Algebra

Research developed within the Algebra group connects with a variety of (sub)areas of Mathematics and Computer Science. Graph-theoretic, geometric or topological arguments have widespread use.

Automata theory: Descriptive complexity in the average case through the analytic combinatorics of conversion methods between regular expressions and finite automata. Invertibility studies to develop a public-key cryptography system based on linear transducers. Use of pre-grammars for type inhabitation.

Combinatorics: Boolean representable simplicial complexes: applications to Algebra and other areas of Combinatorics. Combinatorics of hyperplane arrangements through the Pak-Stanley labeling of its regions. Algebraic and geometric properties of combinatorial decision and optimization problems.

Dynamical systems: Use of algebraic tools to characterize flow-invariant spaces of dynamical systems given by network (graph) structures. Generic dynamical properties such as bifurcations and heteroclinic behavior are proved using that characterization.

Group theory: Generation of finite groups and some graph-theoretical properties of the generating graph of 2-generated groups. Decidability problems about closures of finitely generated subgroups of a given free group of finite rank under certain topologies. 

Representation and ring theory: Structure of non-commutative rings and Hopf algebras, with some emphasis on affine cellular algebras, generalized Weyl algebras, Ore extensions and quantum groups. Topics of interest are factorization in non-commutative rings, PI theory, automorphism groups, injective hulls of simple modules, Hochschild (co)homology and deformation theory. Non-associative and Poisson algebras.

Semigroup theory: Relatively profinite semigroups versus symbolic dynamics and classification of pseudovarieties. Profinite approach to decision problems for pseudovarieties. Profinite topologies. Nilpotent semigroups (in the sense of Mal'cev) and representation theory of finite monoids. Structure of regular and locally inverse semigroups. The Holonomy Theorem and applications to Markov chain mixing problems.