Lectures by Jacob Fox, Tali Kaufman, Roy Meshulam, Janos ...borel2015.unige.ch/Borel2015.pdf · Benny Sudakov: Extermal Results and Problems on Hypergraphs33 1. Part 1 (Day 1, Period

Notes from the 2015 Borel Seminar on

High-Dimensional Expanders

Lectures by Jacob Fox, Tali Kaufman, Roy Meshulam,

Janos Pach, Ori Parzanchevski, Benny Sudakov, and Uli

Wagner

Notes by Mark Kim

The following set of notes consists of live-TEXed transcriptions of mini-coursesgiven at the 2015 Borel Seminar on high-dimensional expanders. The conferencetook place at Les Diablerets, Switzerland from March 9, 2015 to March 13, 2015.For more information on the conference, please visit

http://borel2015.unige.ch.

The notes were produced real-time during each lecture. Moreover, not alltalks are represented here. I stress that the transcriptions are not meant to beauthoritative reproductions of the talks, and that all the errors herein are my own.Corrections and comments are gratefully received at

[email protected].

The present document is up to date as of March 12, 2015; the current version ofthese notes can be found at

http://markhkim.com/writings

Mark H. Kim

Courant Institute, NYU

http://borel2015.unige.ch

mailto:[email protected]

http://markhkim.com/writings

Contents

Chapter 1. Jacob Fox: Regularity Lemmas, High-Dimensional GeometricExpanders 1

1. Regularity Lemmas (Day 1, Period 4) 12. Overlapping Simplices and High-Dimensional Expanders (Day 3, Period

2) 5

Chapter 2. Tali Kaufman: High-Dimensional expanders and property testing 71. One-Dimensional Expanders (Day 2, Period 5) 7

Chapter 3. Alex Lubotzky: From Ramanujan Graphs to RamanujanComplexes 9

1. Spectral Theory of Graphs (Day 1, Period 1) 92. Arithmetic Groups (Day 2, Period 1) 133. Bruhat–Tits trees (Day 3, Period 1) 144. Representation Theory (Day 4, Period 1) 16

Chapter 4. Roy Meshulam: Random Simplical Complexes 191. Topology of the Binomial Model (Day 2, Period 4) 19

Chapter 5. Izhar Oppenheim: High-dimensional Expanders from a1-Dimensional Perspective 22

1. Day 4, Period 5 22

Chapter 6. Janos Pach: Semialgebraic Combinatorics 241. Semi-algebraic Extremal Graph Theory (Day 2, Period 2) 24

Chapter 7. Ori Parzanchevski: High-Dimensional Laplacians and Expansion 281. Cohomology and Spectrum (Day 1, Period 2) 282. Spectrum, Cheeger’s Inequalities, and Mixing (Day 4, Period 4) 30

Chapter 8. Benny Sudakov: Extermal Results and Problems on Hypergraphs 331. Part 1 (Day 1, Period 3) 332. Part 2 (Day 1, Period 5) 35

Chapter 9. Uli Wagner: Coboundry Expansion and Topological Overlapping 391. Definitions (Day 2, Period 3) 392. Outline of the Proof (Day 3, Period 3) 413. Proof (Day 4, Period 2) 42

iii

CHAPTER 1

Jacob Fox: Regularity Lemmas, High-DimensionalGeometric Expanders

1. Regularity Lemmas (Day 1, Period 4)

Definition 1.1. A graph G = (V,E) has a vertex set V and a set E of edges,which are pairs of vertices.

e(X,Y ) denotes the number of pairs in X × Y that are edges.

Theorem 1.2 (Expander mixing lemma). If G = (V,E) is a k-regular graphof size n with second largest (in absolute value) eigenvalue λ, then, for all X,Y ⊆,we have the following inequality:∣∣∣∣e(X,Y )− k

n|X||Y |

∣∣∣∣ ≤ λ√|X||Y |.Roughly, Szemeredi’s regularity lemma states that every large graph can be

partitioned into a bounded number of roughly equally-sized parts so that the graphis random-like between almost all pairs of parts. This is considered one of the mostpowerful results in graph theory.

To state the result precisely, we must understand what “random-like” means.To this end, we let X and Y be vertex subsets of a graph G.

Definition 1.3. The density of the pair (X,Y ) is the quantity

d(X,Y ) =e(X,Y )

|X||Y |.

Definition 1.4. The irregularity irreg(X,Y ) of the pair (X,Y ) is the maxi-mum, over all A ⊆ X and B ⊆ Y of the following quantity

|e(A,B)− d(X,Y )|A||B|| .

We say that (X,Y ) is ε-regular if irreg(X,Y ) ≤ ε|X||Y |.

Definition 1.5. The irregularity of a vertex partition P of a graph G = (V,E)is

irreg(P ) =∑

X,Y ∈Pirreg(X,Y ).

Partition P is said to be ε-regular if irreg(P ) ≤ ε|V |2.

The following is a statement equivalent to the original Szmeredi regularitylemma, due to Lovasz and Szegedy:

Theorem 1.6 (Szemeredi’s regularity lemma). For every ε > 0, there is anM(ε) so that every graph G has an ε-vertex partition P with at most M(ε) parts.

1

1. REGULARITY LEMMAS (DAY 1, PERIOD 4) 2

A natural question to ask at this stage is: how big is M(ε)? Tower functionT (n) is given by T (1) = 2 and T (n) = 2T (n−1). Szemeredi’s original proof givesa Tower-function upper bound. Moreover, Gowers proved that M(ε) ≥ T (ε−c) forsome constant c > 0.

Gowers asked in 1997 what the precise order of the tower height of M(ε). Thiswas resolved by Fox, et al. in 2014:

Theorem 1.7 (Fox-L. M. Lovasz 2014). M(ε) = T (Θ(ε−2)).

Let us now sketch the proof of the regularity lemma. To this end, we introducethe following notion:

Definition 1.8. For a vertex partition P : V = V1 ∪ · · · ∪Vk, the mean-squaredensity of P is declared to be the quantity

q(P ) =∑i,j

pipjd2(Vi, Vj),

where pi = |Vi||V | .

We remark that 0 ≤ q(P ) ≤ 1 for all P . We also remark that if P ′ is arefinement of P , then q(P ′) ≥ q(P ).

The key step is as follows:

Claim. If P with |P | = k is not ε-regular, then there is a refinement P ′ intoat most k2k+1 parts such that q(P ′) ≥ q(P ) + ε2.

At most ε−2 iterations are needed before obtaining an ε-regular partition, andthis gives an upper bound. The current available proofs for lower bounds reverse-engineer this proof to obtain a construction.

There are many applications of the regularity lemma. The so-called regularitymethod is useful:

(1) apply szemeredi’s regularity lemma

use a counting lemma for embedding small graphs

Theorem 1.9 (Triangle counting lemma). If each pair of parts is ε-regular, thenumber of triangles across the three parts is ≈ d(X,Y )d(X,Z)d(Y, Z)|X||Y ||Z|.

Some applications are given below.

Theorem 1.10 (Graph removal lemma). For each ε > 0 and graph H on hvertices there is δ > 0 such that every graph on n vertices with at most δnh copiesof H can be made H-free by removing εn2 edges.

Sketch of proof. Apply Szemeredi’s regularity lemma, and delete edges be-tween pairs which are irregular or sparse. If there is a remaining copy of H, thenits edges go between pairs which are both dense and regular. The counting lemmathen implies that there are at least δnh copies of H.

This was proved for triangles by Ruzsa and Szemeredi in 1976, and in generalby Alon, Duke, and Lefmann, Rodl, and Yuster in 1994.

This has many applications in extremal graph theory, additive number theory,and theoretical computer science, and discrete geometry. For example, this impliesRoth’s theorem, which is the k = 3 case of the Szemeredi’s theorem.


Problem 1.11 (Erdos, Alon, Gowers, Tao). Find a new proof which gives betterbounds.

Theorem 1.12 (Fox). δ−1 can be taken to be a tower of twos of height

cH log ε−1.

The best known lower bound on δ−1 is ε−c log ε−1

, due to Bernhard.The key lemma for the new proof is as follows:

Lemma 1.13. If there are at most α|V1||V2||V3| triangles across V1, V2, V3, thenthere are 1 ≤ i ≤ j ≤ 3 and equitable partitions Qi of Vi and Qj of Vj each with at

most 2α−O(1)

parts such that there are at least 110 |Qi||Qj | pairs (X,Y ) ∈ Qi × Qj

with d(X,Y ) < 2α1/3.

Definition 1.14. For a vertex partition P : V = V1∪· · ·∪Vk, the mean entropydensity of P is the quantity

q(P ) =∑i,j

pipjd(Vi, Vj) log d(Vi, Vj),

where pi = |Vi||V | .

Let us now discuss weak regularity lemmas. These give better bounds and arethus more useful for algorithmic applications. Nevertheless, these are not strongenough for the most interesting applications of the regularity lemma.

Here is an important example of a weak regularity lemma.

Theorem 1.15 (Frieze–Kannan, 1999). For ε > 0, there is h = h(ε) such thatevery graph G = (V,E) has an equitable partition P : V = V1 ∪ · · · ∪ Vh such thatfor any subsets S, T ⊆ V ,∣∣∣∣∣∣e(S, T )

∑i,j

|S ∩ Vj ||T ∩ Vj |d(Vi, Vj)

∣∣∣∣∣∣ < ε|V |2.

This is a fundamental algorithmic tool with only a single exponential bound.Unfortunately, the bound cannot be improved:

Theorem 1.16 (Conlon–Fox). h(ε) = 2Θ(ε−2).

Once again, the proof of this result mimics the Friesz–Kannan regularity lemmaby reverse-engineering it.

For an algorithmic graph theory, we consider the problem of counting cliques.To this end, we let G be a graph with n vertices. Unless P = NP, there is nopolynomial-time algorithm that approximates the maximum clique to within a fac-tor better than O(n1−ε) for any ε > 0.

But what about counting the number of K-cliques? The trivial algorithm is intime O(nk). The best known bound is O(nωk/3), where ω ≈ 2.37 . . . is the matrixmultiplication constant.

Theorem 1.17 (Duke–Lefmann–Rodl). In a graph G on n vertices, we can ap-

proximate the count of cliques of order k within an additive εnk in time 2O(k2/ε5)n2.

Theorem 1.18 (Fox–Lovasz–Zhao). In a graph G on n vertices, we can approx-

imate the count of cliques of order k within an additive εnk in time ε( k2

)n−εO(1)nω.


Corollary 1.19 (Fox–Lovasz–Zhao). In a graph G on n vertices, we can

approximate the count of cliques of of order 1000 within an additive n10000−10−6

intime n2.4

The original regularity method is only useful for dense graph, but most prac-tical problems concern sparse graphs. Toward this direction, Kohayakawa, Rodl,and Scott proved a sparse regularity lemma by mimicking the proof of the usualregularity lemma, making sure that there is no dense spot between any pair.

The problem, however, is that this does not give us a counting lemma in sparsegraphs. In fact, a general counting lemma in sparse graphs cannot hold. How aboutwithin subgraphs of sparse pseudorandom graphs?

Theorem 1.20 (Conlon–Fox–Zhao). (A sparse counting lemma in graphs andhypergraphs satisfying some nice pseudorandomness condition.)

An application of this result is a better relative Szemeredi theorem and a simplerproof of the Green–Tao theorem on long arithmetic progressions in primes.

We conclude this section by discussing hypergraph regularity lemmas.

Definition 1.21. A hypergraph H = (V,E) has a vertex set V and a set E ofedges, which are subsets of V . It is r-uniform if each edge has size r.

Let X, Y , Z be vertex subsets of a 3-uniform hypergraph H. e(X,Y, Z) is thenumber of triples in X × Y × Z which are edges.

Definition 1.22. The density of the triple (X,Y, Z) is the quantity.

d(X,Y, Z) =e(X,Y, Z)

|X||Y ||Z|

Definition 1.23. The irregularity irreg(X,Y, Z) of the triple (X,Y, Z) is themaximum, over all A ⊆ X, B ⊆ Y , and C ⊆ Z,

|e(A,B,C)− d(X,Y, Z)|A||B||C|| .

Definition 1.24. The irregularity of a vertex partition P of a 3-uniform hy-pergraph H = (V,E) is

irreg(P ) =∑

X,Y,Z∈Pirreg(X,Y, Z).

Partition P is ε-regular if irreg(P ) ≤ ε|V |3.

Theorem 1.25 (Chung’s hyergraph regularity lemma). For every ε > 0, thereis an M(ε) so that every graph G has an ε-regular vertex partition P with at mostM(ε) parts.

The problem with this result is that it can only be used to count linear hyper-graphs.

A new hypergraph regularity method was developed independently by Nagle–Rodl–Schacht–Skokan, Gowers, and Tao.

Definition 1.26. For a graph G and 3-uniform hypergraph H, we let eG(H)be the number of triangles in G which are edges of H. We also let K3(G) denote thenumber of triangles in G. Finally, we let dG(H) denote the quantity eG(H)/K3(G),the edge density of H with respect to G.

2. OVERLAPPING SIMPLICES AND HIGH-DIMENSIONAL EXPANDERS (DAY 3, PERIOD 2)5

A better definition of irregularity in this context is as follows:

Definition 1.27. The irregularity irregG(H) of H is the maximum, over allsubgraphs G′ ⊆ G, of the quantity

|eG′(H)− dG(H)|K3(G)|| .

The hypergraph regularity lemma gives a vertex partition. Not only that,between each pair of parts, we also have a partition of complete bipartite graph inbetween there, and each of those is very regular.

2. Overlapping Simplices and High-Dimensional Expanders (Day 3,Period 2)

Theorem 2.1 (Boros–Furedi). For any n points in R2, there is a point p ∈ R2

in (2

9+ o(1)

)(n3

)triangles induced by the n points.

Bukh gave a nice proof of the above theorem, using the following lemma:

Lemma 2.2 (Ceder). For any n points in R2, there are three concurrent linesthat divide R2 into six parts each containing at least n

6 − 1 points.

Theorem 2.3 (Barany). For any n points in Rd, there is a point p ∈ Rd in atleast

(c(d) + o(1))

(n

d+ 1

)simplices induced by the n points.

The current best lower bound is due to Gromov; the current best upper boundis due to Bukh–Matousek–Nivasch.

2d

(d+ 1)!(d+ 1)≤ c(d) ≤ d!

(d+ 1)d

Theorem 2.4 (Pach’s selection theorem). For any n points in Rd, there aredisjoint subsets P1, . . . , Pd+1 each of of size at least c′dn and a point p ∈ Rd suchthat every simplex with one point in each Pi contains p.

See Chapter 6, Theorem 1.24 for details on the selection theorem.

Definition 2.5. For a (d+1)-uniform hypergraph H, its overlap number c(H)is the largest C ∈ (0, 1] such that for every embedding f : V → Rd, there existsa point p ∈ Rd which belongs to at least c|E| simplices whose vertex sets arehyperedges of H.

Definition 2.6. An infinite family H of (d+ 1)-uniform hypergraphs is highlyoverlapping if there exists an absolute constant c > 0 such that c(H) > c for everyH ∈ H.

Gromov asked the following question:

Problem 2.7 (Gromov). There is a highly overlapping infinite family of (d+1)-uniform hypergraphs of bounded degree without isolated vertices.

To this end, there are the following results:

2. OVERLAPPING SIMPLICES AND HIGH-DIMENSIONAL EXPANDERS (DAY 3, PERIOD 2)6

Theorem 2.8 (Fox–Gromov–Lafforgue–Naor–Pach). For each d and ε > 0,there is k = k(ε, d) and an infinite family of k-regular (d+ 1)-uniform hypergraphsH with c(H) > c(d)− ε.

Theorem 2.9 (Fox–Gromov–Lafforgue–Naor–Pach). For all d, ∆, and ε > 0,there is n0 such that every (d + 1)-uniform hypergraph H on n ≥ n0 nonisolatedvertices with maximum degree ∆ satisfies c(H) ≤ c(d) + ε.

Let us now sketch the proof.

Definition 2.10. A (d + 1)-uniform hypergraph H = (V,E) is α-uniform if,for all vertex subsets X1, . . . , Xd+1,∣∣∣∣e(X1, . . . , Xd+1)− |E|

( nd+1 )

|X1||X2| · · · |Xd+1|∣∣∣∣ ≤ αe(H).

We can either do an explicit construction via Ramanujan complexes, or just doa random construction via a first moment argument.

Theorem 2.11 (Fox–Gromov–Lafforgue–Naor–Pach). For each ε > 0, there isα > 0 such that every (d + 1)-uniform hypergraph H = (V,E) which is α-uniformhas c(H) = c(d)± ε.

Lemma 2.12 (Semi-algebraic hypergraph regularity lemma). For each t, d, andε > 0, there is K such that the vertex set of every (d + 1)-uniform hypergraph Xof complexity t has an equitable partition into K parts such that all but at most anε-fraction of the (d+ 1)-tuples of parts are complete or empty.

Here’s a sketch of proof for the lower bound of Theorem 2.11. Let p be thepoint in the maximum number of (d+ 1)-simplices induced by points in V . Applythe semi-algebraic hypergraph regularity lemma to the hypergraph Xp on V whoseedges are those (d+ 1)-tuples whose simplex contains p. Use α-uniformity for eachcomplete (d+ 1)-tuple of parts.

CHAPTER 2

Tali Kaufman: High-Dimensional expanders andproperty testing

1. One-Dimensional Expanders (Day 2, Period 5)

We are interested in the following question:

Let M be an n-by-n matrix whose entries are ±1 and diagonalentries are 1. Could we query M in constant many locationsand:• say yes if M is a tensor power M = v · vt for some vectorv;• say no with probability at least ε if M is ε-far from being a

tensor power, viz., if we need to change at least εn2 entriesof M to get a tensor power.

This question arises in the field of property testing.

Definition 1.1. A graph X = (V,E) is said to be an ε-expander if the Cheegerconstant

h(X) = minA(VA6=∅

|E(A, V rA)|min(|A|, |V rA|)

is bounded below by ε.

Example 1.2. Kn, the complete graph of order n, is an expander.

Given a finite set A, a property is a subset P ⊂ An.

Example 1.3. Let A = F2 and P be the set of all f : Fn2 → F2 such that, forall (x, y) ∈ Fn2 ,

f(x) + f(y) + f(x+ y) = 0.

P is (q, ε)-testable if there exists an algorithm that, given f ∈ An, reads qentries of f and outputs

• yes if f ∈ P with probability 1;• no if f /∈ P with probability at least εdist(f, P ).

Here dist(f, P ) is the fraction of entries needed to be changed in f to have P .

Theorem 1.4 (BLR). Linear functions are (3, 2g )-testable. For the tester,

choose x, y ∈ Fn2 and check f(x) + f(y) + f(x+ y) ≥ 0.

Given a graph X = (V,E), we define PV (X), the set of vertices that do notexpand, i.e.,

A : V → 0, 1 : ∀(u, v) ∈ E A(u) +A(v) = 0 mod 2.

7

1. ONE-DIMENSIONAL EXPANDERS (DAY 2, PERIOD 5) 8

X is an ε-expander if and only if PV (X) = ∅, V = CONST and PV (X) is (2, ε)-testable by its canonical test.

In light of this connection, we define a two-dimensional expander as follows.

Definition 1.5. Let X = (V,E, T ) be a 2-dimensional simplicial complex.Define PV (X) to be the set of vertices that do not expand. Define PE(X) to bethe set of edges that do not expand, i.e.,

PE(X) = f : E → 0, 1 : ∀(e1, e2, e3) ∈ T f(e1) + f(e2) + f(e3) = 0.We call the mandatory sets of edges that never expands CUTS.

X is an ε-expander if we have vertex expansion and edge expansion, viz.,

(1) PV (X) = CONST;(2) PV (X) is (2, ε)-testable by its canonical test;(3) PE(X) = CUTS;(4) PE(X) is (3, ε)-testable by its canonical test.

CHAPTER 3

Alex Lubotzky: From Ramanujan Graphs toRamanujan Complexes

1. Spectral Theory of Graphs (Day 1, Period 1)

Let X = (V,E) be a graph. We shall assume that X is finite and k-regular,i.e., the degree of each vertex is k. Let n = |V |. Define the adjacency matrix ofX to be the matrix A = (Auv), where Auv is the number of edges between u andv. We often think of A as a linear operator from L2(V ) to L2(V ), in the followingmanner:

(Af)(v) =∑w∼v

f(w).

We establish the basic properties of the adjacency matrix below; to this end, werecall that the spectrum of an n-by-n symmetric matrix B is the set

spec(B) = µ0 ≥ µ1 ≥ · · · ≥ µk−1,and the spectral radius of B is

r(B) = maxλ∈spec(B)

|λ|.

Proposition 1.1. Let A be the adjacency matrix of a k-regular graph X =(V,E).

(1) A is a symmetric matrix, and so every eigenvalue of A is real.(2) The spectral radius of A is bounded by k.(3) k is an eigenvalue of A and is of multiplicity 1 if and only if X is connected.(4) −k is an eigenvalue of A if and only if X is bipartite. Indeed, if X is bipartite,

then the spectrum spec(A) of A is symmetric, viz., λ is an eigenvalue of A ifand only if −λ is an eigenvalue of A. (The notion of bipartite graphs is recalledbelow.)

Example 1.2 (Bipartite graphs). Let X = (V,E) be a graph. X is said to ber-partite for some fixed integer r ≥ 2 in case there exists a partition of V into rsets V1, . . . , Vn such that xy /∈ E whenever both x and y are in Vi for some i. Inother words, all edges must connect vertices in different parts of the partition. Ifr = 2, we often use the term bipartite instead of 2-partite.

Proof of Proposition 1.1. (1) is just the spectral theorem.(2) Let λ be an eigenvalue of A, and f ∈ L2(V ) a corresponding nonzero eigen-

function. Since X is finite, there exists a v0 ∈ V such that |f(v0)| = maxv∈V |f(v)|.Observe now that

|λf(v0)| = |(Af)(v0)| =

∣∣∣∣∣ ∑w∼v0

f(w)

∣∣∣∣∣ ≤ ∑w∼v0

|f(w)| ≤ k|f(v0)|,

9

1. SPECTRAL THEORY OF GRAPHS (DAY 1, PERIOD 1) 10

Figure 1. Four bipartite graphs

whence the desired result follows.(3) The constant function 1 : V → V given by the formula 1(v) = v for all

v ∈ V is an eigenfunction for k, as X is k-regular.Let f be a nonzero eigenfunction for k. Let C1, . . . , Cl be the connected

components of V . For each 1 ≤ i ≤ l, we define

vi = maxv∈Ci

|f(v)|.

Since

|kf(vi)| = |(Af)(vi)| =

∣∣∣∣∣∑w∼vi

f(w)

∣∣∣∣∣ ≤ ∑w∼vi

|f(w)| ≤∑w∼vi

|f(vi)| = k|f(vi)|,

we see that ∑w∼vi

|f(w)| =∑w∼vi

|f(vi)|.

It then follows that |f(w)| = |f(vi)| for all w ∼ vi, and we can repeat this argumentas needed to conclude that |f(v)| = |f(vi)| for all v ∈ Ci.

Now, we define

vi = maxv∈Ci

f(v).

Since

kf(vi) = (Af)(vi) =∑w∼vi

f(w) ≤∑w∼vi

f(vi) = kf(vi),

we see that ∑w∼vi

f(w) =∑w∼vi

f(vi).

We already know that |f(v)| = |f(vi)| for all v ∈ Ci, and so the above identityholds only if f(w) = f(vi) for all w ∼ vi. We now repeat this argument as neededto conclude that f(v) = f(vi) for all v ∈ Ci.

What we have shown is that every nonzero eigenfunction of A with respect tok must be locally constant, i.e., constant on each connected component. It follows


that the multiplicity of k is precisely the number of the connected components ofX.

(4) Suppose that X is bipartite with parts V1 and V2. Fix an eigenvalue λ ofA and an eigenfunction f ∈ l2(X) corresponding to λ. The function

f(v) =

f(v) if v ∈ V1;

−f(v) if v ∈ V2

is an eigenfunction for −λ, i.e., Af = λf . In particular, −k is an eigenvalue of A.Conversely, we suppose that −k is an eigenvalue of A. This means that∑

w∈vf(w) = −kf(v)

for all v ∈ V . We let V1 = v ∈ V : f(v) = ‖f‖∞ and V2 = v ∈ V : f(v) =−‖f‖∞.

For each v ∈ V1, we have

−k‖f‖∞ = −kf(v) =∑w∼v

f(w).

Since f(w) ≥ −‖f‖∞ for all w ∼ v, the above identity implies that f(w) = −‖f‖∞for all w ∼ v. In other words, if v ∈ V1, then w ∈ V2 for all w ∼ v.

Similarly, we can show that v ∈ V2 implies w ∈ V1 for all w ∼ v.

Definition 1.3. Let X be a connected graph. We define

λ(X) = max |λ| : λ ∈ spec(A) r −k, k .

λ(X) controls the rate of convergence of the random walk on X to the uniformdistribution. To see this, we let ~p be a probability distribution on V , viz., 0 ≤ pi =p(vi) ≤ 1 for all i and

∑i pi = 1. If, in particular, ~p(0) = (1, 0, . . . , 0), then

~p(t+1) = M~p(t),

where M = 1kA.

Now, if α0, . . . , αn−1 is an orthonormal basis of eigenfunctions of M corre-sponding to the eigenvalues

1 = µ0 ≥ µ1 ≥ · · · ≥ µn−1 ≥ −1,

then we can write

~p(0) =

n−1∑i=0

a(0)i αi,

where a(0)i = 〈~p(0), αi〉. Since

α0 = (n−1/2, . . . , n−1/2),

we see that

a(0)0 = 〈~p(0)α0〉 =

n−1∑i=0

1√np

(0)i =

1√n.

Observe now that

M t(~p(0)) = M t

(n−1∑i=0

a(0)i αi

)=

n−1∑i=0

a(0)i µtiαi,


and so

limt→∞

M t(~p(0)) = a(0)0 α0 = (n−1, . . . , n−1),

as was to be shown.Unfortunately, λ(X) is subject to the following restriction:

Theorem 1.4 (Alon–Boppana). If Xr = (Vr, Er)∞r=1 is a family of k-regulargraphs such that |Xr| → ∞ as r →∞, then

lim infr→∞

λ(Xr) ≥ 2√k − 1.

Example 1.5. If X is a k-regular graph, then the k-regular infinite tree Tk isthe universal cover of X with respect to the shortest-path metric topology on X. Aresult of Kesten shows that, ifA is the adjacency matrix of Tk, then r(A) = 2

√k − 1.

See, for example, the following blog post for a proof: https://lucatrevisan.

wordpress.com/2014/08/20/the-spectrum-of-the-infinite-tree/

Definition 1.6. A finite k-regular graph is said to be a Ramanujan graph ifλ(X) ≤ 2

√k − 1.

Example 1.7. Ramanujan graphs exist. The construction of a standard ex-ample requires the classical theorem of Jacobi, which states that

(1.8) r4(n) = |(x0, x1, x2, x3) ∈ Z4 : x20 + x2

1 + x22 + x2

3 = n| = 8∑d|n4-d

d.

We fix two distinct primes p and q such that p, q ≡ 1 mod 4. By the Jacobitheorem, we see that r4(p) = 8(p+1). Note that if p = x2

0 +x21 +x2

2 +x23, then one of

the xi’s is odd and the other three are even. Let S be the set of α = (x0, x1, x2, x3)such that x2

0 + x21 + x2

2 + x23 = p, x0 > 0 is odd, and x1, x2, and x3 are even. Then

|S| = p+ 1.Observe that there exists an ε ∈ Fq such that ε2 = −1. For each α ∈ S, we see

that

α′ =

(x0 + x1ε x2 + x3ε−x2 + x3ε x0 − x1ε

)is in PGL2(Fq).

For each α ∈ S, we see that the quaternion conjugate α is an element of S.Moreover, α′ = (α′)−1. It now follows that

S′ = α′ : α ∈ S

is a symmetric subset of PGL2(Fq).Let H = 〈S′〉, the subgroup of PGL2(Fq) generated by S′. Let Xp,q =

Cay(H;S′), viz., the graph (V,E) obtained by setting V = H and declaring a ∼ as′for each a ∈ H and every s ∈ S′. Then Xp,q is a (p+ 1)-regular Ramanujan graph.Moreover, if the quadratic residue ( pq ) equals 1, then H = PSL2(Fq) and Xp,q isnon-bipartite. On the other hand, if ( pq ) = −1, then H = PGL2(Fq) and Xp,q isbipartite.

In either case, |λ(Xp,q)| ≤ 2√p. The proof of this result is closely related to

the Riemann Hypothesis over finite fields.

https://lucatrevisan.wordpress.com/2014/08/20/the-spectrum-of-the-infinite-tree/

https://lucatrevisan.wordpress.com/2014/08/20/the-spectrum-of-the-infinite-tree/

2. ARITHMETIC GROUPS (DAY 2, PERIOD 1) 13

2. Arithmetic Groups (Day 2, Period 1)

Let G be a Lie group. A discrete subgroup Γ of G is called a lattice if G/Γ hasa finite G-invariant measure.

Example 2.1. Γ is lattice if G/Γ is compact, as compact groups always admitHaar measures. For example, take G = Rn and Γ = Zn.

Example 2.2. SLn(R) is a Lie group, and SLn(Z) is a lattice, even thoughSLn(R)/SLn(Z) is not compact.

Let n = 2 and take G = SL2(R). Each g =

(a bc d

)∈ SL2(R) acts on the

upper half-plane U = x+ iy : x ∈ R and y > 0 by Mobius transformations

z 7→ az + b

cz + d.

This is a transitive action, and stab(i) = SO(2), the maximal compact subgroupof G. We can therefore identify U with the coset space G/K. (Careful: this is nota quotient group!) G/K is commonly referred to as the hyperbolic plane.

If Γ ≤ G is a cocompact lattice, viz., the quotient G/Γ is compact, then Γ\U =Γ\G/K, a Riemann surface.

In lieu of giving a precise definition upfront, we start by considering someexamples of arithmetic lattices.

Example 2.3. Consider the map f : Z[√

2]→ R× R given by the formula

f(a+ b√

2) = (a+√

2b, a−√

2b).

im f is a discrete subgroup of R× R.Analogously, SLn(Z[

√2]) can be thought of as a discrete subgroup of SLn(R)×

SLn(R).

Example 2.4. For each quadratic form Q, we define

SO(Q,R) = A ∈ GLn+1(R) : Q(Av) = Q(v).Consider the quadratic form

f(x1, . . . , xn+1) = x21 + · · ·+ x2

n − x2n+1.

Let us also take another quadratic form

h(x1, . . . , xn+1) = x21 + · · ·+ x2

n −√

2x2n+1.

We shall consider the embedding SO(h,Z[√

2]) → SO(h,R)× SO(hτ ,R). Here

hτ (x1, . . . , xn+1) = x21 + · · ·+ x2

n +√

2x2n+1,

which is a quadratic form of signature (n+ 1, 0).We have that

SO(h,R)× SO(hτ ,R) = SO(n, 1)× SO(n+ 1),

where SO(n+ 1) is compact. Projecting to the first component, we obtain a latticein SO(n, 1). This is a consequence of the following general fact:

If Γ is discrete in G×H and if H is compact, then Γ is discretein G.

3. BRUHAT–TITS TREES (DAY 3, PERIOD 1) 14

Example 2.5. Z is neither discrete nor dense in Qp. If, however, we embedZ[ 1

p ] diagonally into R × Qp, then it is discrete. Consider now the embedding

SLn(Z[ 1p ]) → SLn(R) × SLn(Qp). We shall imitate the construction in Example

2.4 to obtain a lattice.To this end, we let R be a commutative ring and consider the unit quaternion

space

H1(R) = α = x0 + x1i+ x2j + x3k : x0, x1, x2, x3 ∈ R and ‖α‖ = 1,

which is a group with respect to multiplication.We first consider the R = R case. H1(R) ∼= S3 ∼= SU(2), which is a compact

group.Let us now assume that p is a prime number that is congruent to 1 mod 4. In

this case, we have H1(Qp) ∼= SL2(Qp). To see this, we take ε ∈ Qp with ε2 = −1and send

α 7→(x0 + x1ε x2 + x3ε−x2 + x3ε x0 − x1ε

).

We now consider the embedding

H1

(Z[

1

p

])discrete−−−−→ H1(R)×H1(Qp).

Since

H1(R)×H1(Qp) = SU(2)× SL2(Qp),we see that

Γ = H1

(Z[

1

p

])→ SL2(Qp)

is a discrete embedding.

Now, we fix q 6= p and let Γ(2q) be the kernel of the map

(2.6) H1

(Z[

1

p

])→ H1

(Z[

1

p

]/2qZ

[1

p

]).

Γ(2q) is a subgroup of finite index. The 2 in the parameter 2q makes Γ(2q) torsion-free. Now, recall Xp,q from Example 1.7. Here is the connection: we have thatXp,q = Γ(2q)\G/K, where G = PGL2(Qp) and K = PGL2(Zp). We show in thenext section that G/K is a Bruhat–Tits tree.

3. Bruhat–Tits trees (Day 3, Period 1)

Recall that the field of p-adic numbers is the collection

Qp =

∞∑i=−m

aipi : a ≤ ai ≤ p− 1 and m ∈ Z

and that the ring of p-adic integers is the collection

Zp =

∞∑i=0

aipi : a ≤ ai ≤ p− 1

.

We have the following inclusion relations:

Z ⊆ Zp ⊆ Qp

3. BRUHAT–TITS TREES (DAY 3, PERIOD 1) 15

If x ∈ Qp, then pl ∈ Zp for some l. Let V = Q2p, for every basis α, β of V ,

we write L to denote Zpα + Zpβ. This L is called a lattice. (The choice of theterm “lattice” is an unfortunate coincidence, as L not discrete. Since “lattice” thestandard term for this notion in the literature, we begrudgingly use it here as well.)

Two lattices L1 and L2 are equivalent if there exists a nonzero element µ of Qpsuch that

µL1 = L2.

This gives rise to an equivalence relation, whence we can speak of the equivalenceclass [L] of a lattice L.

We say that an equivalence class of lattices [L1] is adjacent to another equiv-alence class of lattices [L2] if there exist representatives L′1 ∈ [L1] and L′2 ∈ [L2]such that

pL′1 ( L′2 ( L′1.

We let

L0 = Zp(

10

)+ Zp

(01

)and observe that every lattice L = Zpα+ Zpβ admits an exponent l such that

plL ⊆ L0.

We also note that

L/pL = (Zp/pZp)α+ (Zp/pZp)β ∼= F2p.

Therefore, every [L] has (p+1) neighbors, i.e., equivalence classes that are adjacentto [L].

Let us now recall that a tree is a graph such that between two vertices thereis a unique path. Indeed, given [L], there exists a unique l such that L0/p

lL is acyclic p-group. We can then connect L0 and p−1(plL) by going one step at a time.

L0

L1 = pL

...

Lp−1 = p−1(plL)

(For a formal proof, see J. P. Serre, Trees.)We now set V = Q2

p. We claim that

[L] : L = Zpα+ Zpβ, where α, β is a basis for V

is the vertex set of the (p + 1)-regular tree Tp+1. G = GL2(Qp) acts transitivelyon bases of V , and the center Z = Z(GL2(Qp)), the subgroup of scalar matri-

ces, preserves the equivalence relation. Therefore, PGL2(Qp) = G/Z acts on thevertices.

4. REPRESENTATION THEORY (DAY 4, PERIOD 1) 16

We set G = PGL2(Qp) and set K = PGL2(Zp) Observe that K = stab([L0])with respect to the action of G, and that K is a maximal compact subgroup of G.It follows that G/K ∼= Tp+1.

We now recall the Jacobi theorem (1.8). Let

S = α = x0 + x1i+ x2j + x3k : x0 > 0, x0 is odd, x1, x2, x3 are even..

Then |S| = p+ 1 and ε =√−1 ∈ Qp.

Take α ∈ S. Then α ∈ H(Z[ 1p ]), ‖α‖ = p, and p is invertible in Z[ 1

p ]. It follows

that α is invertible in H(Z[ 1p ]ee, ); indeed:

α−1 =α

‖α‖.

We also note that α ≡ 1 mod 2.We now set Γ = H∗(Z[ 1

p ])/Z, so that S ⊆ Γ(2); see (2.6) for the definition of

Γ(2).

Claim. Γ(2) = 〈S〉.

〈S〉 acts simply transitively on the tree G/K, and so 〈S〉 ·K = G and 〈S〉 is afree group on S. It now follows from the above claim that

Tp+1∼= Cay(Γ(2);S),

whence

Γ(2q)\Tp+1∼= Γ(2q)\G/K = Cay (Γ(2q)\Γ(2);S) .

Now would be a good time to go back and read Example 1.7 again!

4. Representation Theory (Day 4, Period 1)

4.1. Ramanujan Graphs and Tempered Representations. We now trans-late the combinatorial problem of constructing Ramanujan graphs to a problem inrepresentation theory.

Proposition 4.1. Let Λ be a cocompact subgroup of G. Λ\G/K is Ramanujanif and only if every infinite-dimensional, irreducible, spherical subrepresentation ofL2(Λ\G) is tempered.

This gives us the desired result, as the next theorem furnishes precisely therequisite conditions:

Theorem 4.2 (Deligne). If Λ = Γ(2q) is a congruent subgroup as before, thenevery infinite-dimensional, irreducible, spherical subrepresentation of L2(Γ(2q)\G)is tempered.

Let C = Cc(K\G//K) be the complex bi-K-invariant functions in G withcompact support, i.e.,

f(k1gk2) = f(g).

C is an algebra with respect to convolution:

(f1 ∗ f2)(x) =

∫G

f1(xg)f2(g−1) dg.

Let us record some basic facts about Cc(K\G/K).


(1) C is a commutative algebra. Indeed, any unitary representation ρ : G→ U (H)of G ϕ : C → End(H) gives rise to ϕ : C → End(H) such that

ϕ(f)(v) =

∫G

f(g)ρ(g)(v) dy.

(2) If HK is the K-invariant subspace of H, then

ρ(C)(Hk) ⊆ Hk.

(3) If (H, ρ) is a unitary representation of G, then dim(HK) ≤ 1. ρ is calledspherical if dimHK = 1.

(4) If ρ is spherical, then ρ is completely determined by ρ.

(1), (2), and (3) imply that C is a Hecke algebra. ϕ can be thought of as a mapinto C; in this manner, ϕ can be thought of as a character.

Let δ = 1K(P 0

0 1)K

, which is in C.

Claim. C is generated as an algebra by δ.

C acts on L2(G/K). Indeed, if f1 ∈ C and f2 ∈ L2(G/K), then f2 ∗ f1 ∈L2(G/K). Now, if δ(f2) = f2 ∗ δ, then

δ(f)(v) =∑w∼v

f(w).

Now, if Λ is a discrete cocompact discrete subgroup of G, then L2(Λ\G)K =L2(Λ\G/K). Therefore, there exists a one-to-one correspondence between spher-ical irreducible subrepresentations of L2(Λ\G) and eigenvectors of the adjacencyoperator δ acting on L2(Λ\G/K). In fact, there is a one-to-one correspondencebetween tempered representations and eigenvalues of δ on HK that are less thanor equal to 2

√p in absolute value.

4.2. Bruhat–Tits Buildings. We begin by considering a simpler case. LetU = Fdp. Define X(0) to be all proper subspaces of U . W0, . . . ,Wt is a t-cell if itis a flag, i.e.,

W0 ⊆W1 ⊆ · · · ⊆Wt.

Now, W0, . . . ,Wd−1 is a maximal flag, and so

dim(S(Fp, d)) = d− 2.

We now consider F = Qp/Fp(t). We define a Bruhat–Tits building Bd(F ) = Bas follows. Given a basis α1, . . . , αd of U = F d, we let L = Zpα1 + · · ·+Zpαd be alattice. Declare L1 ∼ L2 in case there exists a nonzero µ ∈ F such that µL1 = L2.We define the vertex set of B to be the collection [L] : L is a lattice.

We say that [L0], . . . , [Lt] is a t-dimensional cell if there exist representativesLi ∈ [Li] such that [L0], . . . , [Lt] form a “flag”:

pL′0 ⊆ L′t ⊆ · · · ⊆ L′2 ⊆ L′1 ⊆ L0.

Observe that L′0/pL′0∼= Fdp. Similarly as above, we see that

dim(B) = d− 1.

Set G = PGLd(F ). G acts transitively on bases. Z = Z(G), the scalar

matrices, preserve equivalence classes. G = G/Z(G) = PGLd(F ) acts on the


vertices of B as automorphisms of the building. If L0 = Zpe1 + · · · + Zped is thestandard lattice, then

K = PGLd(Zp) = stabG([L0]).

Definition 4.3. The vertices of B come with a color in Z/dZ defined as follows:given L = g(L0), g ∈ GLd(F ), we set

τ([L]) = val(det(g)) mod d ∈ Z/dZ.

The action of G does not preserve the colors. Also, if [L1]e−→ [L2], then τ(e) =

τ(e+) − τ(e−) ∈ Z/dZ. Finally, every camber (maximum cell) contains a uniquepoint of every color. We also remark that the color of edges is preserved by theautomorphisms.

4.3. Outline of Friday’s lecture. Let G = PGLd(Qp), PGLd(Fp(t)). LetK = PGLd(Zp). G/K is a Bruhat–Tits building, a (d − 1)-dimensional simplicialcomplex

If Λ is a cocompact lattice in G, then Λ\G/K is a finite simplicial complex.

Theorem 4.4 (Lafforgue). If Λ = Γ(I) is a congruent subgroup of Γ ≤ G =PGLd(Fp(t)), then every infinite-dimensional, irreducible, spherical subrepresenta-tion of L2(Γ(I)\G) is tempered.

CHAPTER 4

Roy Meshulam: Random Simplical Complexes

1. Topology of the Binomial Model (Day 2, Period 4)

1.1. Random graphs. Let G(n, p) be the space of all random graphs on [n] =1, . . . , n with independent edge probabilities p. Random graphs are ubiquitous.They serve as models for a multitude of natural phenomena, e.g., phase transitionproblems of statistical physics. They are also useful for showing existence results.

Theorem 1.1 (Erdos, 1948). A typical G ∈ G(n, 12 ) will contain neither a

clique nor an independent set on k = 2 log2 n vertices.

Current best explicit constructions achieve only k = 2(logn)ε (for each fixedε > 0).

Theorem 1.2 (Erdos–Renyi, 1958). For any function ω(n) that tends to infin-ity1,

limn→∞

P[G ∈ G(n, p) : G connected] =

0 if p = logn−ω(n)

n

1 if p = logn+ω(n)n

Moreover,

limn→∞

P[G ∈ G(n, c/n) : G acyclic] =

0 if c > 1√

1− c · e 2c+c2

4 if c < 1.

Remark. The first part of the theorem does not give a definite answer whenp = logn

n .

We shall study high-dimensional generalizations of the above theorem.

1.2. The k-dimensional Erdos-Renyi model. Let Y be a simplicial com-plex. Y (i) denotes the i-dim skeleton of Y . Y (i) denotes the oriented i-dimensionalsimplices of Y . We set fi(Y ) = |Y (i)|. ∆n−1 is the (n− 1)-dimensional simplex onV = [n].

We define Yk(n, p) to be the probability space of all complexes

∆(k−1)n−1 ⊆ Y ⊆ ∆

(k)n−1

with probability distribution

P[Y ] = pfk(Y )(1− p)(nk+1 )−fk(Y ).

Compare the following results with Theorem 1.2.

1extremely slowly, in practice

19

1. TOPOLOGY OF THE BINOMIAL MODEL (DAY 2, PERIOD 4) 20

Theorem 1.3 (Linial–Meshulam, 2003; Meshulam–Wallach, 2006). Fix an in-teger k ≥ 1 and a finite abelian group R. For any function ω(n) that tends toinfinity,

limn→∞

P[Y ∈ Yk(n, p) : Hk−1(Y ;R) = 0] =

0 if p = k logn−ω(n)

n

1 if p = k logn+ω(n)n .

Theorem 1.4 (Hoffman–Kahle–Paquette, 2013). If p > 80k lognn , then asymp-

totically almost surely Hk−1(Y ;Z) = 0.

Here are some details. Let us say that σ ∈ ∆n−1(k − 1) is isolated in Y ∈Yk(n, p) if it is not contained in any τ ∈ Y (k). If σ is isolated in Y , then 1σ ∈Hk−1(Y ), and 1σ 6= 0. Now, if p = k logn−ω(n)

n , then

E[number of isolated σ’s] =

(nk

)(1− p)n−k

tends to ∞ as n→∞. A second moment argument then implies that

P[Hk−1(Y ;R) 6= 0]→ 1.

We can also look at different generalizations of connectivity.

Theorem 1.5 (Babson–Hoffman–Kahle, 2010). For any ε > 0,

limn→∞

P[Y ∈ Y2(n, p) : π1(Y ) = 1] =

0 if p = n−

12−ε;

1 if p = n−12 +ε.

Theorem 1.6 (Meshulam, 2011). If Gn∞n=1 is a sequence of finite groupswith |Gn| ≤ nc, then

limn→∞

P[Y ∈ Y2

(n,

(3c+ 6) log n

n

): Hom(π1(Y ), Gn) = 1

]= 1

Here are some details for the Babson–Hoffman–Kahle bound. For Y ∈ Y2(n, p)and u ∈ [n], we let Yu = StY (u). Then Yu ∩ Yv = StY (uv) ∪ G. It follows that, if

p ≥√

10 lognn , then Yu ∩ Yv is connected for all u and v. It now follows from the

nerve lemma that π1(Y ) = 1.Let us now consider hyperbolic groups. Let F (X) be the free group on a

set X. Fix w ∈ F (X) such that w = 1 is in G = 〈X|R〉, where R is a finite set ofrelations. We define the area A(w) of w to be the minimal n such that

w = (u−11 rε11 u1) · · · (u−1

n rεnn un)

for some ui ∈ F (X), ri ∈ R, and εi ∈ −1, 1. We say that a group G is hyperbolic[fill in later]

Some examples of hyperbolic groups: finite groups, free groups, and fundamen-tal groups of surfaces of genus at least 2, i.e.,

G = 〈a1, b1, . . . , ag, bg | [a1, b1] · · · [ag, bg]〉.We also remark that Z2 is not hyperbolic.

We now let X be a 2-dimensional simplicial complex, γ a simplicial null-homotopic loop in X of length |γ|, and AX(γ) the minimal number of simplicesin a filling of γ. The isoperimetric constant is the quantity

I(X) = inf

|γ|

AX(γ): γ ∼ 1

.

1. TOPOLOGY OF THE BINOMIAL MODEL (DAY 2, PERIOD 4) 21

Theorem 1.7. I(X) > 0 if and only if π1(X) is hyperbolic.

Here is a local-to-global principal that is useful for the proof:

Theorem 1.8 (Gromov). Let c > 0 and let X be a 2-dimensional complex thatsatisfies I(S) ≥ c for all pure S ⊆ X such that f2(S) ≤ 106c−2. Then I(X) ≥10−2c.

There is also a density invariant:

Theorem 1.9 (Babson–Hoffman–Kahle). Define

µ(X) = min

f0(Z)

f2(Z): Z ⊆ X

.

If µ(X) > 12 , then X is homotopic to a wedge of circles, 2-spheres, and projective

planes. Moreover, for each ε > 0, there exists a cε > 0 such that µ(X) > 12 + ε

implies I(X) > cε.

Theorem 1.10 (Babson–Hoffman–Kahle). p = o(n−12−ε) implies

P[I(Y ) > 0] = 1− o(1).

Sketch of proof. Let S be a pure complex such that f2(S) ≤ 106c−2ε . If

µ(S) ≤ 12 +ε, then there exists a subcomplex Z ⊆ S such that f0(Z) ≤ ( 1

2 +ε)f2(Z)and hence

P[Y ⊇ S] ≤ nf0(Z)pf2(Z) = n( 12 +ε)f2(Z) · o(n− 1

2−ε)f2(Z) = o(1).

Therefore Y asymptotically almost surely satisfies the following condition: S ⊆ Yand f2(S) ≤ 106c−2

ε imply µ(S) > 12 + ε, which then implies I(S) > cε. It now

follows from the local-to-global principle that I(Y ) > 10−2cε.

CHAPTER 5

Izhar Oppenheim: High-dimensional Expandersfrom a 1-Dimensional Perspective

1. Day 4, Period 5

Let us first review the basic notions and results of the theory of expanders. SeeChapter

Let X = (V,E) be a graph. Define ∆0 : l2(V )→ l2(V ) by setting

∆0φ(v) = φ(v) =1

deg(v)

∑u∼w

φ(u).

Then ∆0 is positive-definite, with eigenvalues

0 = µ0(X) ≤ µ1(X) ≤ · · · ≤ µn−1(X).

Recall that X has λ-spectral expansion if µ1(X) ≥ λ. This implies that H0(X,R) =0 by Cheeger’s inequality.

Recall also that X has two-sided (λ, κ)-expansion if µ1(X) ≥ λ and µn−1(X) ≤κ. This leads to the mixing lemma (Chapter 1, Theorem 1.2).

Finally, we recall that if X has λ-spectral expansion and if X is bipartite, thenwe obtain the bi-partite mixing lemma.

Definition 1.1. Let X be a simplicial complex of dimension d. For eachτ ∈ X(k), we define Xτ to be the collection of all σ ∈ X(i) such that σ ∩ τ = ∅and that σ ∪ τ ∈ X(k + i+ 1).

From now on, X is a pure d-dimensional complex with all the links of X di-mension at least 1 are connected.

Definition 1.2. For λ > d−1d , we say that X has λ-local spectral expansion if,

for all τ ∈ X(d− 2),µ1(Xτ ) ≥ λ.

This implies that, for all 0 ≤ k ≤ d− 1, we have Hk(X,R) = 0, a Cheeger-typeinequality.

Definition 1.3. For λ > d−1d and κ < 2, we say that X has λ-two-sided-local

spectral expansion if, for all τ ∈ X(d− 2),

µ1(Xτ ) ≥ λ and µn−1(Xτ ) ≤ κ.

This gives us a mixing-type result. We also note that if X has λ-local spectralexpansion and if X is (d + 1)-partite, then we obtain the (d + 1)-partite mixinglemma.

The above results are derived using the following methods:

(1) Weights

22

1. DAY 4, PERIOD 5 23

(2) Garland’s method(3) Spectral decent

CHAPTER 6

Janos Pach: Semialgebraic Combinatorics

1. Semi-algebraic Extremal Graph Theory (Day 2, Period 2)

Here are the cornerstones of extremal graph theory:

• Ramsey theorem (1930) and Erdos–Szekeres theorem (1935)• Turan theorem (1941) and Kovari–Sos–Tura theorem (1954)• Szemeredi regularity lemma (1978; Chapter 1, Theorem 1.6)

1.1. Ramsey theory. Ramsey theory is discussed in detail in Chapter 8,Section 1.1 for basic terminology.

Theorem 1.1 (Erdos–Szekeres, 1935).

1

2log n ≤ r2(n) ≤ 2 log n.

Theorem 1.2 (Erdos–Hajnal–Rado, 1965).

c log log n ≤ r3(n) ≤ c′(log n)1/2

Here is an application:

Theorem 1.3. Let s = s(n) denote the largest number such that every set ofn segments in R2 has s elements that are disjoint or all intersect. Then

s(n) ≥ r2(n) ∼ c log n.

In this case, however, a non-Ramsey-theory proof gives a much better bound(LMPT 1994):

s(n) ≥ n1/5.

Here is another application:

Theorem 1.4. We let c(n) denote the largest number c such that every set ofn points in R2 has c points that form a convex polygon. Then

c(n) ≥ r3(n) ≥ c log log n.

A different proof that makes use of a “hidden version” of the Ramsey theoremgives a better bound (Erdos–Szekeres, 1935):

c(n) ≥ 1

2log n

24

1. SEMI-ALGEBRAIC EXTREMAL GRAPH THEORY (DAY 2, PERIOD 2) 25

1.2. Turan-type problems. Turan-type problems are discussed in detail inChapter 8, Section 1.2 and Section 2.1

Theorem 1.5 (Kovar–Sos–Turan, 1954). Let G = (V (G), E(G)) be a graphwith n vertices such that G 6⊇ Kr,r (the complete bipartite graph with r vertices ineach class). Then

|E(G)| ≤ c2−1/rr .

Corollary 1.6. Let I(n) denote the maximum number of incidences betweenn points and n lines in R2. Then

I(n) ≤ c2n2−1/2 = c2n3/2

A newer proof that makes use of the Kovar–Sos–Turan theorem in a roundaboutway gives the optimal bound (Szemeredi–Trotter, 1983):

(1.7) I(n) ∼ cn4/3

1.3. Semi-algebraic graphs and hypergraphs.

Definition 1.8. A subset A ⊆ Rd is semi-algebraic if there exist polynomialsf1, . . . , ft : Rd → R of degree at most t and a Boolean formula φ such that

A = x ∈ Rd : φ[f1(x) ≥ 0, . . . , ft(x) ≥ 0].The description complexity of A is the quantity max(d, t).

Definition 1.9. A semi-algebraic graph is a graph G = (V,E) with a semi-algebraic set A ⊆ Rd × Rd such that xy ∈ E if and only if (x, y) ∈ A.

See Chapter 8, Definition 1.1 for the definition of a k-graph.

Definition 1.10. A semi-algebraic k-graph is a k-graph G = (V,E) with asemi-algebraic set A ⊆ (Rd)k such that (x1, . . . , xk) ∈ E if and only if (x1, . . . , xk) ∈A.

Example 1.11. Intersection graphs of segments, cubes, balls, and so on aresemi-algebraic graphs.

Example 1.12. Clockwise orientations of point-triples in R2 are semi-algebraic3-graphs.

Example 1.13. A collection of triples of balls in R3 admitting a line transversalis a semi-algebraic graph.

We now introduce Ramsey-theoretic notions for this framework.

Definition 1.14. Rk(n) is the minimum R such that in every 2-coloring of allk-tuples of an R-element set, there is a monochromatic subset of size n.

Definition 1.15. Rtk(n) is the minimum R such that in every 2-semi-algebraic-colorings with description complexity bounded above by t of all k-tuples of anR-element set, there is a monochromatic subset of size n.

Here is a result by Erdos–Rado, 1952:

Rtk(n) ≤ Rk(n) ≤ T (k)cn

Theorem 1.16 (Conlon–Fox–Pach–Sudakov–Suk, 2013). (statement?)


Theorem 1.17 (Alon–Pach–Pinchasi–Radoicic–Sharir, 2005). Let G = (V,E)be a semi-algebraic graph of description complexity t with |V | = n. There existV1, V2 ⊆ V with |V1|, |V2| ≥ εn such that V1 × V2 ⊆ E or (V1 × V2)∩E = ∅. (Hereε = ε(t) > 0 depends only on t.

Corollary 1.18 (Pach–Solymosi, 2001). Rt(n) ≤ nc(t)

We now turn to semi-algebraic Turan theory.

Theorem 1.19 (Mantel, 1907). Let ex(n,K3) denote the maximum number ofedges that a K3-free graph of n vertices can have. Then ex(n,K3) = bn2/4c.

Let ext(n,K3) denote the maximum number of edges that a K3-free semi-algebraic graphs of description complexity bounded above by t of n vertices canhave.

Theorem 1.20 (Kovari–Sos–Turan, 1954). Let G = (V1 ∪V2, E) be a Kr,r-freebipartite graph with |V1| = |V2| = n. Then

|E(G)| ≤ crn2−1/r.

Theorem 1.21 (Fox–Pach–Sheffer–Suck–Zahl, 2014). Let G = (V1 ∪ V2, E) bea Kr,r-free semi-algebraic bipartite graph with V1, V2 ⊆ Rd, |V1| = |V2| = n. Then

|E(G)| ≤ cr,tn4/3

if d = 2, and|E(G)| ≤ cr,t,εn2−2/(d+1)+ε

if d ≥ 2 and ε > 0. Here t is the description complexity of G.

This result implies the Szemeredi–Trotter theorem (1.7).

1.4. Regularity theory. Regularity lemmas are discussed in detail in Chap-ter 1, Section 1.

Theorem 1.22 (Szemeredi regularity lemma, 1978). For any ε > 0, thereexists an integer Kε such that the vertex set of every sufficiently large graph G hasa partition into K almost equal parts V1∪· · ·∪Vk, K ≤ Kε with the property that allbut at most εK2 pairs (Vi, Vj) are ε-regular if and only if they behave like randomgraphs with an ε error.

Theorem 1.23 (Semi-algebraic regularity lemma). For any ε > 0 and t, thereexists an integer Kε,t such that the vertex set of every sufficiently large semi-algebraic graph of description complexity at most t has a partition into K almostequal parts V1 ∪ · · · ∪ Vk, K ≤ Kε,t with the property that all but at most εK2 pairs(Vi, Vj) induce an empty or a complete bipartite graph if and only if Vi×Vj ⊆ E(G)or (Vi ∩ Vj) ∩ E(G) = ∅.

Now, Theorem 1.17, combined with the semi-algebraic regularity lemma, yieldsthe following selection theorem:

Theorem 1.24 (Alon–Pach–Pinchasi–Radoicic–Sharir, 2005). For any sets ofvectors U, V ⊆ Rd, one can select U ′ ⊆ U , V ′ ⊆ V with |U ′| ≥ 1

2d+1 |U |, |V ′| ≥1

2d+1 |V | such that

• 〈u, v〉 ≥ 0 for all u ∈ U ′ and v ∈ V ′, or• 〈u, v〉 < 0 for all u ∈ U ′ and v ∈ V ′.


We refer the reader to Chapter 1, Lemma 2.12 for the semi-algebraic hypergraphregularity lemma.

CHAPTER 7

Ori Parzanchevski: High-Dimensional Laplaciansand Expansion

1. Cohomology and Spectrum (Day 1, Period 2)

1.1. Dimension 1. Let X = (V,E) be a graph. For our purposes, E consistsof nonoriented edges. In contrast, we write E± to denote the collection of orientededges, i.e., the set of ordered pairs (v, w) where v, w ∈ E. It follows at once that|E±| = |E|.

The space of 1-forms (also called flows, vector fields, or cochains) on X is theset of all functions f : E± → R such that f(wv) = −f(vw) for all wv = (w, v) ∈ E±.The dimension of Ω1(X) is defined to be the cardinality |E| of the edge set E.

For notational purposes, we define Ω0 to be the set of all functions f : V → Rand Ω−1 to be the set of all functions g : ∅ → R. We shall identify Ω−1 with R.

The coboundary map d0 : Ω0(X)→ Ω1(X) takes f : Ω0(X)→ R to f : Ω1(X)

given by the formula f(ab) = f(a)− f(b). The boundary map ∂1 : Ω1(X)→ Ω0(X)takes f : Ω1(X)→ R to f : Ω0(X)→ R given by the formula

We define an inner product on Ω1(X) as follows:

〈f, g〉 =∑e∈E±

f(e)g(e).

With this inner product, we see that d0 = ∂∗1 , in the sense that

〈d0f, g〉 = 〈f, ∂1g〉for all f ∈ Ω0.

Similarly, we define the coboundary map d−1 : Ω−1(X) → Ω0(X) to be themap that sends each constant c ∈ R to the constant map c1. Considering d−1 as amatrix operator, we can define the boundary map ∂0 : Ω0(X)→ Ω−1(X) to be thetranspose of d−1.

Observe that im d−1 is the set of all constant maps on V . Since ker d is the setof all locally constant maps on V , viz., constant on every connected component ofX, it follows at once that im d−1 ⊆ ker d0, with the equality achieved if and only ifX is connected. The zeroth cohomology module of X is defined to be the quotientspace

H0(X) = ker d0/ im d−1.

Observe that the dimension of H0 is dim ker d0 − dim im d−1.We define the Laplacian ∆0 = ∂1d0 : Ω0(X)→ Ω0(X). Observe that

(∆0f)(v) = deg(v)f(v)−∑w∼v

f(w).

28

1. COHOMOLOGY AND SPECTRUM (DAY 1, PERIOD 2) 29

In particular, if X is k-regular, then ∆0 = kI −A. Observe that

ker ∆0 = ker ∂1d0 = ker d∗0d0 = ker d0.

To see this, we note that, if f ∈ ker d∗0d0, then

0 = 〈d∗0d0f, f〉 = 〈d0f, d0f〉,and so f ∈ ker d0.

In conclusion, we have the following inclusion relation:

(1.1) c1 : c ∈ R = im d−1 ⊆ ker d0 = ker ∆0.

Remark (Homology!). We pause here to note that

im ∆0 = im ∂1∂∗1 = im ∂1,

and im ∂1 is the set of all maps f : V → R that sum to zero on every connectedcomponent.

What is in spec ∆0? (1.1) implies that 0 ∈ spec ∆0; this is the trivial part ofspec ∆0. We call spec ∆0 r 0 the nontrivial part of spec ∆0. Since (im d−1)⊥ =ker ∂0, we see that the nontrivial part of spec ∆0 consists of the eigenvalues thatcome from functions that sum to zero. All in all, we have the following in spec ∆0:

• im d−1: 0• ker ∂0: the zero-sum maps.

– ker d0: 0,. . . ,0.– im ∂1: nonzero.

From this, we obtain the following Hodge decomposition of Ω0(X):

Ω0 = im d−1 ⊕ (ker ∂0 ∩ ker d0)⊕ im ∂1.

We define λ0(X) to be the minimum of all elements of spec ∆0 restricted to the“zero-sum maps” part. Note that λ0(X) = 0 if X is disconnected and λ0(X) > 0if X is connected.

1.2. Dimension 2. Let X = (V,E, T ) be a two-dimensional simplicial com-plex, which is a graph (V,E) with a set T of triangles, which are sets v, u, w suchthat v, w, v, u, v, w ∈ E. Similarly as in the 1-dimensional case, we definethe collection T± of oriented triangles to be the ordered variant of T . We note that|T±| = |T |.

The space of 2-forms on X is defined to be the set Ω2(X) of functions f : T± →R such that

f(vuw) = f(uvw)

for all vuw ∈ T±.[fill in later: definition of d1 and ∂2]Observe that

ker d1 =

f ∈ Ω1(X) :

∫∆

f = 0 over all triangles ∆

.

On the other hand,

im d0 =

f ∈ Ω1(X) :

∮f = 0 over all closed paths

.

The first cohomology module of X is defined to be the quotient space

H1(X) = ker d1/ im d0.

2. SPECTRUM, CHEEGER’S INEQUALITIES, AND MIXING (DAY 4, PERIOD 4) 30

ker d1 is commonly known as the space of closed forms; im d0 is commonlyknown as the space of exact forms. In other words, if H1(X) 6= 0, then there existsa function f ∈∈ Ω1(X) such that

∫∆f = 0 over all triangles ∆, but

∮f 6= 0 for

some closed path. This means there is a closed path that is not triangulizable; inother words, there is a closed path on X that cannot be realized as the boundaryof a triangle. Intuitively, this implies that X “has holes”.

We now define the Laplacian to be the map ∆1 = ∂2d1. The trivial part ofspec ∆1 comes from im d0, and the nontrivial part of spec ∆1 comes from (im d0)⊥ =ker ∂1, the Kirchoff flows. Since

im d0 ( ker ∆1 = ker d1

if and only if H1(X) 6= 0, we see that there exists a Kirchoff closed form if and onlyif there are holes on the complex X. This is due to Beno Eckmann.

The Hodge decomposition in this case is as follows:

Ω1 = im d0 ⊕ (ker d1 ∩ ker ∂1)⊕ im ∂2.

1.3. General Case. In general, we consider j-cells Xj± with orientation. We

can then define Ωj(X) and develop an analogous theory with respect to the map

(df)(σ) =∑

v:v∪σ∈Xj+1

f(v ∪ σ).

2. Spectrum, Cheeger’s Inequalities, and Mixing (Day 4, Period 4)

Let X = (V,E, T ) be a simplicial complex and take λ1 = min spec ∆1 |Z1.

Observe that λ1 = 0 if and only if (V,E, T ) has holes, i.e., H1 6= 0. What canwe say about λ1 0? Here is a multi-dimensional generalization of the Cheegerconstant (Chapter 2, Definition 1.1):

h1(X) = minA

∐B

∐C=V

A,B,C=∅

|T (A,B,C)|n|A||B||C|

.

Recall that, if we take the one-dimensional Cheeger constant to be

h0(X) = minA

∐B=V

|E(A,B)|n|A||B|

,

then we have the following Cheeger inequality:

h20

8k≤ λ0 ≤ h0,

where k is the maximum degree of a vertex of X.

Theorem 2.1 (Parzanchevski–Rosenthal–Tessler). If E = ( V2 ), then λ1 ≤ h1.In dimension d, for Xd−1 = ( Vd ), we have that λd−1 ≤ hd−1.

Proof. Given A∐B∐C = V , consider f ∈ Ω1. Given a = |A|, b = |B|, and

c = |C|, we set

f(vw) =

a v ∈ B and w ∈ C...

f ∈ Z1, and so

(∂1f)(v) = cb+ b(−c) = 0.


Therefore,

λ1 ≤〈∆1f, f〉〈f, f〉

,

as λ1 ≤ spec ∆1. It thus suffices to estimate the right-hand side.Observe first that

〈f, f〉 =∑e

f(e)2

=∑

c:A→Bc2 +

∑a:B→C

a2 +∑

b:C→A

b2

= abc2 + ab2c+ a2bc = (a+ b+ c)abc = nabc.

We also see that

〈∆1f, f〉 = 〈∂2d1f, f〉= 〈d1f, d1f〉

=∑t∈T

(d1f)(t)2

= |T (A,B,C)|n2.

It thus follows that

λ1 ≤|T (A,B,C)|n2

nabc≤ h1.

Remark. There is a generalization for non-complete skeletions by Szedlak–Gundert and Parzanchevski–Golubev.

We also recall the expander mixing lemma (Chapter 1, Theorem 1.2): spec ∆0|Z0 ⊆[k − ε, k + ε], and so, for all A and B,∣∣∣∣|E(A,B)| − kab

n

∣∣∣∣ ≤ √ε2ab.

Here is a higher-dimensional generalization of the mixing lemma:

Theorem 2.2. If spec ∆1|Z1⊆ [k − ε, k + ε] for any k > 0 and ε > 0, and if

E = ( V2 ), then ∣∣∣∣|T (A,B,C)| − kabc

n

∣∣∣∣ ≤ (ε3abc)3/2.

Proof. Let

1AB(e) =

1 if e : A→ B

−1 if e : B → A

0 otherwise.

Then

−〈∆1AB , 1AC〉 = −〈∂2d11AB , 1AC〉 = −〈d11AB , d11AC〉

Now,

(d11AB)(t) =

1 if t ∈ T (A,B,C) or t ∈ T (A,B, V r (A,B,C))

0 otherwise.


and

(d11AC)(t) =

1 if t ∈ T (A,C, V r (A ∪ C))

0 otherwise.

Observe that

|T (A,B,C)| = 〈(kI −∆1)1AB , 1AC〉

= 〈(kI −∆1)(1B1

AB + 1Z1

AB), 1AC〉.We estimate the two terms separately. Since spec(kI −∆1)|Z1

⊆ [−ε, ε], Cauchy–Schwarz implies that

|〈(kI −∆1)(1Z1

AB), 1AC〉| ≤ ‖(kI −∆1)1Z1

AB‖‖1AC‖≤ ε‖1AB‖‖1AC‖

= ε√ab√ac = εa

√bc.

As for the other term, we observe that

|〈(kI −∆1)(1B1

AB), 1AC〉| = k〈1B′

AB , 1AC〉.We claim that

ProjB′ =d0∂1

n,

which we call the lower Laplacian ∆−. To see this, we note that

∆−1 Z1 = d0∂1 ker ∂1 = 0.

Observe that

spec ∆−1 |B1=Z⊥1 =ker ∂⊥1 =(ker ∆1)⊥ = spec ∆−1 r 0= spec d0∂1 r 0= spec ∂1d0 r 0= spec ∆0 r 0= n

The last equality follows from the completeness of the graph.Now,

k

n〈∆−1 1AB , 1AC〉 =

k

n〈∂11AB , ∂11AC〉 =

kabc

n.

Finally, ∣∣∣∣|T (A,B,C)| − kabc

n

∣∣∣∣ ≤ εa√bc · εb√ac · εc√ab.

CHAPTER 8

Benny Sudakov: Extermal Results and Problemson Hypergraphs

1. Part 1 (Day 1, Period 3)

Definition 1.1. A k-graph is a hypergraph X = (V,E) such that each e ∈ Eis a subset of V of cardinality k.

Example 1.2. 2-graphs are just graphs:

Here is an example of a 3-graph:

1.1. Ramsey theory. k-graphs arise, for example, in Ramsey theory. Letrk(s, t) be the smallest positive integer N such that every red-blue coloring of k-tuples on [N ] contains either a red s-clique or a blue t-clique.

If k = 2, then we have the following result:

Theorem 1.3 (Erdos–Szekeres). 2n/2 ≤ r(n, n) ≤ 22n.

Sketch of proof. We first establish

r(s, t) ≤ r(s− 1, t) + r(s, t− 1)

and use this bound inductively to show that r(s, t) ≤ 2s+t.As for the lower bound, we let N = 2n/2 and color edges on [N ] randomly with

probability 1/2 for red and 1/2 for blue. Then the probability of the existence ofmonochromatic set of size n is bounded above by(

Nn

)· 2 · 2−(n2 ) 1.

33

1. PART 1 (DAY 1, PERIOD 3) 34

This bound is 70 years old, and there hasn’t been substantial improvementsince then. The current constructions are random.

Conjecture 1.4 (explicit construction?). Take [p] to be the vertex set, wherep is a prime number that is 1 mod 4. We declare x ∼ y if and only if x − y = z2.This is the Paley graph.

Theorem 1.5 (Ajtai–Komlos–Szemeredi). r(3, t) = Θ

(t2

log t

).

Remark. We recall that f = Θ(g) if and only if f ≤ cg and g ≤ c′f for someconstants c and c′.

For k = 3, we can make use of random coloring to establish the lower bound

r3(n, n) ≥ 2cn2

for some constant c.As for the upper bound, we have the following bound:

r3(s, t) ≤ r2(r3(s− 1, t), r3(s, t− 1)).

Of course, this isn’t a terribly nice bound: we end up with an iterated exponentialbound.

Theorem 1.6 (Erdos–Rado, 1950s). r3(s, t) ≤ 2( r(s,t)

2).

In particular,

r3(n, n) ≤ 22cn

.

Conjecture 1.7. r3(n, n) is of order 22cn

. Indeed, if we use 4-coloring, thenthere is the Erdos–Hajnal bound

r3(n, n, n, n) ≥ 22cn

,

and this construction is not random.

We could try and look at a simpler case:

Theorem 1.8 (Conlon–Fox–Sudakov). 2cn logn ≤ r3(4, n) ≤ 2cn2 logn.

1.2. Turan-type problems. We define ex(n,H) to be the maximum numberof edges in k-graph G on n vertices which contains no H.

Example 1.9. Here is a complete bipartite graph.

n2

n2

A 1907 result of Mantel shows that

ex(n,∆) =n2

4.


Definition 1.10. We define the chromatic number χ(H) of a hypergraph Hto be the minimal t such that the vertices of H can be t-colored without monochro-matic edges.

Theorem 1.11 (Erdos–Stone). Let H be a graph (k = 2). If χ(H) = r + 1,then

ex(n,H) =

(1− 1

r

)n2

2+ o(n2).

Let us now move on to the k = 3 case. Consider, for example, K34 , the 3-graph

on 4 vertices that has all 4 triples. Even in a simple case like this, ex is unknown.In fact, let us define

π(H) = limn→∞

ex(n,H)(n3

) ,

so that determination of π(H) is easier than that of ex(H). The following is open,with a bounty promised by Erdos.

Conjecture 1.12 (Turan). π(K34 ) = 5/9.

We conjecture that there are exponentially many “extremal” configurations.If k is an arbitrary positive integer, then we can consider Kk

k+1, the k-graph onk + 1 vertices that has all k + 1 k-tuples. The following bounds are known:

1

k≤ 1− π(Kk+1)k ≤ log k

2k.

2. Part 2 (Day 1, Period 5)

2.1. Turan-type problems, continued.

Definition 2.1. Let f(n, p, q) denote the maximum number of edges in 3-graphon n vertices such that every p vertices span q or more edges.

The most basic question we can ask is the (6,3)-problem. For this, every 6vertices should have 3 or more edges. This, in particular, means there is nothinglike this:

Therefore, we have the following inequality:(n2

)≥ 3e(H).

We also do not have anything like this:

(triangles picture)

Indeed, we have the following result, which follows from the triangle removal lemma(Chapter 1, Theorem 1.9):

Theorem 2.2 (Ruzsa–Szemeredi). f(n, 6, 3) = o(n2).


The above result of Ruzsa–Szemeredi implies the following classical number-theoretic result of Roth:

Theorem 2.3 (Roth). Fix δ. Let A ⊆ [N ] such that |A| = δN . As N → ∞,A contains a 3-term arithmetic progression.

Sketch of proof. Take A ⊆ [N ] such that A contains no 3-term arithmeticprogression. For x ∈ [N ] and a ∈ A, we consider (x, x+a, x+2a) ⊆ [N ]×[2N ]×[3N ],which gives rise to a hypergraph. Since A has no 3-term arithmetic progression,this hypergraph does not have anything of the following form:

(triangles picture)

Now, e(H) = |A|N = o(n2) by the Ruzsa–Szemeredi theorem, whence |A| = o(N).

Since f(n, 6, 4) already yields interesting applications, we expect the proof ofthe following result to come with intriguing new methods.

Conjecture 2.4. f(n, 7, 4) = o(n2).

So far, we have examined f(n, p, q) with p− q = 3. Let us now take a look atthe p− q = 2 case. In this case, we can have a hypergraph with cn2 edges, where cis an absolute constant.

Conjecture 2.5 (Erdos). There exists a 3-graph on n vertices with cn2 edgessuch that no set of size p ≤ p0 spans at most p0 − 2 edges. In fact, there shouldbe a Steiner triple system with this property; an STS is a collection of 1

3 ( n2 ) triplescovering every pair exactly once.

2.2. Cycles in hypergraph.

Definition 2.6. A cycle in a graph is a collection

(v1, v2), . . . , (vk−1, vk), (vk, v1)of edges.

For example,

is a cycle.

Definition 2.7. A Berge cycle in a hypergraph H is [fill in later]

For example,

v1 v2

x v3

l1 = (x, v1, v2), l2 = (x, v2, x3), l3 = (x, v3, v1)


is a Berge cycle.

Definition 2.8. The girth g(H) of a hypergraph H is the length of the shortestBerge cycle in H. The chromatic number χ(H) of H is the minimum t such thatthere exists a t-coloring of vertices with no monochromatic edges.

If the girth is large, then the hypergraph looks locally like a graph.

Now, trees are easy to color, so we are led to the following question: is χ(H) smallif g(H)? Not so, as the next theorem shows:

Theorem 2.9. There exists a 3-graph H with large g(H) and large χ(H).

Let us sketch the proof. To this end, we let n denote the number of vertices.

Put every triple in with probability p = n−2+ 1n+1 . Then the number of short cycles

is ∑2≤t≤e

(nt

)(nt

)pt n2lpl = nl/e+1 < n.

Now, the number x of independent sets in the hypergraph is bounded above by

n1− 12(l+1) polylog n

(nx

)(1− p)( x3 ) 1.

Therefore,

χ(H) ≥ n

x≈ n

12(l+1) = s.

So far, we have g(H) ≥ l and χ(H) ≥ s, with n ≈ s2(l+1). If H is a graph,then g(H) ≥ l and χ(H) ≥ s and at most sl+1 vertices, and we have an explicitconstruction.

Let us call G a (n, d, λ)-graph if G has n vertices, d-regular, and |λi| ≤ λ for alli ≥ 2. (See Chapter 3, Section 1 for notations and terminology.) We would like λ

to be as small as possible: λ ≈√d. The Hofman bound is as follows: if α(H) ≈ λ

dn,

then χ(H) ≥ dλ .

We conclude this section with a problem from coding theory. A mod2-cycle inH is a collection of edges covering every vertex even number of times. The size ofa cycle is the number of vertices covered at least once.

Feige suggested the following. Consider a 4-graph which has quadratic numberof edges, say, ( n2 ). The vertices are ( n2 ) pairs of vertices of H. If (x, y) and (x′, y′)are two disjoint pairs of edges in H, then we declare (x, y) ∼ (x′, y′).


Let N = ( n2 ). There are 12N ways to partition the vertices into disjoint pairs.

Therefore, 3e(H) = 3N , and so G ⊇ G′ is a graph with degree at least 3. It followsthat in O(log n) steps we have a collision, a cycle.

Conjecture 2.10. That even εn2 4-tuple, give a short mod2-cycle (shortpolylog n?)

CHAPTER 9

Uli Wagner: Coboundry Expansion andTopological Overlapping

1. Definitions (Day 2, Period 3)

Let X = (V,E) be a graph. Denote by δS the set E(S, V rS) of edges betweenS and V r S.

Definition 1.1. X is said to be η-edge expanding (η > 0) if, for all S ⊆ V ,

(1.2)|δS||S|≥ ηmin|S|, |V r S|

|V |.

The edge-expansion constant of X is the minimal η such that (1.2) holds for allS ⊆ V .

We consider X as a one-dimensional simplicial complex with underlying space‖X‖. X is η-edge expanding, and so, for all f : ‖X‖ → R1, there exists a p ∈ R1

such that

(1.3) |e ∈ E : f(e) 3 p| > λ|E|,where λ = η

2 . Note that edge expansion is not required for (1.3): a weaker hypoth-esis, such as having large bisection width, would suffice.

Generalizing, we now consider a d-dimensional simplicial complex X with un-derlying space ‖X‖. We let Xk denote the set of all k-simplices of a finite sim-plicial complex X, so that ‖X‖ =

⋃σ∈X σ. We consider the space of k-chains

Ck(X) = Ck(X;F2) and the space of k-cochains Ck(X) = Ck(X;F2), both of which

are linearly isomorphic to FXk2 . We also consider the map ∂ : Ck(X) → Ck−1(X)

given by the formula

∂σ =∑

τ∈Xk−1

τ⊆σ

τ

for all σ ∈ Xk, and the map δ : Ck(X)→ Ck+1(X) given by the formula

(δc)(σ) = c(∂σ).

for all σ ∈ Xk+1. We obtain the following cochains with coefficients in F2; here weset X−1 = ∅ for notational convenience:

F2 = C−1(X)δ−→ C0(X)

δ−→ · · · δ−→ CdimX(X).

We remark that ∂2 = 0 and δ2 = 0. We also define the usual objects of cohomologytheory:

• Bk(X) = im(δ : Ck−1(X)→ Ck(X))• Zk(X) = ker(δ : Ck(X)→ Ck+1(X))• Hk(X) = Zk(X)/Bk(X).

39

1. DEFINITIONS (DAY 2, PERIOD 3) 40

See Chapter 7, Section 1 for details on the theory of cohomology of graphs.Define the Hamming norm of c ∈ Ck(X) by

|c| = |σ ∈ Xk : c(σ) = 1|.The normalized Hamming norm of c is

‖c‖ =|c||Xk|

.

We now introduce three geometric notions.

Definition 1.4. Let η, µ, L > 0. X satisfies an L-coisoperimetric inequality indimension k if, for all b ∈ Bk(X), there exists an a ∈ Ck−1(X) such that a = δband ‖a‖ ≤ L‖b‖. Equivalently, X satisfies an L-coisoperimetric inequality if andonly if every a ∈ Ck−1 satisfies the inequality

‖δa‖ ≥ 1

Lmin

‖a+ z‖ : z ∈ Zk−1(X)

.

We say that a co-fills b if a = δb.

Remark. In dimension 1, L-coisoperimetry is equivalent to 1L -edge expansion

on each connected component.

Definition 1.5. Let η, µ, L > 0. X has η-coboundary expansion in dimensionk if, for each a ∈ Ck−1(X), the following inequality holds:

‖δa‖ ≥ ηmin‖a+ b‖ : b ∈ Bk−1(X).

Remark. In dimension 1, η-coboundary expansion is equivalent to η-edge ex-pansion.

We remark that X has η-coboundary expansion in dimension k if and only if Xsatisfies a 1

δ -coisoperimetric inequality in dimension k and Hk−1(X) = 0. Indeed,

a /∈ Bk−1(X) if and only if

‖[a]‖ = min‖a+ b‖ : b ∈ Bk−1(X) > 0.

Definition 1.6. Fix µ > 0 and k ∈ N. X is said to satisfy the µ-cosystolicinequality in dimension k if, for all z ∈ Zk(X)/Bk(X), we have the inequality

‖z‖ ≥ µ.

Remark. In dimension 0, µ-cosystolic inequality is equivalent to the fact thatevery connected component of X has size at least µ|X0|.

These definitions generalize to arbitrary cell complexes. For intuition, we as-sume that X is a cell decomposition of a d-dimensional PL manifold without bound-ary M . Letting Y denote the dual cell complex, we see that there is a one-to-onecorrespondence between Xk and Yd−k. We then have the following commutativediagram:

Ck−1(X) Ck(X)

Cd−k+1(Y ) Cd−k(Y )

δ

∼= ∼=

∂

For example, for the 1-dimensional case (edges), the expansion / coisoperimetryof X correspond to the top-dimensional isoperimetry in Y . For the 2-dimensional

2. OUTLINE OF THE PROOF (DAY 3, PERIOD 3) 41

case, the expansion / coisoperimetry of X corresponds to codimension-1 isoperime-try in Y .

We now state the main result of this chapter. To this end, we shall make useof the following technical condition1:

Definition 1.7. X is locally ε-sparse if, for each k ∈ 1, . . . ,dimX and everyv ∈ X0, the following inequality holds:

|σ ∈ Xk : v ∈ σ| ≤ ε|Xk|.

Theorem 1.8 (Gromov’s topological overlap theorem). Let X be a finite sim-plicial complex with underlying space ‖X‖ that is locally ε-sparse for some ε > 0.Let L, µ > 0 and d ∈ N and fix ε > 0 that is sufficiently small with respect toL, µ, and d. Suppose that X has L-isoperimetry in dimensions k = 1, . . . , d andsatisfies the µ-cosystolic inequality in dimensions k = 1, . . . , d− 1. Then, for eachcontinuous map f : ‖X‖ → Rd, there exists p ∈ Rd such that

|σ ∈ Xd : f(σ) 3 p| ≥ λ|Xd|,

where λ = λ(L,M, d, ε) > 0. In this case, we say that X has topological λ-overlapfor maps into Rd.

An analogous problem is to consider only linear maps—this goes by the nameof geometric overlap. We will not discuss this topic, however.

2. Outline of the Proof (Day 3, Period 3)

The outline of the proof is as follows:

(1) Without loss of generality, f is a piecewise-linear map in “general position.”(2) Observe that Rd ∼= Rd × 1 ⊆ Rd+1 View f as a piecewise-linear map in

“general position”, so that f : ‖X‖ → ∂∆d+1 ∼= Sd. Then take a “sufficientlyfine” triangulation T of ∂∆d+1 in “general position” with respect to f .

(3) Define, for every k-simplex τ of T and every (d − k)-simplex σ of X, f ](τ) tobe the algebraic intersection number between f(σ) and τ : this is given by

|f(σ) ∩ τ | mod 2

if dim τ > 0 and by

|f−1(τ) ∩ σ| mod 2

if dim τ = 0. We can then extend linear to obtain f ] : Ck(T )→ Cd−k(X). Thekey fact is that the following diagram commutes:

Ck(T ) Cd−k(X)

Ck−1(T ) Cd−k+1(X).

f]

∂ δ

f]

(4) We need to make the notion of “sufficiently fine” precise: for each k > 0 andevery τ ∈ Xk,

‖f ](τ)‖ < d2ε|Xd−k|.

1Wagner: “I am not sure to what extent this condition is needed.”

3. PROOF (DAY 4, PERIOD 2) 42

3. Proof (Day 4, Period 2)

Lemma 3.1. Let [T ] =∑τ∈Td

τ . Then f ]([T ]) = 1 ∈ C0(X).

Lemma 3.2. There exists a v ∈ T0 such that ‖f ](v)‖ ≥ λ.

We now establish the following commutative diagram:

0 Cd(T ) Cd−1(T ) · · · C1(T ) C0(T ) 0

0 C0(X) C1(X) · · · Cd−1(X) Cd(X) 0

∂

g f]

∂

f]gh

∂ ∂

f]gh

f]gh

δ δ δ

To this end, we begin by observing that f ] is a chain-cochain map (Section 2, Step3). Define another chain-cochain map g : Ck(T ) → Cd−k(X) by setting g(c) = 0for all c ∈ Ck(T ). Observe the following:

Lemma 3.3. It follows from Lemma 3.2 that there exists a map h : Ck(T ) →Cd−k−1(X) such that

f ](c) = δh(c) + h(∂c)

for all c ∈ Ck(T ) and that ‖h(τ)‖ ≤ Sk for all τ ∈ Tk.

Suppose Lemma 3.3 is true and Lemma 3.2 is false. Consider

1 = f ]([T ]) = δ(h[T ]) + h(∂[T ]),

but the right-hand side is 0.We now construct h by induction on k. What we need is that, for τ ∈ Tk,

δh(τ) = f ](τ) + h(∂τ).For k = −1, we set h = 0.For k = 0, we consider v ∈ T0. Observe that

f ](v) = f ](v)− f ](v0) = f ](v − v0) = f ](∂c) ∈ Bd(X). = δf ](c),

provided that there exists a 1-chain c ∈ C1(T ) such that ∂c = v − v0. Existencefollows from connectedness of Sd. We now choose h(v) ∈ Cd−1(X) such thatδh(v) = f ](v) and ‖h(v)‖ < L‖f ](v)‖ < δ. We can set S0 = Lλ.

We now let k > 0 and assume that h is defined on Ck−1(T ). Consider τ ∈ Tk.f ](τ) is defined, and h(∂τ) is defined. We also have that ‖f ](τ)‖ ≤ εd2 and that‖h(∂τ)‖ ≤ (k + 1)Sk−1. Observe that

δ(f ](τ)− h(∂τ)) = δf ](τ)− δh(∂τ)

= f ](∂τ)− δh(∂τ)

= f ](∂τ)− h(∂∂τ)

= 0.

Therefore, z = f ](τ) − h(∂τ) ∈ Zd−k(X). If z /∈ Bd−k(X), then M ≤ ‖z‖ ≤d2ε + (k + 1)Sk−1. Else, we choose h(v) such that δh(v) = z = f ](v) − h(∂v). Inthis case,

‖h(v)‖ ≤ L(d2ε+ (k + 1)Sk−1) := Sk.

3. PROOF (DAY 4, PERIOD 2) 43

We now observe that

S0 = Lλ

Sk = d2ε(L+ L2 + · · ·+ Lk) + (k + 1)!Lk+1λ < M for suitable λ.

We can now set

λ = min1

(k + 1)!Ll+1.

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