Research Statement, Steven Heilman

I am interested in applying analytic techniques to probabilistic problems. These problems aresometimes motivated by hardness results in theoretical computer science. A recurring theme of ourresearch is that mathematical objects that were initially motivated by physics (such as semigroups ofoperators, hypercontractivity, Fourier analysis, soap bubbles and minimal surfaces, Bell’s inequalityfrom quantum mechanics, optimization of energy functionals, the calculus of variations, etc.) nowhave renewed motivation from important problems in computer science.

For example, we recently proved a noncommutative version of a nonlinear Central Limit The-orem [HV16], using some techniques from Fourier analysis [MOO10], random matrix theory, andfunctional analysis [Gro72, CL93]. Our result then implies that a very general computational prob-lem, the noncommutative Grothendieck problem, cannot be solved efficiently (assuming the UniqueGames Conjecture; see Section 1.6). That is, our Central Limit-type theorem implies that com-puters cannot run quickly when solving a wide class of problems. Put another way, our result inanalysis in Euclidean space has ramifications for the analysis and probability of discrete functions.Or put another way, the best constant in Bell’s inequality from quantum mechanics provides aprecise quantitative barrier for computation.

For another example, one main focus of my research has been isoperimetric problems in Euclideanspace equipped with the Gaussian measure [HJN13, Hei14, HMN16, Hei15, Hei16b, Hei16a]. Theseoptimization problems ask for sets of fixed Gaussian volume with smallest Gaussian perimeter.Some additional constraints are added to the sets, and various notions of perimeter are used, someof which depend on the entire set and not just its boundary. We have proved some special cases andprovided counterexamples for some of these conjectures. My long term goal is to develop generalmethods for approaching these problems, since such general methods do not exist. Most recently,I have been developing calculus of variations techniques [Hei16b, Hei16a]. The “classical” calculusof variations does not apply to these problems.

Currently, there are many unsolved Gaussian isoperimetric problems, and their resolution willyield countless dividends. The Gaussian measure is most interesting since it is almost inter-changeable with the uniform measure on the discrete hypercube, via Central Limit Theorems[Rot79, Cha06, MOO10, Mos10, IM12]. These Gaussian isoperimetric results therefore imply in-equalities on the discrete hypercube. The discrete inequalities are then applied to machine learningand Grothendieck inequalities [KN09, KN13, HV16], to the Unique Games Conjecture [KKMO07,MOO10, KM16], to semidefinite programming algorithms such as MAX-CUT [KKMO07, IM12], tosocial choice theory [MOO10, IM12], to learning theory [FGRW12], to communication complexity[CR11], etc. So, solving these isoperimetric problems can tell us how quickly computers can run,and how to design elections so that erroneous tabulation of votes does not affect the outcome ofthe election.

I have also studied Lp Poincare inequalities for 1 < p <∞ [HMO14] on general measure spacesthat do not seem to be provable using standard techniques, such as Littlewood-Paley theory. Inparticular, we proved the degree one case of a conjecture of [MN14]. The conjectured Poincareinequalities [MN14] can be used to construct graphs which are expanders in a very general sense[MN14]. The explicit construction of expander graphs are then used in computer science [HLW06].

1. Noncommutative Majorization Principle

1.1. Commutative Invariance Principles. The invariance principles of [Rot79, Cha06, MOO10]are nonlinear versions of the Central Limit Theorem, with error bounds. That is, the invarianceprinciple implies the Berry-Esseen Central Limit Theorem, which we now recall.

Let n be a positive integer. Let x1, . . . , xn be commutative indeterminate variables, and let

Q(x1, . . . , xn) :=x1 + · · ·+ xn√

n.

1

Let b1, . . . , bn be independent identically distributed (i.i.d.) uniform random variables in {−1, 1},and let g1, . . . , gn be i.i.d. standard Gaussian random variables. Letting E denote expected value,we then define the 2-norm of Q to be

(1) ‖Q‖2 := (E |Q(b1, . . . , bn)|2)1/2.

The Berry-Esseen Central Limit Theorem then says

(2) supt∈R|P(Q(b1, . . . , bn) ≤ t)− P(Q(g1, . . . , gn) ≤ t)| ≤ 3 max

i=1,...,n

∥∥∥∥ ∂

∂xiQ

∥∥∥∥2

.

If the rightmost expression looks unfamiliar, note that Q(g1, . . . , gn) has a standard Gaussiandistribution, and ‖ ∂

∂xiQ‖2 = 1/

√n for all i ∈ {1, . . . , n}. The proof of (2) can also be extended to

moments of Q:

(3)∣∣∣E |Q(b1, . . . , bn)|4 − E |Q(g1, . . . , gn)|4

∣∣∣ ≤ 240 maxi=1,...,n

∥∥∥∥ ∂

∂xiQ

∥∥∥∥2

.

A similar statement can be made for higher moments of Q. The commutative invariance principle[Rot79, Cha06, MOO10] implies, among other things, that (3) holds for multilinear polynomials.

Let d ∈ N. Let Q(x1, . . . , xn) be a multilinear polynomial of degree d, so that

Q(x1, . . . , xn) =∑

S⊆{1,...,n} : |S|≤d

cS∏i∈S

xi, cS ∈ R, ∀S ⊆ {1, . . . , n}.

Assume that ‖Q‖2 ≤ 1. Then the commutative invariance principle [MOO10] says that

(4)∣∣∣E |Q(b1, . . . , bn)|4 − E |Q(g1, . . . , gn)|4

∣∣∣ ≤ 24 · 10d maxi=1,...,n

∥∥∥∥ ∂

∂xiQ

∥∥∥∥2

.

The commutative invariance principle (4) in [MOO10] is proven by a combination of the Lin-deberg replacement argument and the hypercontractive inequality [Sta59, Fed69, Bon70, Nel73,Gro75, Bec75] (see (17) below). That is, one replaces one argument of Q at a time, adding upthe resulting errors and controlling them via the hypercontractive inequality. The invariance prin-ciple (4) has seen many applications [O’D, O’D14] in recent years. Here is a small sample ofsuch applications and references: isoperimetric problems in Gaussian space and in the hypercube[MOO10, IM12], social choice theory, Unique Games hardness results [KKMO07, IM12], analysis ofalgorithms [BR15], random matrix theory [MP14], free probability [NPR10], optimization of noisesensitivity [Kan14], etc. We anticipate that our noncommutative version of (4) (see (7) below)could have similar applications as well .

1.2. Grothendieck’s Inequality. The first application of (4) we will describe is computationalhardness for the commutative Grothendieck inequality [RS09]. For any x = (x1, . . . , xn), y =

(y1, . . . , yn) ∈ Cn, define 〈x, y〉 :=∑n

i=1 xiyi and define ‖x‖2 :=√〈x, x〉. Let {aij}ni,j=1 be a real

matrix. Proven first in 1953, Grothendieck’s Inequality [Gro53, LP68, AN06, BMMN13] saysthere exists a constant K > 0 which does not depend on n or on {aij}ni,j=1 such that

(5) supw1,...,wn,r1,...,rn∈R2n−1

‖wi‖2=‖ri‖2=1, ∀ i=1,...,n

n∑i,j=1

aij〈wi, rj〉 ≤ K · supu1,...,un,ν1,...,νn∈{−1,1}

n∑i,j=1

aijuiνj .

That is, for a general optimization problem (corresponding to the left side of (5)), it is possibleto “round” the unit vectors wi, ri, i = 1, . . . , n to a one-dimensional set of unit vectors ui, νi,i = 1, . . . , n. And the weighted sum of inner products of the vectors does not decrease very muchafter we perform this rounding procedure. It is known that K < π/(2 log(1 +

√2)) [BMMN13],

and that a rounding procedure can establish the best constant in Grothendieck’s inequality [NR14].2

However, it remains a major open problem to find this optimal rounding procedure and to find thesmallest possible constant K in Grothendieck’s inequality.

Finding the smallest possible constant K in (5) has several interpretations beyond mathematics.From the physics perspective, the best constant K in (5) is also the smallest constant in certainBell inequalities in quantum mechanics [Pis12]. More specifically, Bell’s inequality says that thesmallest K possible in (5) satisfies K > 1. From the computer science perspective, assuming theUnique Games Conjecture (see Section 1.6), it is impossible to approximate the right side of (5),in time polynomial in n, within a multiplicative factor smaller than K, where K is the smallestpossible constant in the inequality (5) [RS09]. Mathematically, (5) can be rewritten as a ratiobetween two tensor product norms. One could consider this breadth of interpretation as evidencefor the difficulty and importance of finding the smallest possible value of K in (5).

1.3. The Noncommutative Grothendieck Inequality. Since (5) is very general, and sincerounding procedures appear often in theoretical computer science [AN06, LOGT12, ABS10], in-equality (5) has many applications. Similarly, the noncommutative version of (5) has many ap-plications [NRV14]. The noncommutative Grothendieck inequality was conjectured in [Gro53] andproven in [Pis78, Kai83]. The noncommutative Grothendieck inequality was also given an algorith-mic interpretation in [NRV14]. This algorithmic interpretation gives constant factor approximationalgorithms for variants of the Principle Component Analysis problem and a generalized Procrustesproblem.

Let n,m be positive integers and let S be a set. We denote Mm(S) as the set of m × m ma-trices (Uij)

mi,j=1 such that Uij ∈ S for all i, j ∈ {1, . . . ,m}. We write U ∈ Um(Cn) if U ∈ Mn(Cn)

and∑m

k=1〈Uik, Ujk〉 =∑m

k=1〈Uki, Ukj〉 = 1{i=j}. In particular, Um(C) is the set of m × m com-plex unitary matrices. Let Mijk` ∈ Mm(Mm(C)). Then the Noncommutative GrothendieckInequality for Complex Scalars [Pis78, Kai83] says

(6) supU,V ∈Um(Cn)

∣∣∣∣∣∣m∑

i,j,k,`=1

Mijk`〈Uij , Vk`〉

∣∣∣∣∣∣ ≤ 2 · supX,Y ∈Um(C)

∣∣∣∣∣∣m∑

i,j,k,`=1

Mijk`XijYk`

∣∣∣∣∣∣ .If Mijk` = 0 for all i, j, k, ` ∈ {1, . . . ,m} such that either i 6= j or k 6= `, then (6) becomes (5)(replacing R by C) [NRV14]. In this way, (6) generalizes (5). Also, (6) can be proven by somehow“rounding” the vector-valued matrices U, V to scalar-valued matrices X,Y .

The constant 2 is the smallest possible constant in (6), in stark contrast to (5), where the smallestconstant is still unknown. As in [RS09], we seek an algorithmic interpretation of the sharp constant2 in (6) in terms of the Unique Games Conjecture.

1.4. Our Contribution: Noncommutative Moment Majorization. The technical heart ofthe computational hardness result of [RS09] for (5) is the invariance principle (4). So, in order togeneralize the result of [RS09] to the noncommutative inequality (6), we need to prove a noncom-mutative version of (4). That is, we need to allow the polynomials Q to be matrix-valued, and weneed to replace the random variables in (4) with random matrices.

We now describe our noncommutative version of the invariance principle (4) [HV16], which then

implies computational hardness for (6). If Y is a complex matrix, we write |Y | := (Y Y ∗)1/2.We first define a noncommutative multilinear polynomial. Let X1, . . . , Xn be noncommutative

m ×m indeterminate variables. Let Q(X1, . . . , Xn) be a noncommutative multilinear polynomialof degree d ∈ N. That is, for any S ⊆ {1, . . . , n}, there exists m×m complex matrices cS such that

Q(X1, . . . , Xn) =∑

S⊆{1,...,n} : |S|≤d

cS∏i∈S

Xi.

3

Also, the product terms are in increasing order, so e.g.∏i=1,2,3Xi = X1X2X3. If p > m, we can

extend Q to take as input p× p indeterminate variables X1, . . . , Xn by defining

Qι(X1, . . . , Xn) :=∑

S⊆{1,...,n} : |S|≤d

(cS 00 0

)∏i∈S

Xi.

Let p > m, p,m ∈ N. Let d ∈ N. Let Im denote the m ×m identity matrix and let ‖·‖ denotethe operator norm of a matrix. We say that random matrices G1, . . . , Gn are i.i.d. if the matricesare independent, and the distribution of Gi is equal to the distribution of Gj for any 1 ≤ i, j ≤ n.(This condition is more general than the matrices having i.i.d. entries.)

Let Q(X1, . . . , Xn) be a noncommutative multilinear polynomial of degree d. Assume that‖Q‖2 ≤

√m, where ‖Q‖2 is defined in (1). Let H1, . . . ,Hn be i.i.d. uniformly random p×p unitary

matrices. Let G1, . . . , Gn be i.i.d. random m×m matrices that satisfy EG1G∗1 = Im, EG1 = 0, and

‖E |G1|4 ‖ ≤ 10. Then the Noncommutative Majorization Principle for Fourth Moments[HV16] says

(7)

E1

mTr

∣∣∣∣Qι((G1 00 0

)H1, . . . ,

(Gn 00 0

)Hn

)∣∣∣∣4≤ E

1

mTr |Q(b1, . . . , bn)|4 +m4240d max

i=1,...,n

∥∥∥∥ ∂

∂xiQ

∥∥∥∥2

+Om,n

(1

p

).

Comparing (7) with (4), we see that (7) is a weaker assertion. Fortunately, (7) is sufficient forapplications. Moreover, (4) is false in the generality of (7), so we cannot hope to improve (7) to(4). We can prove more general statements than (7), addressing higher moments of Q and theoperator norm of Q. The proof of (7) uses the Lindeberg replacement argument, Fourier analysisand noncommutative hypercontractive inequalities [Gro72, CL93].

The idea of (7) is that as p → ∞ and m,n remain fixed, Om,n (1/p) → 0. Then (7) resembles(4), using normalized trace moments in place of the fourth moment.

1.5. Our Application: Hardness for the Noncommutative Grothendieck Problem. In-equality (7) implies the following computational hardness result [HV16]:

Assuming the Unique Games Conjecture, no polynomial time algorithm (in m) can approximatethe quantity

supU,V ∈Um(C)

∣∣∣∣∣∣m∑

i,j,k,`=1

Mijk`UijVk`

∣∣∣∣∣∣within a multiplicative factor smaller than 2. That is, the left side of (6) (which can be computedin polynomial time, since it is a semidefinite program) is the best way to approximate the rightside of (6) in polynomial time.

We should mention that this hardness result was proven recently in [BRS15] using entirelydifferent techniques. However, our approach also proves computational hardness for other versionsof the noncommutative Grothendieck inequality [BKS16], to which the approach of [BRS15] doesnot seem to apply.

Over the last two decades, many sharp computational hardness results of this type have beenproven [KKMO07, KR08, KN09, KN13, KN12, IM12, RS09]. And essentially all of these resultsused the commutative invariance principle (4). So, our hardness result appears to be the first useof a noncommutative invariance-type principle.

1.6. The Unique Games Conjecture. The Unique Games Conjecture [Kho02] is a standardassumption in theoretical computer science. This conjecture can be considered a contemporaryproxy for the assumption that P 6=NP. That is, proving or disproving the Unique Games Conjectureis expected to have similar significance and consequences to proving or disproving P6=NP. Moreover,

4

both problems are closely related. As we will describe below, the Unique Games Conjecture canbe succinctly stated as: approximate linear algebra is hard.

Definition 1.1 (Γ-MAX-2LIN(p)). Let p ≥ 2 be a prime number. We define the Γ-MAX-2LIN(p) problem. In this problem, we are given n ∈ N and 2n variables xi ∈ Z/pZ, i ∈ {1, . . . , 2n}.We are also given a matrix {aij}2ni,j=1 with aij ≥ 0 for all i, j ∈ {1, . . . , 2n} and a set E ⊆ {1, . . . , n}×{1, . . . , n} with n elements. An element (i, j) ∈ E corresponds to one of n linear equations of theform xi − xj = cij(mod p), where cij ∈ Z/pZ. The goal of the Γ-MAX-2LIN(p) problem is to findthe following quantity:

(8) max(x1,...,x2n)∈(Z/pZ)2n

∑(i,j)∈E :

xi−xj=cij(mod p)

aij .

That is, we need to maximize the (weighted) number of equations xi − xj = cij(mod p) that aresatisfied.

Conjecture 1.2 (Unique Games Conjecture, [Kho02, KKMO07]). For every ε ∈ (0, 1), thereexists a prime number p(ε) such that no polynomial time algorithm (with respect to the parametern) can distinguish between the following two cases, for instances of Γ-MAX-2LIN(p(ε)) with aij = 1for all i, j ∈ {1, . . . , 2n}:

(i) (8) is larger than (1− ε)n, or(ii) (8) is smaller than εn.

If (8) were equal to n, then we could find (x1, . . . , x2n) achieving the maximum in (8) by Gauss-ian elimination. One can therefore interpret the Unique Games Conjecture as an assertion thatapproximate linear algebra is hard. Recently, it has been shown that there is a subexponentialtime algorithm that can distinguish case (i) from case (ii) above [ABS10], leading some to believethat the Unique Games Conjecture could be false. On the other hand, there is recent evidence thatthe conjecture could still be true [KM16, KS16]. The truth or falsity of this conjecture remains amajor open problem.

2. The Standard Simplex Conjecture

2.1. Euclidean Isoperimetry. Classical isoperimetry can be traced to ancient times, though itsfull understanding still remains incomplete. Generally speaking, we look for an object with leastperimeter among all objects of fixed volume. And we expect that the smallest perimeter object hasa simple structure.

Isoperimetric problems and minimal surface theory have a long history in differential geometry[Ste38, Sch72, Min96, Wei27, Hur02, Lev51, Sim68, Law70, Bor75, Alm76, Gro83, Sim83, BdC84,EH89, Tal95, BL96, HMRR02, CM12, CIMW13], with some motivation from the physics of soapbubbles [Pla73, Tay76]. In particular, soap bubbles we encounter in the real world are minimalsurfaces. Many deep results have come from these investigations, and we hope to continue thistradition.

To see that our knowledge is still limited, note that we still do not know the three disjoint setsof fixed Euclidean volume with minimum total surface area. This problem is known as a triplebubble, or multi bubble problem [HMRR02, Rei08, CCH+08]. Further below, we will discuss aGaussian version of this problem, known as the Standard Simplex Conjecture, which is also stillunsolved. Gaussian isoperimetric problems have gained recent interest due to their applicationsin theoretical computer science. For background motivation, we begin with the usual Euclideanisoperimetric inequality.

Let n ≥ 1 be an integer, let A ⊆ Rn be a Borel set with smooth boundary ∂A. Let voln(A)denote the Euclidean volume of A, and let voln−1(∂A) denote the Euclidean surface area of ∂A.

5

Let r > 0 and let B(0, r) := {(x1, . . . , xn) ∈ Rn : x21 + · · ·+ x2

n ≤ r2} be a Euclidean ball such thatvoln(A) = voln(B(0, r)). The Classical Isoperimetric Inequality says that the Euclidean ballhas the smallest boundary among all sets of fixed volume:

(9) voln(A) = voln(B(0, r)) =⇒ voln−1(∂A) ≥ voln−1(∂B(0, r)).

Let x = (x1, . . . , xn) ∈ Rn, and recall that ‖x‖2 := (x21 + · · · + x2

n)1/2 = 〈x, x〉1/2. Let dx be

Lebesgue measure on Rn, and let dγn(x) := e−‖x‖22/2(2π)−n/2dx. Let f : Rn → R be a bounded

function and let t ≥ 0. Let ∆ := −∑n

i=1 ∂2/∂x2

i . Let e−t∆f(x) denote the classical heatsemigroup applied to f . That is, for any x ∈ Rn,

e−t∆f(x) :=

∫Rn

f(x+ y√

2t )dγn(y).

The classical isoperimetric inequality (9) is a consequence of the following inequality for the heatsemigroup e−t∆, which can be proven via symmetrization [Led96].

(10) voln(A) = voln(B(0, r)) =⇒ ∀ t ≥ 0,

∫Rn

1A · e−t∆1Adx ≤∫Rn

1B(0,r) · e−t∆1B(0,r) dx.

So, as heat flows out of a given set, the one that preserves the most of its heat is the ball. Toget (9) from (10), let t → 0 in (10). The quantity

∫Rn 1A · e−t∆1Adx is sometimes called the heat

content of A. We will focus on statements of the form (10) below.

2.2. Gaussian Isoperimetry. For applications to theoretical computer science, it is more usefulto have a Gaussian version of (10). To this end, we first replace the heat semigroup by the Ornstein-Uhlenbeck semigroup. Define the operator L by Lf(x) := 〈x,∇f(x)〉+ ∆f(x), for any x ∈ Rn. Forany x ∈ Rn and t ≥ 0, define the Ornstein-Uhlenbeck semigroup applied to f : Rn → R by

e−tLf(x) :=

∫Rn

f(xe−t + y√

1− e−2t )dγn(y).

Let A ⊆ Rn be a Borel set. Denote γn(A) :=∫A dγn(x). Let H ⊆ Rn be a half space. That is,

H is the region that lies on one side of a hyperplane. For brevity, we will not state the Gaussiananalogue of (9), and we instead state the Gaussian analogue of (10) [Bor85], since the latter impliesthe former.

(11) γn(A) = γn(H) =⇒ ∀ t ≥ 0,

∫Rn

1A · e−tL1Adγn ≤∫Rn

1H · e−tL1Hdγn.

The quantity∫Rn 1A · e−tL1Adγn is referred to as the noise stability of the set A with parameter

e−t. Borell’s original result (11) from [Bor85] has been highly influential, and we wish to duplicatehis success by generalizing (11).

2.3. Social Choice Theory. By combining (11) with the invariance principle (4), the work[MOO10] solved the Majority is Stablest problem. This problem says that the most noise-stableway to determine the winner of an election between two candidates is to take the majority. Thisresult assumes that no one person has too much influence over the election’s outcome. The rigorousstatement of this problem uses functions with domain {−1, 1}n (see Section 5.1). Finally, inequality(11) should be considered well-understood, due to recent proofs [MN15, Eld15] of stability versionsof (11).

Applications of mathematics to the analysis of elections arguably began with Marquis de Con-dorcet in the 1700s, with further developments by Game Theorists such as Shapley, Shubik andBanzhaf in the 1950s and 1960s [SS54, Ban65]. In the last three decades, discrete Fourier anal-ysis has provided new insights into social choice theory for mathematics and computer science[KKL88, Kal02, MOO10, Mos12]. Yet, the analogue of the Majority is Stablest Theorem for threecandidates is still unresolved [IM12], since the three set version of (11) is still unresolved (see (12)below).

6

2.4. Gaussian Isoperimetry for multiple sets. Borell’s inequality (11) gives a sharp compu-tational hardness result for the MAX-CUT problem [KKMO07]. The MAX-CUT problem asks forthe partition of the vertices of an undirected graph into two disjoint sets that maximizes the numberof edges going between the two sets. One can find a partition of the vertices achieving .87856 timesthe maximum number of cut edges in polynomial time [GW95]. And assuming the Unique GamesConjecture, the constant .87856 is the best possible. When we modify the MAX-CUT problem toallow a partition of the vertices of the graph into k > 2 disjoint sets, we get the MAX-k-CUT prob-lem. And for this problem, there is only a conjecture for the best possible approximation that canbe done in polynomial time. This same conjecture would also solve the analogue of the Majorityis Stablest problem for more than two candidates. This conjecture, formulated by Isaksson andMossel in [IM12], is called the Standard Simplex Conjecture.

The Standard Simplex Conjecture asks for the minimum total Gaussian perimeter of a partitionof Rn into k > 2 sets, each of Gaussian measure 1/k. This conjecture, stated in (12), does not seemto follow from a symmetrization argument [BS01, IM12], or from the methods of [MN15, Eld15].

Let A1, . . . , Ak ⊆ Rn with 3 ≤ k ≤ n + 1, n ≥ 2, ∪ki=1Ai = Rn, γn(Ai) = 1/k. We nowdescribe the conjectured minimizer of the total Gaussian perimeter. Let z1, . . . , zk ∈ Rn be thevertices of a regular k-simplex, which is centered at the origin of Rn. For any i = 1, . . . , k, letBi := {x ∈ Rn : 〈x, zi〉 = maxj=1,...,k〈x, zj〉}. Then {Bi}ki=1 is a partition of Rn into k regularsimplicial cones. Generalizing (11), the Standard Simplex Conjecture says

(12)

γn(Ai) = 1/k, ∀ i = 1, . . . , k ∧ ∪ki=1Ai = Rn

=⇒ ∀ t ≥ 0,k∑i=1

∫Rn

1Ai · e−tL1Aidγn ≤k∑i=1

∫Rn

1Bi · e−tL1Bidγn.

Morally speaking, the results of [Eva93] imply that, if equality holds in (12), then for all i = 1, . . . , k,the set ∂Ai should have constant mean curvature, except on a negligible subset. It is difficult toturn this intuition into a proof (unless equality holds for all t > 0 [MS02]), but this intuitionexplains why the Standard Simplex Conjecture (12) is believed to be true.

2.5. Our Contribution. For any n ≥ 2, there exists t(n) > 0 such that for any t(n) < t < ∞and k = 3, the conjecture (12) holds, as I show in [Hei14]. I use geometric and Fourier analyticarguments to show the first variation of (12) defines a contractive mapping, when restricted topartitions that almost achieve equality in (12).

In [HMN16], together with Mossel and Neeman, we show that if the measure restriction of(12) is changed, then the most natural restatement of the conjecture (12) is false. Specifically, if

a1, . . . , ak are real numbers with 0 < ai < 1 for all i = 1, . . . , k and∑k

i=1 ai = 1 with (a1, . . . , ak) 6=(1/k, . . . , 1/k), and if t > 0, then the inequality (12) does not hold if we try to replace the sets{Bi}ki=1 with any set of simplicial cones that partition Euclidean space. This negative result impliesthat all of our known proofs [Bor85, BS01, MN15, Eld15] of the case k = 2 of the Standard SimplexConjecture (which is equivalent to Borell’s inequality (11)) do not work in the cases k ≥ 3. Thatis, new methods are needed.

2.6. Potential Developments. Improvements on the result [Hei14] would yield: a solution of amulti-bubble problem in Gaussian space [CCH+08, IM12, Led96], the Plurality is Stablest Con-jecture [KKMO07],[IM12, Theorem 1.10], and optimal computational hardness results for approxi-mating the MAX-k-CUT problem [IM12, Theorem 1.13]. The former problem follows from (12) byletting t→ 0. The latter problem assumes the Unique Games Conjecture, from Section 1.6. Also,the Plurality is Stablest Conjecture says that the most noise-stable way to determine the winner ofan election between k candidates is to take the plurality. This result assumes that no one personhas too much influence over the election’s outcome.

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3. The Propeller Conjecture

In an effort to better understand Grothendieck’s inequality (5), and to approximate the optimaof kernel clustering problems from machine learning, Khot and Naor [KN09, KN13] investigatedGrothendieck’s inequality (5) for {aij}ni,j=1 that are positive semidefinite.

3.1. Generalized Positive Semidefinite Grothendieck Inequalities. Suppose {aij}ni,j=1 is apositive semidefinite matrix, i.e. a real symmetric matrix with all eigenvalues nonnegative. Letν1, . . . , νk ∈ Rk with k ≥ 2, and let B = {bij}ki,j=1 be the symmetric positive semidefinite matrix

with bij := 〈νi, νj〉. Then the Generalized Positive Semidefinite Grothendieck Inequality[KN13, Theorem 3.1, Theorem 3.3] says that there exists C(B) > 0 which does not depend on nor on {aij}ni,j=1 such that

(13) maxw1,...,wn∈`2

‖wi‖2=1, ∀ i=1,...,n

n∑i,j=1

aij〈wi, wj〉 ≤1

C(B)· maxσ : {1,2,...,n}→{1,2,...,k}

n∑i,j=1

aij〈νσ(i), νσ(j)〉.

Instead of rounding the vectors wi, i = 1, . . . , n to 1 or −1 as in Grothendieck’s inequality (5),Khot and Naor prove (13) by rounding the vectors wi, i = 1, . . . , n to k vectors νi, i = 1, . . . , k.Also, the best constant 1/C(B) is found by finding the best rounding procedure.

3.2. The Sharp Constant of Grothendieck Inequalities. Let Ik denote the k × k identitymatrix. In [KN09, KN13], it is shown that C(B) can be found by solving a finite dimensionaloptimization problem, whose parameters depend on B. However, this optimization problem isnon-convex in general, so standard methods cannot compute C(B). The Propeller Conjectureguesses the value of C(B) for B = Ik, k ≥ 4:

(14) C(Ik) := supA1,...,Ak : ∪ki=1Ai=Rk−1,

γk−1(Ai∩Aj)=0, ∀ i,j∈{1,...,k}, i 6=j

k∑i=1

∥∥∥∥∫Ai

xdγk−1(x)

∥∥∥∥2

2

=9

8π= C(I3).

The constant C(I3) = 9/(8π) is computed in [KN09] using Lagrange multipliers, but C(I4) doesappear to be computable using this technique.

3.3. Our Contribution. Together with Naor and Jagannath, using some theoretical results anda brute force search, we give a computer-assisted proof of the Conjecture (14) in the case k = 4[HJN13]. The analytic results use a connection between the maximization problem (14) and discreteharmonic maps into the sphere. Intuition derived from this connection allows several ad hocarguments to rule out candidates for partitions that maximize (14).

3.4. Potential Developments. Solving (14) for all k ≥ 4 would yield better understanding ofsemidefinite programming algorithms and potential insight into computing the best constant inGrothendieck’s inequality (5).

4. The Symmetric Gaussian Problem

4.1. Gaussian Perimeter of Symmetric Sets. The Symmetric Gaussian Problem (15) belowwas first mentioned in [Bar01] (for the Gaussian perimeter), and then restated in [CR11, O’D12] inrelation to the Gap-Hamming-Distance problem from communication complexity. The SymmetricGaussian Problem is a conjectural variant of Borell’s Theorem (11) that says the most noise stablesymmetric sets in Euclidean space are balls or their complements. In other words, when we restrict

8

Borell’s Theorem (11) to symmetric sets (so that half spaces are no longer considered), then ball’sor their complements have the smallest conjectured perimeter:

(15)

A = −A ∧ γn(A) = γn(B(0, r)) = γn(B(0, r′)) =⇒ ∀t ≥ 0,∫Rn

1A · e−tL1Adγn ≤ max

(∫Rn

1B(0,r) · e−tL1B(0,r)dγn,

∫Rn

1B(0,r′)c · e−tL1B(0,r′)cdγn

).

This problem exhibits many of the difficulties of the other conjectures (12) and (14), though thepresent problem seems more tractable.

4.2. Our Contribution. In [Hei15], I show that (15) is true for n = 1 and t sufficiently large,by simplifying the argument from my work on the Standard Simplex Conjecture [Hei14]. Whenn ≥ 2, I demonstrate that the Conjecture (15) can sometimes be false, depending on the size of themeasure γn(A). More specifically, I compute the second variation of the ball and its complement,adapting a second variation argument from [CS07]. It turns out that the computation of the secondvariation of noise stability of the ball essentially reduces to proving an L2 Poincare inequality onthe sphere.

The main point of this work is to successfully use a second variation argument for noise stability.Previous work on noise stability problems focused on first variation (i.e. first derivative) arguments.So, the computation of the second variation of noise stability was a largely missing piece from theliterature. And Conjecture (15) served as a test-case for this technique. As discussed in Section 6below, I am currently working on better understanding the second variation of noise stability forother related problems, e.g. from Sections 2 and 3.

5. Strong Contractivity and Kahn-Kalai-Linial

5.1. Discrete Analysis and Hypercontractivity. Expander graphs, i.e. graphs with boundeddegrees and large spectral gaps, have been studied extensively in both pure and applied mathematics[HLW06]. In the paper [MN14], the authors construct a family of graphs that have a spectral gapwith respect to any uniformly convex Banach space. That is, these graphs are expander graphsin a much stronger sense than the usual definition of expander graphs. In order to improve theirexpander graph construction, Mendel and Naor made a conjecture concerning the decay of the heatsemigroup in Lp spaces. The conjecture of [MN14] can be understood as an attempt to developLittlewood-Paley theory for non-doubling metric spaces. For simplicity, we state this conjectureonly in the case of real-valued functions.

Let n be a positive integer. Let f : {−1, 1}n → R be a function. Let µ be the uniform probabilitymeasure on the hypercube, so that µ(x) = 2−n for each x ∈ {−1, 1}n. Any f : {−1, 1}n → R can be

written as f =∑

S⊆{1,...,n} f(S)WS , where for all x = (x1, . . . , xn) ∈ {−1, 1}n, WS(x) :=∏i∈S xi

and f(S) :=∫{−1,1}n f(x)WS(x)dµ(x). For any t ≥ 0, define e−tLf :=

∑S⊆{1,...,n} e

−t|S|f(S)WS ,

Lf :=∑

S⊆{1,...,n} |S| f(S)WS , and ∀ 1 ≤ p <∞, define ‖f‖p := (∫{−1,1}n |f(x)|p dµ(x))1/p. For all

i ∈ {1, . . . , n}, define the influence Iif of the ith variable on f by

Iif :=∑

S⊆{1,...,n} : i∈S

(f(S))2.

Let k ≥ 1, k ∈ Z. The Conjecture [MN14, Remark 5.5] says: ∀ p > 1, ∃ c(p) > 0 such that

(16) f(S) = 0 ∀S ⊆ {1, . . . , n} with |S| < k =⇒ ∀ t > 0, ‖e−tLf‖p ≤ e−tkc(p)‖f‖p.Equivalently, Conjecture (16) is a “higher order” Poincare inequality: ∀ p > 1, ∃ c(p) > 0such that

f(S) = 0 ∀S ⊆ {1, . . . , n} with |S| < k =⇒∫|f |p−1 sign(f)Lfdµ ≥ kc(p)

∫|f |p dµ.

9

A weaker form of (16) with the term e−tkc(p) replaced by e−min(t,t2)kc(p) can be proven [MN14,Lemma 5.4] using Holder’s inequality and the hypercontractive inequality [Sta59, Fed69, Bon70,Nel73, Gro75, Bec75]:

(17) ∀ 1 < p < q <∞, ∀ t > 1

2log

(q − 1

p− 1

), ‖e−tLf‖q ≤ ‖f‖p.

Using the hypercontractive inequality (17), Kahn, Kalai and Linial proved the following famousinequality [KKL88], resolving a conjecture of Ben-Or and Linial.

Theorem 5.1 (Kahn-Kalai-Linial). [KKL88, Theorem 3.1] There exists a universal constant c >0 such that, for any f : {−1, 1}n → {−1, 1}, we have maxi=1,...,n Iif ≥ c(

∫[f−

∫fdµ]2dµ)(log n)/n.

This Theorem says that a discrete function with values in {−1, 1} must have some asymmetry inits Fourier coefficients. To see this, note that it is easy to construct a function f : {−1, 1}n → R such

that maxi=1,...,n Iif ≤ 10(∫

[f −∫fdµ]2dµ)/n, just by choosing f such that f(S) = 2/(n(n−1)) for

all |S| = 2 and such that f(S) = 0 for all other S ⊆ {1, . . . , n}. Note that the Fourier coefficientsof f are then symmetric with respect to permutations on the set {1, . . . , n}, but f does not takevalues {−1, 1}, so Theorem 5.1 does not apply.

5.2. Our Contribution. In [HMO14], together with Mossel and Oleszkiewicz, we prove the casek = 1 of the Conjecture (16) of Mendel and Naor. In fact, we prove (16) for any probability spacewith a symmetric Markov semigroup Pt whose generator L := − d

dtPt|t=0+ satisfies an L2 Poincareinequality. We then answer a question of Hatami and Kalai, showing that Theorem 5.1 cannot bestrengthened unless a logarithmic number of Fourier coefficients of the function f vanish. That is,

Theorem 5.2 ([HMO14]). There exists c > 0 such that, for all n ∈ N, there exists f : {−1, 1}n →{−1, 1} with f(S) = 0 for all S ⊆ {1, . . . , n} with |S| ≤ log n such that maxi=1,...,n Iif ≤ c(log n)/n.

In the case that f(S) = 0 for all S ⊆ {1, . . . , n} with |S| ≤ C(n) log n, the equality∑n

i=1 Iif =∑S⊆{1,...,n} |S| (f(S))2 shows that maxi=1,...,n Iif ≥ C(n)(log n)/n. So, there is a phase transition

in the possible behavior of the maximum influence maxi=1,...,n Iif , which occurs when C(n) > 0 isbounded or unbounded as n→∞.

Finally, we demonstrated a generalization of Talagrand’s inequality for functions f : {−1, 1}n →{−1, 1} with f(S) = 0 for all S ⊆ {1, . . . , n} with |S| < k. The usual Talagrand inequality thencorresponds to the case k = 1.

Theorem 5.3 ([HMO14]). Let k ≥ 1. Let f : {−1, 1}n → R with f(S) = 0 for all S ⊆ {1, . . . , n}with |S| < k. ∀ i = 1, . . . , n, let ∂

∂xif(x) := [f(x1, . . . , xn) − f(x1, . . . , xi−1,−xi, xi+1, . . . , xn)]/2,

where x = (x1, . . . , xn) ∈ {−1, 1}n. Then

(18) ‖f‖22 ≤ 6n∑i=1

‖ ∂∂xif‖22

k + log(‖ ∂∂xif‖2/‖ ∂

∂xif‖1) .

5.3. Potential Developments. Proving the case k > 1 of the Conjecture (16) of Mendel andNaor would give improved understanding of analysis in non-Euclidean spaces, going beyond (orsupplementing) Littlewood-Paley theory in non-doubling metric measure spaces. Also, the solutionof this problem would improve our understanding of expander graphs.

6. Forthcoming Work

6.1. Noise Stability and the Calculus of Variations. My first goal [Hei16b, Hei16a] is toextend the variational methods of [Hei15] and [BBJ16] to the periodic isoperimetric problem from[KM16]. Recently, [KM16] essentially showed that the Unique Games Conjecture for the field of

10

two elements (that is, using p = p(ε) = 2 in Section 1.6) follows from the following conjecture. LetA ⊆ Rn with −A = Ac and such that A+ e = Ac for every standard basis vector e ∈ Rn. That is,let A be a “periodized set.” Then the noise stability of A is at most the noise stability of the set

H :=

{x = (x1, . . . , xn) ∈ Rn : sin(π

n∑i=1

xi) ≥ 0

}.

Note that the boundary ∂H of H is a set of hyperplanes of the form

{x = (x1, . . . , xn) ∈ Rn : ∃ k ∈ Z such that x1 + · · ·+ xn = k}.

For this reason, H is called a “periodized half space.”The Unique Games Conjecture for the field of two elements is expected to contain the same

difficulties as the full conjecture [KM16]. So, working on this isoperimetric problem is tantamountto working directly on the Unique Games conjecture itself.

I also hope to extend the variational methods of [Hei15] and [BBJ16] to the Standard Sim-plex Conjecture (12) and to the Propeller Conjecture (14). Previous investigations of Gaussianisoperimetry and Gaussian noise stability all use “global” methods [Bor85, Bor03, MN15, Eld15].That is, these investigations all use that the translation of a half space is still a half space. Manynoise stability problems lack this translational symmetry. For example, in conjecture (15), if a setA ⊆ Rn satisfies A = −A, then a translation of A no longer satisfies A = −A. Sometimes, thislack of translational symmetry is unexpected, as we showed in [HMN16]. However, a recent work[BBJ16] managed to use the calculus of variations to prove the Gaussian isoperimetric inequality((11) as t→ 0). Since the calculus of variations is an inherently local method, there is hope for thework [BBJ16] to perhaps apply to other related problems.

Already, it appears that my main result [Hei14] for the Standard Simplex Conjecture can berecovered with a relatively straightforward second variation argument. That is, the rather com-plicated argument of [Hei14] can be replaced by a rather intuitive an automatic calculation. Itremains to try to improve the range of parameters of this result.

A key aspect of [BBJ16] is explicitly writing down variations that increase the noise stabilityof a set. These variations happen to be the eigenfunctions of an Ornstein-Uhlenbeck operatoron the boundary of the set. These eigenfunctions were identified in [CM12], where a maximalversion of the Gaussian perimeter called an “entropy” was studied, for surfaces that are self-similarunder the mean curvature flow. The Ornstein-Uhlenbeck operator they study comes from thesecond-variation formula for the Gaussian perimeter. The idea of studying the eigenfunctions ofan operator restricted to the surface seems to be due to Simons [Sim68]. And the idea of studyingan “entropy” associated to surfaces seems to have originated in Perelman’s proof of the PoincareConjecture, where he introduced his entropy functional. And Perelman’s investigation of “entropy”was influenced by the hypercontractive inequality (17).

6.2. Hypercontractivity for Markov Chains. My second goal is to prove hypercontractive in-equalities on general finite graphs [GRST14, EM12, LV09]. Other than those methods developed in[DSC96], there is currently no reliable way to prove a reasonably sharp hypercontractive inequalityon a given graph. One of the main ways to prove a hypercontractive inequality on a manifold isto use curvature bounds. Some theories of Ricci curvature for metric-measure spaces have receivedrecent attention [Stu06, LV09, Oll09, Mie13, AGS14, JL14, GRST14, EKS15, BM15, Pau16, FS15,KKRT16], but it is unclear what role these theories can play in proving hypercontractive inequali-ties. For example, perhaps one might be able to show that a graph has some curvature bound, butit may be more technically feasible to just prove a hypercontractive inequality directly, instead offirst proving a curvature bound. With this issue in mind, there have been a great deal of recentattempts to define and study discrete versions of curvature on graphs [Oll09]. Just as curvature on

11

manifolds is associated with differential operators (by Bochner’s equality and semigroup theory),curvature theories on graphs are associated with Markov chains.

As an intermediate goal, it would be desirable to prove that a certain class of random graphs,e.g. random Cayley graphs, satisfies a good hypercontractive inequality with positive probability.I am currently working on this problem with Professor Georg Menz [HM16]. Also, we demonstratethat on the discrete square, essentially all Markov chains have negative curvature, while theysatisfy hypercontractive inequalities. That is, certain curvature theories seem ineffective for provinghypercontractive inequalities, even in the simple example of a square.

6.3. Noncommutative Hypercontractivity and Free Probability. My third goal is to inves-tigate noncommutative hypercontractive inequalities in general settings. This project is an out-growth of my previous joint work [HV16]. Noncommutative hypercontractive inequalities [Gro72,CL93, Bia97, JPP+15, JPPP15, JZ15, RX16] have received a good deal of recent attention, due toapplications in quantum information theory and free probability. I am currently working on thisproblem with Professor Dimitri Shlyakhtenko. We are considering some noncommutative versionsof Bezout’s Theorem [DTW76], and hypercontractive constants for the free group von Neumannalgebra [Bia97, JPPP13, JPP+15, RX16].

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