Kuperberg Group

Primes, Polynomials, and Patterns

Prime numbers, which are natural numbers that cannot be expressed as a product of two smaller numbers, are fundamental objects in modern mathematics, used throughout number theory and in applications such as encryption systems. Prime numbers exhibit a striking dichotomy: they are defined very precisely, and demonstrate algebraic properties (for example, even numbers larger than 2 are never prime), and yet in many ways their distribution throughout the integers appears random.

The Kuperberg group works on understanding the ways that the primes “look random”—and the ways in which they do not—using analytic, probabilistic, and combinatorial methods. The primes can be fairly precisely modeled by certain randomly generated sequences. Understanding properties of these sequences can lead us to new insights about prime numbers. At the same time, techniques such as sieve methods, analytic tools, and methods from additive combinatorics or algebraic geometry can help resolve questions about prime numbers themselves. Moreover, investigating other arithmetic sequences (such as the sequence of sums of two squares) or analogs of the prime numbers in other settings (such as irreducible polynomials in a polynomial ring) increases our understanding of the primes while leading to new and exciting discoveries in their own right.