## Singularities and applications to Geometry, Topology, Algebra and Cryptography

MTM2013-45710-C2-1-P

The research project MTM2013-45710-C2-1-P fosuses on several aspects of **Singularity Theory** from a transversal perspective. Our purpose is the resolution of well-known problems which can be interpreted or addressed from Singularity Theory. Such problems are theoretical as well as calculation effectiveness associated. Applications go from Group Theory to Cryptography including Topology of Algebraic Varieties, low-dimensional Geometry or Combinatorial problems such as lattice-point counting.The different questions are organized as follows:

**Local theory of singularities.**Immersed Jung's method for hypersurfaces, the monodromy Conjecture for surfaces, local invariants of curves in surface singularities, local structure of non-reduced curve singularities, jumping numbers on surface singularities, topology of Milnor fibers of quasi-ordinary singularities, Zeta functions of Newton non-degenerate germs of quotient singularities, generalizations of curvettes to higher dimension, the kite of a plane curve singularity, arcs, cylinders and equisingular families, symplectic methods and the Heegard-Floer Cohomology.**Global aspects of singularities and polynomial maps.**Jacobian Conjecture, study of cyclic coverings of weighted projective plane and their quotients, modules of logarithmic derivations on the plane, Albanese varieties of quasi-projective varieties, generalizations of Terao's conjecture, affine surfaces and classification of rational cuspidal curves.**Topology of Algebraic Varieties**. Fundamental groups of weighted-quasi-projective varieties, cohomology algebra for toric varieties, CW-complex structures of plane projective curves, explicit Heegaard splittings of graph manifolds, generating series for varieties and applications of the power structure, orbifold pencils, geometry, CM-type singularities, and essential coordinate characteristic varieties.**Birational and low-dimensional geometry**. including Nash modification for toric varieties, codimension-one foliations on three-manifolds and foliations of the 2-sphere, p-adic Matchbox manifolds, continuous deformation between some three-dimensional geometries, and singular geometric structures (orbifolds) in manifolds obtained by Dehn surgery in some algebraic knots.**Applications to Group Theory**. such as the study of the homology groups of kernels of Artin groups, and quasi-projectivity of link groups.**Applications to Cryptography**. Post-quatum cryptography with multivariate systems (MS), new multivariate cryptographic primitives, and MS-cryptoanalysis using Groebner bases.**Other Applications**. Counting lattice points and library developing in SAGE for coordinate components, braid monodromy, zeta functions, Bernstein polynomials, Alexander modules, and representations of infinite groups.

### Leading researcher:

Leading researcher:

Initial date:

January, 2014

End date:

December, 2016

Funding agency:

MINECO

Fund:

43.099 euros

Researchers of the IUMA: