Mathematical marvels - CUNY Graduate Center

DR ALEX GAMBURD
Mathematical marvels
Dr Alex Gamburd describes his research involving expander graphs and prime
numbers, which has applications to quasi-crystals and quantum computation,
exemplifying fruitful interactions between pure and applied mathematics
To begin, could you defi ne expander graphs?
Expanders are highly connected sparse graphs
widely used in computer science. Clearly high
connectivity is desirable in any communication
network. The necessity of sparsity is perhaps
best seen in the case of the network of neurons
in the brain: since the axons have fi nite thickness
their total length cannot exceed the quotient of
the average volume of one’s head and the area of
axon’s cross-section. In fact, this is the context in
which expander graphs fi rst implicitely appeared
in the work of Barzdin and Kolmogorov in 1967.
What questions about expander graphs
interest you?
There are basically two sources of raw material
for constructing mathematical structures:
randomness and number theory. It was observed
early on that random regular graphs are
expanders. The explicit construction of optimal
expanders – Ramanujan graphs – used deep
number-theoretic results from the theory of
automorphic forms to construct expanders as
Cayley graphs of groups with respect to some
very special choices of generators.
70 INTERNATIONAL INNOVATION
A basic question that arose in the mid-90s – at
the time when I was starting my PhD – is to what
extent the expansion is the property of groups
alone, independent of the choice of generators.
I became fascinated/obsessed with this problem
and obtained some partial results in my thesis
under the direction of Peter Sarnak. Several years
ago, in joint work with Jean Bourgain, we were
able to fi nally resolve the problem in many cases,
by bringing in some recently developed tools
from additive combinatorics. Few experiences in
life are as frustrating as being stuck on a problem
for 10 years. Fewer still are those which can
match the joy of fi nally solving it.
Can you describe some of the tools from
additive combinatorics used in your work?
One of the basic results is the ‘sum-product
phenomenon’, which is the following. When
you study addition and multiplication tables for
numbers from one to nine (which my sevenyear
old daughter is doing now) you might
notice that there are many more numbers in
the multiplication table. This basically has to do
with the fact that the numbers from one to nine
form an arithmetic progression. If you take a set
forming an arithmetic progression (or a subset
of it) and add it to itself it will not grow much; if
you take a set forming a geometric progression
(or a subset of it) and multiply it by itself it
will also not grow much. However a subset
of integers cannot be both a arithmetic and a
geometric progression and so it will grow either
when multiplied or added with itself.
What are some of the applications of your
work on expanders?
The new methods developed in the joint work
with Jean Bourgain had a number of applications,
in particular in quantum computation and
theory of quasi-crystals. But the most exciting
and unexpected development was that using
a suitable generalisation of our results, in
joint work with Peter Sarnak, we were able to
obtain novel sieving results pertaining to the
distribution of prime numbers, thus, in part,
repaying the debt of computer science to
number theory.
Can you outline why prime numbers are so
fundamentally interesting?
Prime numbers are basic building blocks of
integers: as was known already to Euclid, any
number can be uniquely expressed as a product
of one or more primes. (Obtaining an explicit
factorisation of a large composite integer
appears to be extremely hard – this fact is at
the foundation of much of today’s computer
security protocols.) Euclid also showed that
there are infi nitely many primes, but many of
the basic questions, which already fascinated
the Greeks, remain open. For example, are there
infi nitely many twin primes, that is, primes
separated by two?
What we did in joint work with Bourgain and
Sarnak is to sieve for primes in problems with
hyperbolic fl avour. If you look carefully at the
Escher’s pictures (or at the picture of integral
Apollonian packing in Figure 4) you will notice
that in contrast to Euclidean case, the boundary
of a ball in a hyperbolic plane is roughly of
the same size as the area of the ball – it is this
feature that necessitates the application of
expander result in the hyperbolic setting.
Some of our results probably could have
been appreciated by Euclid and Apollonius,
and certainly by Gauss and Dirichlet. So
mathematics involved in the study of prime
numbers is, on the one hand, of immense
practical importance in contemporary
applications. On the other hand, it has a
transcendent and timeless quality, being part of
the millennia-old tradition which includes some
of the fi nest achievements of the human spirit.
It is this double nature of mathematics which I
fi nd very appealing, and which I try to convey to
my students.

Expanding interactions
EXPANDERS ARE HIGHLY connected sparse
graphs widely used in computer science, in areas
ranging from parallel computation to complexity
theory and cryptography. There are several ways
of making the intuitive notions of connectivity
and sparsity precise, the simplest and most widely
used is the following. Given a subset of vertices,
its boundary is the set of edges connecting the
set to its complement. The expansion of a subset
is a ratio of the size of a boundary to the size of a
set. The expansion of a graph is a minimum over
all expansion coeffi cients of its subsets.
The expansion coeffi cient captures the notion of
being highly-connected, the bigger the expansion
coeffi cient, the more highly-connected is the
graph. Of course one can simply connect all the
vertices but in this case the number of edges
grows as a square of the number of vertices. The
problem of constructing expanders is nontrivial
because we put the second constraint: the graphs
are to be sparse, ie. the number of edges should
grow linearly with the number of vertices. The
simplest way to accomplish this is to demand
that the graphs be regular, that is, each vertex
has the same number of neighbours (say 3). A
family of regular graphs is said to form a family
of expanders if the expansion coeffi cient of all
the graphs in the family is bounded from below
by some positive constant.
“The expansion coeffi cient is a notion which is
very easy to grasp but it is diffi cult to compute
numerically or to estimate analytically, as the
number of subsets grows exponentially with
the number of vertices,” explains Alex Gamburd,
Professor of Mathematics at the Graduate
Center of the City University of New York and
the University of California, Santa Cruz. “The
starting point of most current work on expanders
is that expansion coeffi cient has a spectral
interpretation: to put it sonorously, if you hit
a graph with a hammer, you can determine
how highly-connected it is by listening to the
bass note. In more
technical terms,
high connectivity
is equivalent to
establishing a
spectral gap for
an averaging (or
Laplace) operator on
the graph”.
CAYLEY GRAPHS
OF GROUPS
Explicit construction
of expanders is
given in terms of
Cayley graphs –
globally-symmetric
graphs defi ned by simple local rules. A Cayley
graph of a group with respect to a fi xed
generating set is a graph whose vertices are
elements of the group; the neighbours of an
element are determined by multiplying it by all
the generators, which is a fi xed small number,
say 3. The simplest example is furnished by the
group of two by two matrices of determinant
one with entries in integers modulo a prime. It
is a consequence of a deep spectral gap result of
Selberg, proved in 1965, that Cayley graphs of
this group with respect to standard generators
are expanders. Here is a simple related example:
take as vertices the integers modulo a prime
and connect them if they differ by plus/minus
one or if their product has remainder 1 when
divided by this prime (in other words, they
are inverses modulo that prime), the resulting
family is a family of expanders. Figure 1
exhibits such graphs for primes 101, 499, 997.
PSEUDO-RANDOMNESS
The crucial feature underlying expansion of
graphs in Figure 1 is pseudo-randomness:
if you take a set of integers from one to
DR ALEX GAMBURD
Research by mathematicians on expander graphs, originating in computer science, turns out to have
unexpected and far-reaching applications to quasi-crystals, quantum computing, and number theory
FIGURE 1. SIMPLE RULES, COMPLEX STRUCTURES
FIGURE 2. CUBIC GRAPH ON 80 VERTICES (FULLERENE C-80) WITH AN
EXPANSION COEFFICIENT OF ¼ REPRESENTED BY THE SHADED SUBSET
some prime (say 7) and invert them modulo
that prime, the resulting sequence looks
random (for example we obtain 1, 4, 5, 2,
3, 6 when inverting 1, 2, 3, 4, 5, 6, modulo
7). “This is a basic example of the pseudorandomness
phenomenon: by performing a
simple deterministic operation on some set
of increasing size, a sequence is obtained
which looks increasingly random,” explains
Gamburd. “Quantitative statements about
pseudorandom phenomena (which are
of great interest in natural and computer
sciences) are expressed in terms of the
spectral gap for the associated averaging
operator; in this case, Selberg’s theorem,
which gives expanders with respect to very
special choices of generators.” In 2005,
resolving a problem posed by Dr Alex
Lubotzky in 1994, Dr Jean Bourgain and
Gamburd proved that ‘almost any’ choice
of generators give rise to expanders. In their
work they developed novel methods for
proving spectral gap results, which turned
out to have a wide range of applications.
QUASI-CRYSTALS
This application is related to the theory of
quasi-crystals. Generalising two-dimensional
aperiodic tiling, Drs John Conway and Charles
Radin constructed a self-similar (hierarchical)
tiling of three dimensional space with a single
prototile, such that the tiles occur in an infi nite
number of different orientations in the tiling.
The tile is a prism, which when scaled up by 2
is subdivided into 8 copies of itself (‘daughter
tiles’). If one iterates this same subdivision
procedure over and over, one creates in the
limit the desired tiling of three dimensional
space by prisms (Figure 3). Conway and Radin
showed that the orientations of tiles in the
tiling are uniformly distributed and posed
WWW.RESEARCHMEDIA.EU 71

INTELLIGENCE
EXPANDER GRAPHS: INTERACTIONS
BETWEEN ARITHMETIC, GROUP THEORY
AND COMBINATORICS
OBJECTIVES
The fi rst project will be devoted to addressing
the question to what extent expansion is a
property of groups alone, independent of the
choice of generators. New robust families of
expanders using recently developed tools from
additive combinatorics will be constructed.
The second project builds on the recent joint
work with Bourgain and Sarnak, in which
expanders were used to obtain novel sieving
results towards non-abelian generalisations
of Dirichlet’s theorem on primes in arithmetic
progressions. The general problem addressed
in the second project involves sieving for
primes (or almost-primes) on an orbit of a
group generated by fi nitely many polynomial
maps; application of combinatorial Brun sieve
depends crucially on the expansion property
of the ‘congruence graphs’ associated with the
orbit.
KEY COLLABORATORS
Jean Bourgain, School of Mathematics,
Institute for Advanced Study, Princeton, NJ
Peter Sarnak, Princeton University and
Institute for Advanced Study, Princeton, NJ
FUNDING
National Science Foundation –
award no. 0645807
CONTACT
Dr Alexander Gamburd
Presidential Professor of Mathematics
Mathematics PhD Program
The CUNY Graduate Center
365 Fifth Avenue
New York, NY 10016-4309, USA
T +1 212 817 8539
E agamburd@gmail.com
Videotaped lectures:
www.msri.org/web/msri/online-videos/-/
video/showVideo/14588
www.msri.org/web/msri/online-videos/-/
video/showVideo/14561
ALEX GAMBURD is a recipient of the
Presidential Early Career Award for Scientists
and Engineers (PECASE), which is the highest
honour that a beginning scientist or engineer
can receive in the US. Gamburd has been on
the faculty of UCSC since 2004. In 2007 he
received a Sloan Research Fellowship and
Von Neumann Fellowship from the Institute
for Advanced Study in Princeton. In 2011
he was appointed Presidential Professor
of Mathematics at the CUNY Graduate
Center. Gamburd earned his BS degree in
Mathematics from the Massachusetts Institute
of Technology and MA and PhD degrees in
Mathematics from Princeton University.
72 INTERNATIONAL INNOVATION
the question of how fast this convergence to
uniform distribution takes place. This question
reduces to the study of the spectral gap for
the averaging operator associated with eight
rotations giving orientations of daughter tiles.
A consequence of the work of Bourgain and
Gamburd on the spectral gap of averaging
operators on a sphere is that this convergence
takes place exponentially fast.
QUANTUM COMPUTATION
Another application of this result is of
importance in quantum computing. In the
context of quantum computation elements of
three dimensional rotation group are viewed
as ‘quantum gates’ and a set of elements
generating a dense subgroup is called
‘computationally universal’ (since any element
of rotation group can be approximated
by some word in the generating set to an
arbitrary precision). A set of elements is called
‘effi ciently universal’ if any element can be
approximated by a word of length which is
logarithmic with respect to the inverse of the
chosen precision (this is the best possible).
A consequence of the result of Bourgain
and Gamburd is that many computationally
universal sets are effi ciently universal.
AFFINE LINEAR SIEVE
In the joint work of Bourgain, Gamburd and
Dr Peter Sarnak the new spectral gap results
were applied to obtain novel sieving results
pertaining to distribution of prime numbers.
“The general belief is that apart from the
obvious local structure of primes (for example,
that they, apart from two, are all odd) they
behave as if they are randomly distributed,”
Gamburd reveals. “This intuition is developed
in sieve methods, to prove, for example,
the following approximation to twin prime
conjecture: there are infi nitely many integers
separated by 2, one of which is a prime and
Few experiences in life are as frustrating
as being stuck on a problem for
10 years. Fewer still are those
which can match the joy
of fi nally solving it
FIGURE 4. INTEGRAL APOLLONIAN PACKING
another a product of two prime. What we did in
joint work with Bourgain and Sarnak is to sieve
for primes in problems with hyperbolic fl avour.”
A simple example illustrating this line of research
is related to Integral Apollonian packings (Figure
4). A classical result of Apollonius asserts that
given three mutually tangent circles there are
exactly two circles tangent to all three. Given
a set of four mutually tangent circles, one can
construct (using Apollonius theorem) four new
circles, each of which is tangent to three of the
given ones. Continuing to repeatedly fi ll in the
lunes between mutually tangent circles with
further tangent circles we arrive at infi nite
circle packing. A remarkable fact is that if you
start with four mutually tangent circles having
integral curvatures (curvature is the inverse of
the radius) all the circles in the packing will have
integral curvatures as well – hence the name
‘Integral Apollonian packings’. Affi ne linear
sieve, developed by Bourgain, Gamburd and
Sarnak, begins to probe the properties of prime
numbers appearing in such packings.
FIGURE 3. QUAQUAVERSAL TILING