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 Algebra, Number Theory and Combinatorics Group at Sabancı University include several aspects of the theory of finite fields, in particular, algebraic varieties and curves over finite fields, finite geometries, and their applications to coding theory, the generation and analysis of pseudorandom numbers, as well as integer partitions and q-series.
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.
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.
Commutative and Computational Algebra, Combinatorics and Algebraic Statistics:
Investigation of questions related to binomial ideals, toric rings, Koszul Algebras and their filtrations, lattice ideals, combinatorics of partially ordered sets, application of Gröbner bases, and algebraic and homological properties of powers of ideals.
Finite Geometry, Galois geometry, Incidence geometry, Semifields and non-associative algebras, Finite fields, Computer algebra, Coding theory, Combinatorics, Geometry of tensor products, Segre varieties, Algebraic Geometry.