Sobre la estructura de las singularidades de las superficies algebroides sumergidas

Esta memoria esta dedicada a estudiar el proceso de resolucion puntual de una superficie algebroide sumergida y su relacion con la estructura de la singularidad en la 1a parte se demuestran unos resultados sobre equisingularidad en codim, 1 que permiten describir el proceso de resolucion de la singularidad mediante un arbol de niveles finitos. En la 2a parte se demuestra que eliminando las transformadas cuadraticas en direcciones tangentes a curvas permitidas se obtiene un subarbol finito. En la 3a parte se demuestra que este subarbol contiene suficiente informacion acerca de la singularidad viendo que cuando existen exponentes caracteristicos (singularidades cuasiordinarias) la igualdad de estos exponentes equivale a la de los subarboles

Aplicación de MAPLE a la Investigación

Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEpu

FPGA implementation of post-quantum DME cryptosystem

The rapid development of quantum computing constitutes a significant threat to modern Public-Key Cryptography (PKC). The use of Shor's algorithm with potential powerful quantum computers could easily break the two most widely used public key cryptosystems, namely, RSA and Elliptic Curve Cryptography (ECC), based on integer factorization and discrete logarithm problems. For this reason, Post-Quantum Cryptography (PQC) based on alternative mathematical features has become a fundamental research topic due to its resistance against quantum computers. The National Institute of Standards and Technology (NIST) has even opened a call for proposals of quantum-resistant PKC algorithms in order to standardize one or more PQC algorithms. Cryptographic systems that appear to be extremely difficult to break with large quantum computers are hash -based cryptography, lattice -based cryptography, code -based cryptography, and multivariate -quadratic cryptography. Furthermore, efficient hardware implementations are highly required for these alternative quantum -resistant cryptosystems

Equivariant Versions of Higher Order Orbifold Euler Characteristics

There are (at least) two different approaches to define an equivariant analogue of the Euler characteristic for a space with a finite group action. The first one defines it as an element of the Burnside ring of the group. The second approach emerged from physics and includes the orbifold Euler characteristic and its higher order versions. Here we give a way to merge the two approaches together defining (in a certain setting) higher order Euler characteristics with values in the Burnside ring of a group. We give Macdonald type equations for these invariants. We also offer generalized (“motivic”) versions of these invariants and formulate Macdonald type equations for them as well

On the pre-lambda-ring structure on the Grothendieck ring of stacks and the power structures over it

We describe a pre-lambda-structure on the Grothendieck ring of stacks (originally studied by Torsten Ekedahl) and the corresponding power structures over it, discuss some of their properties and give some explicit formulae for the Kapranov zeta-function for some stacks. In particular, we show that the nth symmetric power of the class of the classifying stack BGL(1) of the group GL(1) coincides, up to a power of the class L of the affine line, with the class of the classifying stack BGL(n)

Monodromy conjecture for some surface singularities

In this work we give a formula for the local Denef–Loeser zeta function of a superisolated singularity of hypersurface in terms of the local Denef–Loeser zeta function of the singularities of its tangent cone. We prove the monodromy conjecture for some surfaces singularities. These results are applied to the study of rational arrangements of plane curves whose Euler–Poincaré characteristic is three

On rational cuspidal projective plane curves

In 2002, L. Nicolaescu and the fourth author formulated a very general conjecture which relates the geometric genus of a Gorenstein surface singularity with rational homology sphere link with the Seiberg--Witten invariant (or one of its candidates) of the link. Recently, the last three authors found some counterexamples using superisolated singularities. The theory of superisolated hypersurface singularities with rational homology sphere link is equivalent with the theory of rational cuspidal projective plane curves. In the case when the corresponding curve has only one singular point one knows no counterexample. In fact, in this case the above Seiberg--Witten conjecture led us to a very interesting and deep set of `compatibility properties' of these curves (generalising the Seiberg--Witten invariant conjecture, but sitting deeply in algebraic geometry) which seems to generalise some other famous conjectures and properties as well (for example, the Noether--Nagata or the log Bogomolov--Miyaoka--Yau inequalities). Namely, we provide a set of `compatibility conditions' which conjecturally is satisfied by a local embedded topological type of a germ of plane curve singularity and an integer $d$ if and only if the germ can be realized as the unique singular point of a rational unicuspidal projective plane curve of degree $d$. The conjectured compatibility properties have a weaker version too, valid for any rational cuspidal curve with more than one singular point. The goal of the present article is to formulate these conjectured properties, and to verify them in all the situations when the logarithmic Kodaira dimension of the complement of the corresponding plane curves is strictly less than 2