge: Prove cov(n)  O(n) Thm[R,KW] cov(n) is a circuit lower bound for Determinant n-bit primes composite Primality [Razborov, Karchmer,W] Approximation method Finite ultrafilters Approximation method Finite ultrafilters Fractional cover O(n)

16. Why do Lower bounds manifest themselves in so many ways? Computation is everywhere!

17. Happy 60 th B’day Noga