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**Theoretical Computer Science methods in asymptotic geometry**

Avi Wigderson IAS, Princeton For Vitali Milman’s 70th 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 L2 sections of L1

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**Parallel Repetition of Games and Periodic Foams**

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**Minimize surface area given volume.**

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

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**Minimize surface area given volume.**

Many bubbles Isoperimetric problem: Minimize surface area given volume. Why? Physics, Chemistry, Engineering, Math… Best solution?: Consider R3 Kelvin Optimal… Wearie-Phelan Even better

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

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**Bounds in d dimensions [Kindler,O’Donnell, Rao,Wigderson] ≤OPT≤**

“Spherical Cubes” exist! Probabilistic construction! (simpler analysis [Alon-Klartag]) OPEN: Explicit?

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**Randomized Rounding 1 Round points in Rd to points in Zd**

such that for every x,y 1. 2. x y 1

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Spine Surface blocking all cycles that wrap around Torus

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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: z1,z2,…,zn 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**

Game G Draw j [m] at random Xij Yij Alice Bob αj βj Check if αj + βj = bj Pr [YES] ≤ 1- 2n variables: X1,X2,…,Xn, Y1,Y2,…,Yn m linear equations: Xi1 + Yi1 = b1 Xi2 + Yi2 = b2 ….. Xim + Yim = bm Promise: no setting of the Xi,Yi satisfy more than (1-)m of all equations

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**Hardness amplification by parallel repetition**

Game Gk Draw j1,j2,…jk [m] at random Xij1Xij2 Xijk Yij1Yij2 Yijk Alice Bob αj1αj2 αjk βj1βj2 βjk Check if αjt + βjt = bjt t [k] Pr[YES] ≤ (1-2)k [Raz,Holenstein,Rao] Pr[YES] ≥ (1-2)k 2n variables: X1,X2,…,Xn, Y1,Y2,…,Yn m linear equations: Xi1 + Yi1 = b1 Xi2 + Yi2 = b2 ….. Xim + Yim = bm Promise: no setting of the Xi,Yi satisfy more than (1-)m of all equations [Feige-Kindler-O’Donnell] Spherical Cubes X [Raz] [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 SS-1. G = SL2(p) : group 2 x 2 matrices of det 1 over Zp. S = { M1 , M2 } : M1 = ( ) , M2 = ( ) 1 1 0 1 1 0 Theorem. [LPS] Cay(G,S) is an expander family.

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**Algebraic Constructions (cont.)**

[Margulis] SLn(p) is expanding (n≥3 fixed!), via property (T) [Lubotzky-Philips-Sarnak, Margulis] SL2(p) is expanding [Kassabov-Nikolov] SLn(q) is expanding (q fixed!) [Kassabov] Symmetric group Sn is expanding. …… [Lubotzky] All finite non-Abelian simple groups expand. [Helfgot,Bourgain-Gamburd] SL2(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 SL3(pZ)/SL3(pnZ) ) expand with O(1) generating sets (again relies on property T). AB – Alon Boppana LPS – Lubotzky-Philips-Sarnak M- Margulis KN Kassabov-Nikolov K – Kassabov L – Lubotzky H – Helfgot BG Bourgain-Gamburd

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**Explicit Constructions (Combinatorial) -Zigzag Product [Reingold-Vadhan-W]**

K an [n, m, ]-graph. H an [m, d, ]-graph. Definition. K z H has vertices {(v,h) : vK, hH}. v u (v,h) Edges G- Gromov – iterated products of cubes. Madras & Randall, Martin & Randall – decomposition theorems of Markov Chains Thm. [RVW] K z H is an [nm, d2, +]-graph, K z H is an expander iff K and H are. Combinatorial construction of expanders.

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**Iterative Construction of Expanders**

K an [n,m,]-graph. H an [m,d,] -graph. [RVW] K z H is an [nm,d2,+]-graph. The construction: A sequence K1,K2,… of expanders Start with a constant size H a [d4, d, 1/4]-graph. K1 = H2 Ki+1 = Ki2 z H [RVW] Ki is a [d4i, d2, ½]-graph.

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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 <T> = B, <S> = A , S = sB (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 G1, G2 ,… of groups, with generating sets T1,T2, … such that Cay(Gn,Tn) are expanders. Challenge: Define Gn+1,Tn+1 from Gn,Tn

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**Constant degree expansion in iterated wreath-products [Rosenman-Shalev-W]**

Start with G1 = SYMd, |T1| ≤ √d. [Kassabov] Iterate: Gn+1 = SYMd x Gnd Get (G1 ,T1 ), (G2 ,T2),…, (Gn ,Tn ),... Gn: automorphisms of d-regular tree of height n. Cay(Gn,Tn ) expands few expanding orbits for Gnd d n Theorem [RSW] Cay(Gn, Tn) constant degree expanders.

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**Near-constant degree expansion in solvable groups [Meshulam-W]**

Start with G1 = T1 = Z Iterate: Gn+1 = Gn x Fp[Gn] Get (G1 ,T1 ), (G2 ,T2),…, (Gn ,Tn ),... Cay(Gn,Tn ) expands few expanding orbits for Fp[Gn] Conjecture (true for Gn’s): Cay(G,T) expands G has ≤exp(d) irreducible reps of every dimension d. Theorem [Meshulam-W] Cay(Gn,Tn) with near-constant degree: |Tn| O(log(n/2) |Gn|) (tight! [Lubotzky-Weiss] )

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**Belief Propagation in Codes and L2 sections of L1**

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**Random Euclidean sections of L1N**

Classical high dimensional geometry [Kashin 77, Figiel-Lindenstrauss-Milman 77]: For a random subspace X RN with dim(X) = N/2, L2 and L1 norms are equivalent up to universal factors |x|1 = Θ(√N)|x|2 xX L2 mass of x is spread across many coordinates #{ i : |xi| ~ √N||x||2 } = Ω(N) Analogy: error-correcting codes: Subspace C of F2N with every nonzero c C has (N) Hamming weight.

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**Euclidean sections applications:**

Low distortion embedding L2 L1 Efficient nearest neighbor search Compressed sensing Error correction over the Reals. …… Challenge [Szarek, Milman, Johnson-Schechtman]: find an efficient, deterministic section with L2~L1 X RN dim(X) vs. istortion(X) (X) = Maxx 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(N2 ) 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 Rd is (t,)-spread if every x L, S [d], |S|≤t ||xS||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 L2 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 Rd : a random subspace X(H,L) = { xRE : xE(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(logdn) 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” It follows that W Z V By expander mixing lemma, |Z| < |W|/d Iterating this logd n times… 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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