# Number Theory

Number theory is the study of the integers (i.e. whole numbers) and related objects. Topics studied by number theorists include the problem of determining the distribution of prime numbers within the integers and the structure and number of solutions of systems of polynomial equations with integer coefficients. Many problems in number theory, while simple to state, have proofs that involve apparently unrelated areas of mathematics. A beautiful illustration is given by the use of complex analysis to prove the “Prime Number Theorem,” which gives an asymptotic formula for the distribution of prime numbers. Yet other problems currently studied in number theory call upon deep methods from harmonic analysis.

In addition, conjectures in number theory have had an impressive track record of stimulating major advances even outside the subject. For example, attempts to prove “Fermat’s Last Theorem” resulted in the development of large areas of algebra over the course of three centuries, and its recent proof involved a profound unifying force in modern mathematics known as the Langlands program.

At Duke, number theory is represented by Jayce Getz, Heekyoung Hahn, and Lillian Pierce, with Richard Hain, Leslie Saper and Chad Schoen working in related areas. Our specialties include analytic number theory, the Langlands program, the geometry of locally symmetric spaces, arithmetic geometry and the study of algebraic cycles.

## Faculty

#### Robert Calderbank

###### Charles S. Sydnor Professor of Computer Science

**Keywords in this area**

error-correcting codes, data storage, discrete harmonic analysis, sphere packing, algorithms, representation theory

#### Jayce Robert Getz

###### Assistant Professor in the Department of Mathematics

**Keywords in this area**

Automorphic representations, arithmetic geometry, Trace formulae

#### Heekyoung Hahn

###### Assistant Research Professor of Mathematics

**Keywords in this area**

relative trace formula, automorphic L-functions, modular forms and elliptic curves, partitions

#### Richard Hain

###### Professor of Mathematics

**Keywords in this area**

arithmetic

#### William L. Pardon

###### Professor of Mathematics

**Keywords in this area**

Commutative algebra, Quadratic forms

#### Lillian Beatrix Pierce

###### Assistant Professor in the Department of Mathematics

**Keywords in this area**

the circle method, character sums & exponential sums, class numbers, sieve numbers

#### Leslie Saper

###### Professor of Mathematics

**Keywords in this area**

automorphic forms, arithmetic of algebraic varieties

#### Caroline Turnage-Butterbaugh

###### William W. Elliott Assistant Research Professor

**Keywords in this area**

multiplicative number theory, Riemann zeta-function, L-functions, distribution of primes