To begin, could you define 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 finite 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 first 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. 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 finally 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 finally 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 seven- year 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 infinitely many primes, but many of the basic questions, which already fascinated the Greeks, remain open. For example, are there infinitely 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 flavour. 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 finest achievements of the human spirit. It is this double nature of mathematics which I find very appealing, and which I try to convey to my students. 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 Mathematical marvels 70 INTERNATIONAL INNOVATION DR ALEX GAMBURD
Transcript
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.
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 seven-year 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.
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
Mathematical marvels
70 INTERNATIONAL INNOVATION
DR
ALEX
GAM
BURD
WWW.RESEARCHMEDIA.EU 71
DR ALEX GAMBURD
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
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 pseudo-randomness 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
Expanding interactionsResearch 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 2. CUBIC GRAPH ON 80 VERTICES (FULLERENE C-80) WITH AN EXPANSION COEFFICIENT OF ¼ REPRESENTED BY THE SHADED SUBSET
FIGURE 1. SIMPLE RULES, COMPLEX STRUCTURES
INTELLIGENCE
72 INTERNATIONAL INNOVATION
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 3. QUAQUAVERSAL TILING
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
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.
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 GamburdPresidential Professor of Mathematics
Mathematics PhD ProgramThe CUNY Graduate Center365 Fifth AvenueNew York, NY 10016-4309, USA
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.
