RESEARCH ACTIVITY

My research lies at the intersection of finite geometry, algebraic geometry over finite fields, coding theory, and cryptography. A common theme is the use of algebraic and geometric methods to study existence, classification, and exceptional phenomena over finite fields.

Scattered structures and rank-metric codes

A major part of my recent research concerns scattered linear sets, linearized polynomials, and their higher-dimensional generalizations. This includes maximum scattered and exceptional scattered structures, scattered sequences, evasive subspaces, higher-scattered and (h)-scattered spaces, and their connections with rank-metric and MRD codes.

Particular attention is devoted to classification and equivalence problems, and to the interaction between geometric invariants and code-theoretic invariants.

Polynomial functions over finite fields

I study polynomial and rational functions over finite fields with special algebraic or combinatorial properties, including permutation polynomials, complete permutation polynomials, planar and APN functions, PcN and APcN functions, and related notions of differential uniformity.

A central problem is the classification of exceptional functions, namely functions for which a given property persists over infinitely many extensions of the base field.

Algebraic geometry over finite fields

Algebraic curves and varieties over finite fields provide a fundamental tool throughout my research. I use rational-point estimates, irreducibility, function fields, automorphism groups, singularities, coverings, Galois groups, and monodromy groups to investigate problems arising from finite geometry, coding theory, and cryptography.

These techniques are particularly effective when a combinatorial property can be translated into the existence or non-existence of rational points on an associated algebraic variety.

Finite geometry

My work in finite geometry includes the construction and classification of extremal configurations in finite projective and affine spaces.

Among the objects considered are arcs, caps, blocking sets, saturating sets, semiovals and semiarcs, linear sets, ovoids, Hermitian varieties, and configurations in finite polar spaces.

A recurring objective is to understand how algebraic structures determine geometric configurations and, conversely, how finite-geometric constructions produce new algebraic and coding-theoretic objects.

Coding theory

I am interested in codes arising from finite geometry and algebraic geometry, including MDS and NMDS codes, algebraic-geometric codes, quantum codes, minimal codes, locally recoverable codes, subspace codes, and rank-metric codes.

Geometric structures such as blocking sets, cutting blocking sets, scattered spaces, and algebraic curves often lead naturally to codes with extremal parameters or special structural properties.

Cryptographic functions

Another research direction concerns functions over finite fields with applications in symmetric cryptography.

This includes APN and differential functions, permutation properties, bent and related Boolean functions, and algebraic approaches to differential and nonlinear properties. The emphasis is often on classification, exceptional behaviour, and on the use of algebraic geometry to rule out or characterize infinite families.