Group of Singular group

Department of Algebra, Geometry and Topology
Faculty of Mathematical Science

We are interested in different aspects of Singularity Theory from a transversal perspective. Our purpose is twofold: on the one hand, the resolution of well-known problems which can be interpreted or addressed from Singularity Theory, and on the other hand the understanding of the deep relations connecting the different areas of knowledge around Singularity Theory. Such problems are theoretical, but they are also, and in an increasing way, practical, related to effectiveness and computational aspects. For instance, this is especially evident in our approach to Cryptography, to Group Theory (braid groups, Artin groups), the Topology of Algebraic Varieties, low-dimensional Geometry or other combinatorial problems. We continue to intensify our ties to national and international research groups, bringing together over 15 universities and other research institutions, which we have collaborations in the recent years. These colaborations include members of the K.U. Leuven (UNAEUROPA), Bordeaux Univ., Moscow Univ., the Inst. Math Jussieu (Paris), Univ. Lille, Univ. Seville, Univ. Valladolid, Univ. Zaragoza and BCAM. These collaborations have resulted into research stays, publications, conference organization, and organization of other international meetings. This perspective has shown to be fruitful and it allows us to progress in the knowledge of the problems, to advance in the achievement of excellence, and to prepare young researchers for the continuation of this direction.

Most of the members of our research group are located in the Department of Algebra, Geometry and Topology of the Mathematical Science Faculty of Complutense University of Madrid, and there are other belong to the University of Zaragoza and to Basque Center for Applied Mathematics. In addition, the group benefits from the interaction with the Institute of Interdisciplinary Mathematics (IMI) and the Institute of Mathematical Sciences (ICMAT). It has access to well-equipped and fast computers and also to the mathematical library of the University, which contains one of the most complete collections in mathematical sciences in Spain.

Our main research topics are the following ones:
1. Local theory of singularities. We focus our interest in the study of curve, surface, and hypersurface singularities (monodromy conjecture), singularities of polynomial maps (Milnor fiber), properties of the Nash modification, arc spaces and equisingular families.
2. Global aspects of singularities. This approach includes the study of polynomial maps and foliations, hyperplane arrangements and Terao’s conjecture, as well as classification of rational cuspidal curves.
3. Topology of Algebraic Varieties and in low dimension. Topological aspects of complements to projective hypersurfaces are included here, as well as quasi-projective surfaces in del Pezzo surfaces, K(pi,1) conjecture, quasi-projectivity problem for Artin groups, explicit Heegaard splittings of graph manifolds, generating series for varieties and applications of the power structure, and orbifold pencils.
4. Birational and low-dimensional geometry. We consider from an algebraic point of view generating series for varieties and applications of the power structure; from a geometric point of view (geometric structures); and from a topological point of view (Dehn surgery).
5. Applications to Group Theory, such as the study of the homology groups of kernels of Artin groups, and quasi-projectivity of link groups.
6. Applications to Cryptography, including Post-quatum cryptography with multivariate systems (MS), new multivariate cryptographic primitives, and MS-cryptoanalysis using Gröbner bases.
7. Other Applications, such as counting lattice points in rational polytopes, and library developing in SAGE for coordinate components, braid monodromy, knot invariants, zeta functions, Bernstein polynomials, Alexander modules, and representations of infinite groups.