## Research topics

Members of the Algebra group work among others on

- group theory, including algebraic groups,
- Lie algebras,
- combinatorial identities,
- computational Algebra,
- cryptography, coding theory and other applications of Algebra.

Alessandra Bernardi works in Multilinear Algebra, in particular she deals with Tensor Decomposition. This theme include purely algebraic areas such as zero-dimensional schemes, quotient algebras and syzygies of ideals, areas classically dealt with methods close to algebraic geometry such as secant varieties and parameterization of rational curves, more modern areas related to the study of theoretical and numerical algorithms for the solution of polynomial systems and tensor decomposition, up to more applied areas such as signal processing, phylogenetic and quantum information. She coordinates an interdisciplinary workgroup on "Quantum Information, Algebra and Geometry" which sees the interdisciplinary participation of many members of the departments of mathematics and physics, FBK, DISI and TIFPA. She wrote a book on Linear Algebra and Analytical Geometry (Italian).

Giancarlo Rinaldo works in the area of combinatorial and computational commutative algebra and its applications. In particular he studies algebraic invariants (i.e. depth, Castelnuovo-Mumford regularity, finite free resolution, Hilbert-Poincaré series) of rings associated with combinatorial objects like simplicial complexes, graphs, lattices, and codes. Besides he described in various articles algorithms, implemented in CoCoA, Macaulay 2 and Singular, to compute such invariants. Finally, he is interested in algebraic cryptographic protocols and their safety.

Willem de Graaf works in the field of computational algebra, in particular on algorithms in Lie Theory (Lie algebras, algebraic groups, Lie groups and related structures). He is the author of two books on these subjects, and coauthor of the functionality for Lie Theory of the systems of computational algebra GAP and Magma.

Massimiliano Sala heads the CryptoLabTN, which collaborates with public and private entities on questions related to security and cryptography. His recent research interest include work on the evaluation of multivariate polynomials over finite fields (with E. Ballico and M. Elia), and various aspects of error-correcting codes, including computational ones (with several students and collaborators). In a recent work with R. Aragona, A. Caranti and F. Dalla Volta, it is shown that the group generated by the round functions of an AES-like cryptosystem over an arbitrary finite field is large.

Andrea Caranti has worked in group theory, and on graded, modular group algebras. The recent research interests of Andrea Caranti include the theory of skew braces, and its applications to study Hopf-Galois structures and solutions to the Yang-Baxter equation; the study of multiple holomorphs of abelian, perfect and finite nilpotent groups; constructions of thin Lie algebras from algebras of maximal class.

###### Staff

- Andrea Caranti - Full professor - Head of the group
- Massimiliano Sala - Full professor
- Alessandra Bernardi - Associate Professor
- Willem De Graaf - Associate Professor
- Giancarlo Rinaldo - Assistant Professor