Title: Theoretical Computer Science methods in asymptotic geometry
1Theoretical Computer Science methods in
asymptotic geometry
 Avi Wigderson
 IAS, Princeton
 For Vitali Milmans 70th birthday
2Three topics Methods and Applications
 Parallel Repetition of games and
 Periodic foams
 Zigzag Graph Product and
 Cayley expanders in nonsimple groups
 Belief Propagation in Codes and
 L2 sections of L1
3 Parallel Repetition of Games and Periodic
Foams
4 Isoperimetric problem Minimize surface
area given volume.
One bubble. Best solution Sphere
5 Many bubbles Isoperimetric problem
Minimize surface area given volume.
Why? Physics, Chemistry,
Engineering, Math
Best solution? Consider R3
Kelvin 1873 Optimal
WeariePhelan 1994 Even better
6Our Problem
Minimum surface area of body tiling Rd with
period Zd ? d2 area
4
gt4
Choe89 Optimal!
7Bounds in d dimensions
OPT
OPT
Spherical Cubes exist! Probabilistic
construction! (simpler analysis
AlonKlartag) OPEN Explicit?
8Randomized Rounding
Round points in Rd to points in Zd such that for
every x,y 1. 2.
x
y
1
9Spine
Surface blocking all cycles that wrap around
Torus
10Probabilistic 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.
11Linear 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? NPhard PCP Theorem
AS,ALMSS 3) Does there exist z satisfying .5m
equations? Easy YES! Hastad ??gt0, it
is NPhard to distinguish (A,b) which are not
(½?)satisfiable, from those (1?)satisfiable!
12Linear equations as Games
Game G Draw j ? m at random Xij
Yij Alice
Bob aj
ßj Check if aj ßj 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
13Hardness amplification by parallel repetition
Game Gk Draw j1,j2,
jk ? m at random Xij1Xij2
Xijk Yij1Yij2 Yijk
Alice Bob aj1aj2 ajk
ßj1ßj2 ßjk Check if ajt ßjt bjt
?t? k PrYES (1?2)k Raz,Holenstein,Rao
PrYES (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
FeigeKindlerODonnell
Spherical Cubes ?
Raz
X
KORWSpherical Cubes ?
14Zigzag Graph Product and Cayley expanders in
nonsimple groups
15Expanding Graphs  Properties
 Geometric high isoperimetry
 Probabilistic rapid convergence of random walk
 Algebraic small second eigenvalue ? 1
Theorem. Cheeger, Buser, Tanner, AlonMilman,
Alon, JerrumSinclair,
All properties are
equivalent!
Numerous applications in CS Math! Challenge
Explicit, low degree expanders H n,d, ?graph
n vertices, degree d, ?(H)? ?lt1
16Algebraic explicit constructions Margulis
73,GaberGalil,AlonMilman,LubotzkyPhilipsSarna
k,
Nikolov,Kassabov,
,BourgainGamburd 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?S1.
Theorem. LPS Cay(G,S) is an expander family.
17Algebraic Constructions (cont.)
 Margulis SLn(p) is expanding (n3 fixed!), via
property (T)  LubotzkyPhilipsSarnak, Margulis SL2(p) is
expanding  KassabovNikolov SLn(q) is expanding (q fixed!)
 Kassabov Symmetric group Sn is expanding.

 Lubotzky All finite nonAbelian simple groups
expand.  Helfgot,BourgainGamburd SL2(p) with most
generators.  What about nonsimple groups?
 Abelian groups of size n require gtlog n
generators  ksolvable gps of size n require gtlog(k)n gens
LW  Some pgroups (eg SL3(pZ)/SL3(pnZ) ) expand with
 O(1) generating sets (again relies on property
T).
18Explicit Constructions (Combinatorial) Zigzag
Product ReingoldVadhanW
K an n, m, ?graph. H an m, d, ?graph.
Edges
Combinatorial construction of expanders.
19Iterative Construction of Expanders
K an n,m,?graph. H an m,d,? graph.
The construction A sequence K1,K2,
of expanders
Start with a constant size H a d4, d, 1/4graph.
RVW Ki is a d4i, d2, ½graph.
20Semidirect Product of groups
A, B groups. B acts on A. Semidirect product
A x B
Connection semidirect product is a special
case of zigzag Assume ltTgt B, ltSgt A , S sB
(S is a single Borbit)
AlonLubotzkyW Expansion is not a group
property
MeshulamW,RozenmanShalevW Iterative
construction of Cayley expanders in
nonsimple groups.
Construction A sequence of groups G1, G2 ,
of
groups, with generating sets T1,T2,
such that
Cay(Gn,Tn) are expanders. Challenge Define
Gn1,Tn1 from Gn,Tn
21Constant degree expansion in iterated
wreathproducts RosenmanShalevW
Start with G1 SYMd, T1 vd.
Kassabov Iterate Gn1 SYMd x Gnd Get
(G1 ,T1 ), (G2 ,T2),
, (Gn ,Tn ),... Gn
automorphisms of dregular tree of height n.
Cay(Gn,Tn ) expands ? few expanding orbits for
Gnd
Theorem RSW Cay(Gn, Tn) constant degree
expanders.
22Nearconstant degree expansion in solvable
groups MeshulamW
Start with G1 T1 Z2. Iterate Gn1
Gn x FpGn Get (G1 ,T1 ), (G2 ,T2),
, (Gn ,Tn
),... Cay(Gn,Tn ) expands ? few expanding orbits
for FpGn Conjecture (true for Gns)
Cay(G,T) expands ? G has exp(d) irreducible
reps of every dimension d.
Theorem MeshulamW Cay(Gn,Tn) with
nearconstant degree Tn ? O(log(n/2) Gn)
(tight! LubotzkyWeiss )
23Belief Propagation in Codes and L2 sections of L1
24Random Euclidean sections of L1N
 Classical high dimensional geometry
 Kashin 77, FigielLindenstraussMilman
77  For a random subspace X ? RN with dim(X) N/2,
 L2 and L1 norms are equivalent up to universal
factors 
 x1 T(vN)x2 ?x?X
 L2 mass of x is spread across many coordinates

 i xi vNx2
O(N)  Analogy errorcorrecting codes Subspace C of
F2N with every nonzero c ? C has ?(N) Hamming
weight.
25Euclidean sections applications
 Low distortion embedding L2 ? L1
 Efficient nearest neighbor search
 Compressed sensing
 Error correction over the Reals.

 Challenge Szarek, Milman, JohnsonSchechtman
find an efficient, deterministic section with
L2L1  X ? RN dim(X) vs. ?istortion(X)
 ?(X) Maxx?
X(vNx2)/x1 
 We focus on dim(X)?(N) ?(X) O(1)
26Derandomization results ArsteinMilman
 For
 dim(X)N/2 ?(X) (vNx2)/x1 O(1)
 X ker(A)

random bits  Kashin 77, GarnaevGluskin 84
O(N2 )  A a random sign matrix.
 ArsteinMilman 06
O(N log N)  Expander walk on As columns
 LovettSodin 07
O(N)  Expander walk kwise independence
 GuruswamiLeeW 08 ?(X) exp(1/?) N?
??gt0  Expander codes belief propagation
27Spread subspaces
 Key ideas GuruswamiLeeRazborov
 L ? Rd is (t,?)spread if every x ? L,

 ?S ? d, St xS2 (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
28Constant distortion construction GLW (like
Tanner codes)
Ingredients for XX(H,L)  H(V,E) a dregular
expander  L ? Rd a random subspace X(H,L)
x?RE 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.
29Distortion/spread analysis
 GLW If H is an (n, d, vd)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 lt n/20 vertices
30Belief / Mass propagation
 Define Z z ? W z has gt d/10 neighbors in W
 By local (d/10, 1/10)spread, mass in W \ Z
leaks out
It follows that
By expander mixing lemma, Z lt
W/d Iterating this logd n times
Completely analogous to iterative decoding of
binary codes, which extends to errorcorrection
over Reals. Alon This myopic analysis cannot
be improved! OPEN Fully explicit Euclidean
sections
31Summary
 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