Mathematical marvels - CUNY Graduate Center
Mathematical marvels - CUNY Graduate Center
Mathematical marvels - CUNY Graduate Center
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DR ALEX GAMBURD<br />
<strong>Mathematical</strong> <strong>marvels</strong><br />
Dr Alex Gamburd describes his research involving expander graphs and prime<br />
numbers, which has applications to quasi-crystals and quantum computation,<br />
exemplifying fruitful interactions between pure and applied mathematics<br />
To begin, could you defi ne expander graphs?<br />
Expanders are highly connected sparse graphs<br />
widely used in computer science. Clearly high<br />
connectivity is desirable in any communication<br />
network. The necessity of sparsity is perhaps<br />
best seen in the case of the network of neurons<br />
in the brain: since the axons have fi nite thickness<br />
their total length cannot exceed the quotient of<br />
the average volume of one’s head and the area of<br />
axon’s cross-section. In fact, this is the context in<br />
which expander graphs fi rst implicitely appeared<br />
in the work of Barzdin and Kolmogorov in 1967.<br />
What questions about expander graphs<br />
interest you?<br />
There are basically two sources of raw material<br />
for constructing mathematical structures:<br />
randomness and number theory. It was observed<br />
early on that random regular graphs are<br />
expanders. The explicit construction of optimal<br />
expanders – Ramanujan graphs – used deep<br />
number-theoretic results from the theory of<br />
automorphic forms to construct expanders as<br />
Cayley graphs of groups with respect to some<br />
very special choices of generators.<br />
70 INTERNATIONAL INNOVATION<br />
A basic question that arose in the mid-90s – at<br />
the time when I was starting my PhD – is to what<br />
extent the expansion is the property of groups<br />
alone, independent of the choice of generators.<br />
I became fascinated/obsessed with this problem<br />
and obtained some partial results in my thesis<br />
under the direction of Peter Sarnak. Several years<br />
ago, in joint work with Jean Bourgain, we were<br />
able to fi nally resolve the problem in many cases,<br />
by bringing in some recently developed tools<br />
from additive combinatorics. Few experiences in<br />
life are as frustrating as being stuck on a problem<br />
for 10 years. Fewer still are those which can<br />
match the joy of fi nally solving it.<br />
Can you describe some of the tools from<br />
additive combinatorics used in your work?<br />
One of the basic results is the ‘sum-product<br />
phenomenon’, which is the following. When<br />
you study addition and multiplication tables for<br />
numbers from one to nine (which my sevenyear<br />
old daughter is doing now) you might<br />
notice that there are many more numbers in<br />
the multiplication table. This basically has to do<br />
with the fact that the numbers from one to nine<br />
form an arithmetic progression. If you take a set<br />
forming an arithmetic progression (or a subset<br />
of it) and add it to itself it will not grow much; if<br />
you take a set forming a geometric progression<br />
(or a subset of it) and multiply it by itself it<br />
will also not grow much. However a subset<br />
of integers cannot be both a arithmetic and a<br />
geometric progression and so it will grow either<br />
when multiplied or added with itself.<br />
What are some of the applications of your<br />
work on expanders?<br />
The new methods developed in the joint work<br />
with Jean Bourgain had a number of applications,<br />
in particular in quantum computation and<br />
theory of quasi-crystals. But the most exciting<br />
and unexpected development was that using<br />
a suitable generalisation of our results, in<br />
joint work with Peter Sarnak, we were able to<br />
obtain novel sieving results pertaining to the<br />
distribution of prime numbers, thus, in part,<br />
repaying the debt of computer science to<br />
number theory.<br />
Can you outline why prime numbers are so<br />
fundamentally interesting?<br />
Prime numbers are basic building blocks of<br />
integers: as was known already to Euclid, any<br />
number can be uniquely expressed as a product<br />
of one or more primes. (Obtaining an explicit<br />
factorisation of a large composite integer<br />
appears to be extremely hard – this fact is at<br />
the foundation of much of today’s computer<br />
security protocols.) Euclid also showed that<br />
there are infi nitely many primes, but many of<br />
the basic questions, which already fascinated<br />
the Greeks, remain open. For example, are there<br />
infi nitely many twin primes, that is, primes<br />
separated by two?<br />
What we did in joint work with Bourgain and<br />
Sarnak is to sieve for primes in problems with<br />
hyperbolic fl avour. If you look carefully at the<br />
Escher’s pictures (or at the picture of integral<br />
Apollonian packing in Figure 4) you will notice<br />
that in contrast to Euclidean case, the boundary<br />
of a ball in a hyperbolic plane is roughly of<br />
the same size as the area of the ball – it is this<br />
feature that necessitates the application of<br />
expander result in the hyperbolic setting.<br />
Some of our results probably could have<br />
been appreciated by Euclid and Apollonius,<br />
and certainly by Gauss and Dirichlet. So<br />
mathematics involved in the study of prime<br />
numbers is, on the one hand, of immense<br />
practical importance in contemporary<br />
applications. On the other hand, it has a<br />
transcendent and timeless quality, being part of<br />
the millennia-old tradition which includes some<br />
of the fi nest achievements of the human spirit.<br />
It is this double nature of mathematics which I<br />
fi nd very appealing, and which I try to convey to<br />
my students.
Expanding interactions<br />
EXPANDERS ARE HIGHLY connected sparse<br />
graphs widely used in computer science, in areas<br />
ranging from parallel computation to complexity<br />
theory and cryptography. There are several ways<br />
of making the intuitive notions of connectivity<br />
and sparsity precise, the simplest and most widely<br />
used is the following. Given a subset of vertices,<br />
its boundary is the set of edges connecting the<br />
set to its complement. The expansion of a subset<br />
is a ratio of the size of a boundary to the size of a<br />
set. The expansion of a graph is a minimum over<br />
all expansion coeffi cients of its subsets.<br />
The expansion coeffi cient captures the notion of<br />
being highly-connected, the bigger the expansion<br />
coeffi cient, the more highly-connected is the<br />
graph. Of course one can simply connect all the<br />
vertices but in this case the number of edges<br />
grows as a square of the number of vertices. The<br />
problem of constructing expanders is nontrivial<br />
because we put the second constraint: the graphs<br />
are to be sparse, ie. the number of edges should<br />
grow linearly with the number of vertices. The<br />
simplest way to accomplish this is to demand<br />
that the graphs be regular, that is, each vertex<br />
has the same number of neighbours (say 3). A<br />
family of regular graphs is said to form a family<br />
of expanders if the expansion coeffi cient of all<br />
the graphs in the family is bounded from below<br />
by some positive constant.<br />
“The expansion coeffi cient is a notion which is<br />
very easy to grasp but it is diffi cult to compute<br />
numerically or to estimate analytically, as the<br />
number of subsets grows exponentially with<br />
the number of vertices,” explains Alex Gamburd,<br />
Professor of Mathematics at the <strong>Graduate</strong><br />
<strong>Center</strong> of the City University of New York and<br />
the University of California, Santa Cruz. “The<br />
starting point of most current work on expanders<br />
is that expansion coeffi cient has a spectral<br />
interpretation: to put it sonorously, if you hit<br />
a graph with a hammer, you can determine<br />
how highly-connected it is by listening to the<br />
bass note. In more<br />
technical terms,<br />
high connectivity<br />
is equivalent to<br />
establishing a<br />
spectral gap for<br />
an averaging (or<br />
Laplace) operator on<br />
the graph”.<br />
CAYLEY GRAPHS<br />
OF GROUPS<br />
Explicit construction<br />
of expanders is<br />
given in terms of<br />
Cayley graphs –<br />
globally-symmetric<br />
graphs defi ned by simple local rules. A Cayley<br />
graph of a group with respect to a fi xed<br />
generating set is a graph whose vertices are<br />
elements of the group; the neighbours of an<br />
element are determined by multiplying it by all<br />
the generators, which is a fi xed small number,<br />
say 3. The simplest example is furnished by the<br />
group of two by two matrices of determinant<br />
one with entries in integers modulo a prime. It<br />
is a consequence of a deep spectral gap result of<br />
Selberg, proved in 1965, that Cayley graphs of<br />
this group with respect to standard generators<br />
are expanders. Here is a simple related example:<br />
take as vertices the integers modulo a prime<br />
and connect them if they differ by plus/minus<br />
one or if their product has remainder 1 when<br />
divided by this prime (in other words, they<br />
are inverses modulo that prime), the resulting<br />
family is a family of expanders. Figure 1<br />
exhibits such graphs for primes 101, 499, 997.<br />
PSEUDO-RANDOMNESS<br />
The crucial feature underlying expansion of<br />
graphs in Figure 1 is pseudo-randomness:<br />
if you take a set of integers from one to<br />
DR ALEX GAMBURD<br />
Research by mathematicians on expander graphs, originating in computer science, turns out to have<br />
unexpected and far-reaching applications to quasi-crystals, quantum computing, and number theory<br />
FIGURE 1. SIMPLE RULES, COMPLEX STRUCTURES<br />
FIGURE 2. CUBIC GRAPH ON 80 VERTICES (FULLERENE C-80) WITH AN<br />
EXPANSION COEFFICIENT OF ¼ REPRESENTED BY THE SHADED SUBSET<br />
some prime (say 7) and invert them modulo<br />
that prime, the resulting sequence looks<br />
random (for example we obtain 1, 4, 5, 2,<br />
3, 6 when inverting 1, 2, 3, 4, 5, 6, modulo<br />
7). “This is a basic example of the pseudorandomness<br />
phenomenon: by performing a<br />
simple deterministic operation on some set<br />
of increasing size, a sequence is obtained<br />
which looks increasingly random,” explains<br />
Gamburd. “Quantitative statements about<br />
pseudorandom phenomena (which are<br />
of great interest in natural and computer<br />
sciences) are expressed in terms of the<br />
spectral gap for the associated averaging<br />
operator; in this case, Selberg’s theorem,<br />
which gives expanders with respect to very<br />
special choices of generators.” In 2005,<br />
resolving a problem posed by Dr Alex<br />
Lubotzky in 1994, Dr Jean Bourgain and<br />
Gamburd proved that ‘almost any’ choice<br />
of generators give rise to expanders. In their<br />
work they developed novel methods for<br />
proving spectral gap results, which turned<br />
out to have a wide range of applications.<br />
QUASI-CRYSTALS<br />
This application is related to the theory of<br />
quasi-crystals. Generalising two-dimensional<br />
aperiodic tiling, Drs John Conway and Charles<br />
Radin constructed a self-similar (hierarchical)<br />
tiling of three dimensional space with a single<br />
prototile, such that the tiles occur in an infi nite<br />
number of different orientations in the tiling.<br />
The tile is a prism, which when scaled up by 2<br />
is subdivided into 8 copies of itself (‘daughter<br />
tiles’). If one iterates this same subdivision<br />
procedure over and over, one creates in the<br />
limit the desired tiling of three dimensional<br />
space by prisms (Figure 3). Conway and Radin<br />
showed that the orientations of tiles in the<br />
tiling are uniformly distributed and posed<br />
WWW.RESEARCHMEDIA.EU 71
INTELLIGENCE<br />
EXPANDER GRAPHS: INTERACTIONS<br />
BETWEEN ARITHMETIC, GROUP THEORY<br />
AND COMBINATORICS<br />
OBJECTIVES<br />
The fi rst project will be devoted to addressing<br />
the question to what extent expansion is a<br />
property of groups alone, independent of the<br />
choice of generators. New robust families of<br />
expanders using recently developed tools from<br />
additive combinatorics will be constructed.<br />
The second project builds on the recent joint<br />
work with Bourgain and Sarnak, in which<br />
expanders were used to obtain novel sieving<br />
results towards non-abelian generalisations<br />
of Dirichlet’s theorem on primes in arithmetic<br />
progressions. The general problem addressed<br />
in the second project involves sieving for<br />
primes (or almost-primes) on an orbit of a<br />
group generated by fi nitely many polynomial<br />
maps; application of combinatorial Brun sieve<br />
depends crucially on the expansion property<br />
of the ‘congruence graphs’ associated with the<br />
orbit.<br />
KEY COLLABORATORS<br />
Jean Bourgain, School of Mathematics,<br />
Institute for Advanced Study, Princeton, NJ<br />
Peter Sarnak, Princeton University and<br />
Institute for Advanced Study, Princeton, NJ<br />
FUNDING<br />
National Science Foundation –<br />
award no. 0645807<br />
CONTACT<br />
Dr Alexander Gamburd<br />
Presidential Professor of Mathematics<br />
Mathematics PhD Program<br />
The <strong>CUNY</strong> <strong>Graduate</strong> <strong>Center</strong><br />
365 Fifth Avenue<br />
New York, NY 10016-4309, USA<br />
T +1 212 817 8539<br />
E agamburd@gmail.com<br />
Videotaped lectures:<br />
www.msri.org/web/msri/online-videos/-/<br />
video/showVideo/14588<br />
www.msri.org/web/msri/online-videos/-/<br />
video/showVideo/14561<br />
ALEX GAMBURD is a recipient of the<br />
Presidential Early Career Award for Scientists<br />
and Engineers (PECASE), which is the highest<br />
honour that a beginning scientist or engineer<br />
can receive in the US. Gamburd has been on<br />
the faculty of UCSC since 2004. In 2007 he<br />
received a Sloan Research Fellowship and<br />
Von Neumann Fellowship from the Institute<br />
for Advanced Study in Princeton. In 2011<br />
he was appointed Presidential Professor<br />
of Mathematics at the <strong>CUNY</strong> <strong>Graduate</strong><br />
<strong>Center</strong>. Gamburd earned his BS degree in<br />
Mathematics from the Massachusetts Institute<br />
of Technology and MA and PhD degrees in<br />
Mathematics from Princeton University.<br />
72 INTERNATIONAL INNOVATION<br />
the question of how fast this convergence to<br />
uniform distribution takes place. This question<br />
reduces to the study of the spectral gap for<br />
the averaging operator associated with eight<br />
rotations giving orientations of daughter tiles.<br />
A consequence of the work of Bourgain and<br />
Gamburd on the spectral gap of averaging<br />
operators on a sphere is that this convergence<br />
takes place exponentially fast.<br />
QUANTUM COMPUTATION<br />
Another application of this result is of<br />
importance in quantum computing. In the<br />
context of quantum computation elements of<br />
three dimensional rotation group are viewed<br />
as ‘quantum gates’ and a set of elements<br />
generating a dense subgroup is called<br />
‘computationally universal’ (since any element<br />
of rotation group can be approximated<br />
by some word in the generating set to an<br />
arbitrary precision). A set of elements is called<br />
‘effi ciently universal’ if any element can be<br />
approximated by a word of length which is<br />
logarithmic with respect to the inverse of the<br />
chosen precision (this is the best possible).<br />
A consequence of the result of Bourgain<br />
and Gamburd is that many computationally<br />
universal sets are effi ciently universal.<br />
AFFINE LINEAR SIEVE<br />
In the joint work of Bourgain, Gamburd and<br />
Dr Peter Sarnak the new spectral gap results<br />
were applied to obtain novel sieving results<br />
pertaining to distribution of prime numbers.<br />
“The general belief is that apart from the<br />
obvious local structure of primes (for example,<br />
that they, apart from two, are all odd) they<br />
behave as if they are randomly distributed,”<br />
Gamburd reveals. “This intuition is developed<br />
in sieve methods, to prove, for example,<br />
the following approximation to twin prime<br />
conjecture: there are infi nitely many integers<br />
separated by 2, one of which is a prime and<br />
Few experiences in life are as frustrating<br />
as being stuck on a problem for<br />
10 years. Fewer still are those<br />
which can match the joy<br />
of fi nally solving it<br />
FIGURE 4. INTEGRAL APOLLONIAN PACKING<br />
another a product of two prime. What we did in<br />
joint work with Bourgain and Sarnak is to sieve<br />
for primes in problems with hyperbolic fl avour.”<br />
A simple example illustrating this line of research<br />
is related to Integral Apollonian packings (Figure<br />
4). A classical result of Apollonius asserts that<br />
given three mutually tangent circles there are<br />
exactly two circles tangent to all three. Given<br />
a set of four mutually tangent circles, one can<br />
construct (using Apollonius theorem) four new<br />
circles, each of which is tangent to three of the<br />
given ones. Continuing to repeatedly fi ll in the<br />
lunes between mutually tangent circles with<br />
further tangent circles we arrive at infi nite<br />
circle packing. A remarkable fact is that if you<br />
start with four mutually tangent circles having<br />
integral curvatures (curvature is the inverse of<br />
the radius) all the circles in the packing will have<br />
integral curvatures as well – hence the name<br />
‘Integral Apollonian packings’. Affi ne linear<br />
sieve, developed by Bourgain, Gamburd and<br />
Sarnak, begins to probe the properties of prime<br />
numbers appearing in such packings.<br />
FIGURE 3. QUAQUAVERSAL TILING