Algebra and Number Theory

Algebra and Number Theory

The theory of finite fields has a long tradition in mathematics. Originating from problems in number theory (Euler, Gauss), the theory was first developed purely out of mathematical curiosity. For a long time, this theory was used exclusively in pure mathematics, in areas such as number theory, algebraic geometry, group theory, and so on, without any relevance to applications. The situation changed dramatically with the development of modern information technologies. Finite fields were recognized to provide a natural framework for a wide variety of applications, particularly in information transmission and data security.

The research areas of the Number Theory/Algebra Group at Sabancı University include several aspects of the theory of finite fields, in particular, curves over finite fields and their applications to coding theory, the generation and analysis of pseudorandom numbers, as well as integer partitions and q-series.

There has been tremendous research activity in the last decades focusing on curves over finite fields, or equivalently algebraic function fields over finite fields. Interesting results were obtained, such as the construction (by A. Garcia and H. Stichtenoth) of explicit towers of function fields meeting the Drinfeld-Vladut bound.

The active, international collaboration of the Number Theory/Algebra Group is reflected by recent events such as "September Research on Curves over Finite Fields", the "SU Lecture Series on Coding Theory" and "Semester on Curves, Codes and Cryptography".

Current Research Topics

Arithmetic of Finite Fields:
Permutation polynomials, polynomial factorization.

Function Fields and Curves over Finite Fields:
Rational points, maximal curves, towers of function fields, automorphisms, modular curves, Drinfeld modular curves.

Coding Theory:
Cyclic and quasi-cyclic codes, algebraic geometry codes, asymptotically good codes.

Sequences and stream ciphers, cryptographically significant functions (bent, plateaued, almost perfect nonlinear), secret sharing schemes.

Enumerative Combinatorics and Applications:
Integer partitions, permutations and permutation statistics. Basic hypergeometric series and their identities. Bijective and sieve methods.

Algebra and Number Theory Alumni