GALOIS GEOMETRIES AND THEIR APPLICATIONS: CONNECTING ALGEBRA AND GEOMETRY
Algebraic geometry is a branch of mathematics based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems and viceversa. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. The power of the connection between Algebra and Geometry shows itself in a number of applications. Leading examples are Coding theory and Cryptography.
My main research interests concern Galois Geometries, their applications to Coding Theory and Cryptography, and their interactions with Algebraic Curves over Finite Fields (both from purely algebraic and purely geometrical points of view).
(1) Algebraic Geometry in positive characteristic (automorphism groups of algebraic curves, birational invariants, maximal curves, quotient curves)
(2) Coding Theory (functional codes, AG codes, quantum codes, convolutional codes)
(3) Linear sets and their applications (scattered polynomials, MRD codes)
(4) Permutation polynomials over finite fields (bent functions)