Research Interests

My general area of research is Number Theory.

Specifically, I work with algebraic dynamical systems and with integral lattice constructions. If you already know what number theory is, skip to here.

Number Theory

An oversimplified description of number theory is that it is the study of the natural numbers {1, 2, 3, 4,...}, or more generally of the rational numbers (the numbers that are ratios of integers). Here are a few examples of classical problems in number theory.

Prime Conjectures
A prime number is a positive integer that has exactly two factors, 1 and itself. The first few prime numbers are

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,...
In the Elements, written 23 centuries ago, Euclid gives a famous proof that there are infinitely many primes. The Fundamental Theorem of Arithmetic states that every integer greater than 1 factors uniquely into prime numbers.

No even number other than 2 is prime, so other than 2 and 3, there are no pairs of consecutive primes. However, there are lots of pairs of primes that differ by 2: {3,5}, {5,7}, {11,13}, {17,19}, {29,31}.... These pairs are called twin primes. It is conjectured but not known that there are infinitely many twin primes.

Another famous unsolved conjecture about primes is the Goldbach Conjecture, which states that every even number greater than 4 is a sum of two primes. For instance,
6 = 3 + 3
8 = 3 + 5
10 = 3 + 7
12 = 5 + 7
14 = 3 + 11 = 7 + 7
16 = 3 + 13 = 5 + 11
18 = 5 + 13 = 7 + 11
20 = 3 + 17 = 7 + 13

The Goldbach Conjecture is featured in the recent novel Uncle Petros and Goldbach's Conjecture by Apostolos Doxiadis.

Diophantine Problems
A Diophantine problem, named after the ancient Greek mathematician Diophantus, is an equation in one or more variables for which we seek either integer or rational solutions. The most famous of these problems is Fermat's Last Theorem, conjectured by Fermat in the seventeenth century and finally proven in 1994, which states that

xn + yn = zn
has no integer solutions for n > 2 where none of x, y, and z is zero. In addition to the link above, there are many places to read about the history of this theorem online.

In the case where n = 2, positive solutions to the equation x2 + y2 = z2, called Pythagorean triples, are the integers which form the side lengths of a right triangle. There are infinitely many Pythagorean triples, and there is an elegant geometric construction using the unit circle that gives a general formula for all of them. If we choose two postive integers m and n with no common prime factors, one of which is even, and if m > n then
x = m2 - n2
y = 2mn
z = m2 + n2
is a Pythagorean triple, and every Pythagorean triple can be scaled into one of this form. For instance, m = 2 and n = 1 gives {3,4,5}, and m = 3 and n = 2 gives {5,12,13}.

Applications
Number theory was once notorious for having no real-world applications whatsoever. However, in recent decades, both prime factorization (in the RSA algorithm and related algorithms) and Diophantine problems (in algorithms using elliptic curves) have become essential components of public-key encryption systems. When you send your credit card number over the web, number theory keeps hackers from stealing it.

Algebraic Dynamics

In the latter half of the twentieth century, the mathematical theory of chaos exploded onto the scene. Chaos most commonly manifests itself in dynamical systems.

The simplest way to form a dynamical system is to start with a function from a set to itself. For instance,

f(x) = x2
is a function that takes any real number as an input and returns its square, another real number, as an output. To turn f into a dynamical system, we apply f to the real numbers over and over again and study the long term behavior of the system.

If we start with a number x, the orbit of x under f is the set of points

{x, f(x), f 2(x), f 3(x),...},
where f 2(x) = f(f(x)), f3(x) = f(f(f(x))), and so on. When f(x) = x2, this orbit is
{x, x2, x4, x8, x16,...}.

In this simple example, we can describe the orbits of various points systematically as follows.

UNDER CONSTRUCTION

Integral Lattices

To get an extremely rough idea of what my lattice work is like, read the section on symmetry groups of polyhedra on this site. A lattice is an array of points in n-dimensional space that is uniform in a certain mathematical sense. A natural way in which lattices arise is as the array of centers of spheres in a sphere packing. (A classical problem of mathematics, which was finally solved in 1998, is proving what is the most efficient way of packing identical spheres in three-dimensional space--for instance, of stacking canon balls efficiently.) One way to describe and construct lattices in higher dimensions through an understanding of their groups of symmetries.

Back to my home page