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Theoretical Computer Science methods in asymptotic geometry Avi Wigderson IAS, Princeton For Vitali Milman’s 70 th birthday

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Three topics: Methods and Applications Parallel Repetition of games and Periodic foams Zig-zag Graph Product and Cayley expanders in non-simple groups Belief Propagation in Codes and L 2 sections of L 1

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Parallel Repetition of Games and Periodic Foams

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Isoperimetric problem: Minimize surface area given volume. One bubble. Best solution: Sphere

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Many bubbles Isoperimetric problem: Minimize surface area given volume. Why? Physics, Chemistry, Engineering, Math… Best solution?: Consider R 3 Kelvin 1873 Optimal… Wearie-Phelan 1994 Even better

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Our Problem Minimum surface area of body tiling R d with period Z d ? d=2 area: 4>4Choe’89: Optimal!

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Bounds in d dimensions ≤ OPT ≤ [Kindler,O’Donnell, Rao,Wigderson] Rao,Wigderson] ≤OPT≤ “Spherical Cubes” exist! Probabilistic construction! (simpler analysis [Alon-Klartag]) OPEN: Explicit?

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Randomized Rounding Round points in R d to points in Z d such that for every x,y x y 1

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Spine Torus Surface blocking all cycles that wrap around

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Probabilistic construction of spine Step 1 Probabilistically construct B, which in expectation satisfies B Step 2 Sample independent translations of B until [0,1) d is covered, adding new boundaries to spine.

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Linear equations over GF(2) m linear equations: Az = b in n variables: z 1,z 2,…,z n Given (A,b) 1) Does there exist z satisfying all m equations? Easy – Gaussian elimination 2) Does there exist z satisfying ≥.9m equations? NP-hard – PCP Theorem [AS,ALMSS] 3) Does there exist z satisfying ≥.5m equations? Easy – YES! [Hastad] >0, it is NP-hard to distinguish (A,b) which are not (½+ )-satisfiable, from those (1- )-satisfiable!

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Linear equations as Games 2n variables: X 1,X 2,…,X n, Y 1,Y 2,…,Y n m linear equations: Xi 1 + Yi 1 = b 1 Xi 2 + Yi 2 = b 2 ….. Xi m + Yi m = b m Promise: no setting of the X i,Y i satisfy more than (1- )m of all equations Game G Draw j [m] at random Xi j Yi j Alice Bob α j β j Check if α j + β j = b j Pr [YES] ≤ 1-

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Hardness amplification by parallel repetition 2n variables: X 1,X 2,…,X n, Y 1,Y 2,…,Y n m linear equations: Xi 1 + Yi 1 = b 1 Xi 2 + Yi 2 = b 2 ….. Xi m + Yi m = b m Promise: no setting of the X i,Y i satisfy more than (1- )m of all equations Game G k Draw j 1,j 2,…j k [m] at random Xi j1 Xi j2 Xi jk Yi j1 Yi j2 Yi jk Alice Bob α j1 α j2 α jk β j1 β j2 β jk Check if α jt + β jt = b jt t [k] Pr[YES] ≤ (1- 2 ) k [Raz,Holenstein,Rao] Pr[YES] ≥ (1- 2 ) k [Feige-Kindler-O’Donnell] Spherical Cubes [Raz] X [KORW]Spherical Cubes

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Zig-zag Graph Product and Cayley expanders in non-simple groups

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Expanding Graphs - Properties Geometric: high isoperimetry Probabilistic: rapid convergence of random walk Algebraic: small second eigenvalue ≤1 Theorem. [Cheeger, Buser, Tanner, Alon-Milman, Alon, Jerrum-Sinclair,…]: All properties are equivalent! Numerous applications in CS & Math! Challenge: Explicit, low degree expanders H [n,d, ]-graph: n vertices, degree d, (H) <1

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Algebraic explicit constructions [Margulis ‘73,Gaber-Galil,Alon-Milman,Lubotzky-Philips- Sarnak,…Nikolov,Kassabov,…,Bourgain-Gamburd ‘09,…] Many such constructions are Cayley graphs. G a finite group, S a set of generators. Def. Cay(G,S) has vertices G and edges (g, gs) for all g G, s S S -1. Theorem. [LPS] Cay(G,S) is an expander family. G = SL 2 (p) : group 2 x 2 matrices of det 1 over Z p. S = { M 1, M 2 } : M 1 = ( ), M 2 = ( )

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Algebraic Constructions (cont.) [Margulis] SL n (p) is expanding (n≥3 fixed!), via property (T) [Lubotzky-Philips-Sarnak, Margulis] SL 2 (p) is expanding [Kassabov-Nikolov] SL n (q) is expanding (q fixed!) [Kassabov] Symmetric group S n is expanding. …… [Lubotzky] All finite non-Abelian simple groups expand. [Helfgot,Bourgain-Gamburd] SL 2 (p) with most generators. What about non-simple groups? -Abelian groups of size n require >log n generators - k-solvable gps of size n require >log (k) n gens [LW] -Some p-groups (eg SL 3 (pZ)/SL 3 (p n Z) ) expand with O(1) generating sets (again relies on property T).

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Explicit Constructions (Combinatorial) -Zigzag Product [Reingold-Vadhan-W] K an [n, m, ]-graph. H an [m, d, ]-graph. Combinatorial construction of expanders. H v u (v,h)(v,h) Thm. [RVW] K z H is an [nm, d 2, + ]-graph, Definition. K z H has vertices {(v,h) : v K, h H}. K z H is an expander iff K and H are. Edges

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Iterative Construction of Expanders K an [n,m, ]-graph. H an [m,d, ] -graph. The construction: A sequence K 1,K 2,… of expanders Start with a constant size H a [d 4, d, 1/4 ]-graph. K 1 = H 2 [RVW] K i is a [d 4i, d 2, ½]-graph. [RVW] K z H is an [nm,d 2, + ]-graph. K i+1 = K i 2 z H

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Semi-direct Product of groups A, B groups. B acts on A. Semi-direct product: A x B Connection: semi-direct product is a special case of zigzag Assume = B, = A, S = s B (S is a single B-orbit) [Alon-Lubotzky-W] Cay(A x B, TsT ) = Cay (A,S) z Cay(B,T) [Alon-Lubotzky-W] Expansion is not a group property [Meshulam-W,Rozenman-Shalev-W] Iterative construction of Cayley expanders in non-simple groups. Construction: A sequence of groups G 1, G 2,… of groups, with generating sets T 1,T 2, … such that Cay(G n,T n ) are expanders. Challenge: Define G n+1,T n+1 from G n,T n

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Constant degree expansion in iterated wreath-products [Rosenman-Shalev-W] Start with G 1 = SYM d, |T 1 | ≤ √d. [Kassabov] Iterate: G n+1 = SYM d x G n d Get (G 1,T 1 ), (G 2,T 2 ),…, (G n,T n ),... G n : automorphisms of d-regular tree of height n. Cay(G n,T n ) expands few expanding orbits for G n d Theorem [RSW] Cay(G n, T n ) constant degree expanders. d n

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Near-constant degree expansion in solvable groups [Meshulam-W] Start with G 1 = T 1 = Z 2. Iterate: G n+1 = G n x F p [G n ] Get (G 1,T 1 ), (G 2,T 2 ),…, (G n,T n ),... Cay(G n,T n ) expands few expanding orbits for F p [G n ] Conjecture (true for G n ’s) : Cay(G,T) expands G has ≤exp(d) irreducible reps of every dimension d. Theorem [Meshulam-W] Cay(G n,T n ) with near-constant degree: |T n | O(log (n/2) |G n |) ( tight! [Lubotzky-Weiss] )

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Belief Propagation in Codes and L 2 sections of L 1

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Random Euclidean sections of L 1 N Classical high dimensional geometry [Kashin 77, Figiel-Lindenstrauss-Milman 77]: For a random subspace X R N with dim(X) = N/2, L 2 and L 1 norms are equivalent up to universal factors |x| 1 = Θ( √ N )|x| 2 x X L 2 mass of x is spread across many coordinates #{ i : |x i | ~ √ N ||x|| 2 } = Ω ( N ) Analogy: error-correcting codes: Subspace C of F 2 N with every nonzero c C has (N) Hamming weight.

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Euclidean sections applications: Low distortion embedding L 2 L 1 Efficient nearest neighbor search Compressed sensing Error correction over the Reals. …… Challenge [Szarek, Milman, Johnson-Schechtman]: find an efficient, deterministic section with L 2 ~L 1 X R N dim(X) vs. istortion(X) (X) = Max x X ( √ N ||x|| 2 )/||x|| 1 We focus on: dim(X)= ( N ) & (X) =O(1)

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Derandomization results [Arstein-Milman] For dim(X)=N/2 (X) = ( √ N||x|| 2 )/||x|| 1 = O(1) X= ker(A) # random bits [Kashin ’77, Garnaev-Gluskin ’84] O(N 2 ) A a random sign matrix. [Arstein-Milman ’06] O(N log N) Expander walk on A’s columns [Lovett-Sodin ‘07] O(N) Expander walk + k-wise independence [Guruswami-Lee-W ’08] (X) = exp(1/ ) N >0 Expander codes & “belief propagation”

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Spread subspaces Key ideas [Guruswami-Lee-Razborov]: L R d is (t, )-spread if every x L, S [d], |S|≤t ||x S || 2 ≤ (1- )||x| “No t coordinates take most of the mass” Equivalent notion to distortion (and easier to work with) –O(1) distortion ( (d), (1) )-spread –(t, )-spread distortion O( -2 · (d/t) 1/2 ) Note: Every subspace is trivially (0, 1)-spread. Strategy: Increase t while not losing too much L 2 mass. –(t, )-spread (t’, ’)-spread

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Constant distortion construction [GLW] (like Tanner codes) Belongs to L Ingredients for X=X(H,L): - H(V,E): a d-regular expander - L R d : a random subspace X(H,L) = { x R E : x E(v) L v V } Note: - N = |E| = nd/2 - If L has O(1) distortion (say is (d/ 10, 1/10 )-spread) for d = n /2, we can pick L using n random bits.

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Distortion/spread analysis [GLW]: If H is an (n, d, √ d)-expander, and L is (d/ 10, 1/10 )-spread, then the distortion of X(H,L) is exp(log d n) Picking d = n we get distortion exp(1/ ) = O(1) Suffices to show: For unit vector x X(H,L) & set W of < n/ 20 vertices W V

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Belief / Mass propagation Define Z = { z W : z has > d/ 10 neighbors in W } By local (d/ 10, 1/10 )-spread, mass in W \ Z “leaks out” By expander mixing lemma, |Z| < |W|/d Iterating this log d n times… It follows that W Z V Completely analogous to iterative decoding of binary codes, which extends to error-correction over Reals. [Alon] This “myopic” analysis cannot be improved! OPEN: Fully explicit Euclidean sections

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Summary TCS goes hand in hand with Geometry Analysis Algebra Group Theory Number Theory Game Theory Algebraic Geometry Topology … Algorithmic/computational problems need math tools, but also bring out new math problems and techniques

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