Recerca · Abdó Roig-Maranges

Articles

1. Morphic cohomology of toric varieties. 2012.
Homology, Homotopy and Applications. 14 (2012) 1, pp. 113--132.
▹ abs | pdf | arxiv | doi |

In this paper we construct a spectral sequence computing a modified version of morphic cohomology of a toric variety (even in the singular case) in terms of combinatorial data coming from the fan of the toric variety.

2. On the complexity of the Whitehead minimization problem. 2007.
International Journal of Algebra and Computation. 17 (2007) 8, pp. 1611--1634.
▹ abs | pdf | arxiv | doi | (amb Enric Ventura i Pascal Weil)

The Whitehead minimization problem consists in finding a minimum size element in the automorphic orbit of a word, a cyclic word or a finitely generated subgroup in a finite rank free group. We give the first fully polynomial algorithm to solve this problem, that is, an algorithm that is polynomial both in the length of the input word and in the rank of the free group. Earlier algorithms had an exponential dependency in the rank of the free group. It follows that the primitivity problem -- to decide whether a word is an element of some basis of the free group -- and the free factor problem can also be solved in polynomial time.

Notes

1. Build your own undecidable proposition. 2012.
▹ abs | pdf |

These are notes of an informal talk about the Gödel incompleteness theorem. We aim to give a reasonably accurate picture of the beautiful ideas behind this theorem, avoiding overwhelming details. This is intended for a technically minded but non-specialist audience.

2. Descent for blowup's on smooth schemes. 2009.
▹ abs | pdf |

These are the notes of a talk in which we translate from the $\mathbb{A}^1$-homotopy theoretic context an argument from Morel showing that a homotopy invariant presheaf of spectra on the category of smooth schemes that satisfies Nisnevich descent automatically satisfies descent for abstract blow-ups.

Presentacions

1. Cohomologia mòrfica de varietats tòriques. 2010.
▹ abs | pdf |

En aquesta xerrada introduiré la cohomologia mòrfica, una teoria cohomològica de varietats algebraiques complexes a mig camí entre la cohomologia singular, i la cohomologia motívica. Aquesta cohomologia, té la particularitat de capturar informació geomètrica de la varietat algebraica, tenint una definició força concreta en termes de cicles algebraics (al contrari que la cohomologia motívica en la versió de Voevodsky). Explicaré per què és interessant estudiar-la i els problemes que presenta, i descriuré una successió espectral que permet fer-ne alguns càlculs explícits per varietats tòriques.