This is the material which I presented at the 60th birthday conference of my good friend Jos\'{e} Luis Vicente in Seville in September 2001. It is based on the nine lectures, now called sections, which were given by me at Purdue in Spring 1997. This should provide a good calculational background for the Galois theory of vectorial (= additive) polynomials and their iterates.

A combinatorial description of the minimal free resolution of a lattice ideal allows us to the connection of Integer Linear Programming and Algebra. The non null reduced homology spaces of some simplicial complexes are the key. The extremal rays of the associated cone reduce the number of variables.

We describe the singularity of all but finitely-many germs in a pencil generated by two germs of plane curve sharing no tangent.

In this paper, we use the graphs as a tool to study nilpotent Lie algebras. It implies to set up a link between graph theory and Lie theory. To do this, it is already known that every nilpotent Lie algebra of maximal rank is associated with a generalized Cartan matrix $A$ and it is isomorphic to a quotient of the positive part $\mathfrak{n}_+$ of the Kac-Moody algebra $\mathfrak{g}(A)$. Then, if $A$ is affine, we can associate $\mathfrak{n}_+$ with a directed graph (from now on, we use the term digraph) and we can also associate a subgraph of this digraph with every isomorphism class of nilpotent Lie algebras of maximal rank and of type $A$. Finally, we show an algorithm which obtains these subgraphs and also groups them in isomorphism classes.

We present a proof of embedded desingularization for closed subschemes which does not make use of Hilbert-Samuel function and avoids Hironaka's notion of normal flatness (see also \cite{EncinasVillamayor2000} page 224). Given a subscheme defined by equations, we prove that embedded desingularization can be achieved by a sequence of monoidal transformations; where the law of transformation on the equations defining the subscheme is simpler then that used in Hironaka's procedure. This is done by showing that desingularization of a closed subscheme $X$, in a smooth sheme $W$, is achieved by taking an algorithmic principalization for the ideal $I(X)$, associated to the embedded scheme $X$. This provides a conceptual simplification of the original proof of Hironaka. This algorithm of principalization (of Log-resolution of ideals), and this new procedure of embedded desingularization discussed here, have been implemented in MAPLE.

Let $k$ be a perfect field of characteristic $p>0$, $k(t)_{per}$ the perfect closure of $k(t)$ and $A$ a $k$-algebra. We characterize whether the ring $$ A\otimes_k k(t)_{per}=\bigcup_{m\geq 0}(A\otimes_k k(t^{\frac{1}{p^m}})) $$ is noetherian or not. As a consequence, we prove that the ring $A\otimes_k k(t)_{per}$ is noetherian when $A$ is the ring of formal power series in $n$ indeterminates over $k$.

We give a new method to compute the centralizer of an element in Artin braid groups and, more generally, in Garside groups. This method, together with the solution of the conjugacy problem given by the authors in \cite{FGM}, are two main steps for solving conjugacy systems, thus breaking recently discovered cryptosystems based in braid groups \cite{AAG}. We also present the result of our computations, where we notice that our algorithm yields surprisingly small generating sets for the centralizers.

Nonassociative algebras can be applied, either directly or using their particular methods, to many other branches of Mathematics and other Sciences. Here emphasis will be given to two concrete applications of nonassociative algebras. In the first one, an application to group theory in the line of the Restricted Burnside Problem will be considered. The second one opens a door to some applications of non-associative algebras to Error correcting Codes and Cryptography.

We study 0-dimensional real rank one valuations centered in a regular local ring of dimension $n\geq 2$ such that the associated valuation ring can be obtained from the regular ring by a sequence of quadratic transforms. We define two classical invariants associated to the valuation (the refined proximity matrix and the multiplicity sequence) and we show that are equivalent data of the valuation.

We report on a partial solution of the conjecture that the class of finite solvable groups can be characterised by 2-variable identities. The proof requires pieces from number theory, algebraic geometry, singularity theory and computer algebra. The computations were carried out using the computer algebra system SINGULAR.

We consider categories of generalized perverse sheaves, with relaxed constructibility conditions, by means of the process of gluing $t$-structures and we exhibit explicit abelian categories defined in terms of standard sheaves categories which are equivalent to the former ones. In particular, we are able to realize perverse sheaves categories as non full abelian subcategories of the usual bounded complexes of sheaves categories. Our methods use induction on perversities. In this paper, we restrict ourselves to the two-strata case, but our results extend to the general case.

We describe the slopes, with respect to the coordinates hyperplanes, of the hypergeometric systems of codimension one, that is when the toric ideal is generated by one element.

This paper deals with valuations of fields of formal meromorphic functions and their residue fields. We explicitly describe the residue fields of the monomial valuations. We also classify all the discrete rank one valuations of fields of power series in two and three variables, according to their residue fields. We prove that all our cases are possible and give explicit constructions.

It is well known that the integral closure of a monomial ideal in a polynomial ring in a finite number of indeterminates over a field is a monomial ideal, again. Let $R$ be a noetherian ring, and let $(x_1,\ldots,x_d)$ be a regular sequence in $R$ which is contained in the Jacobson radical of $R$. An ideal $\mathfrak a$ of $R$ is called a monomial ideal with respect to $(x_1,\ldots,x_d)$ if it can be generated by monomials $x_1^{i_1}\cdots x_d^{i_d}$. If $x_1R+\cdots + x_dR$ is a radical ideal of $R$, then we show that the integral closure of a monomial ideal of $R$ is monomial, again. This result holds, in particular, for a regular local ring if $(x_1,\ldots,x_d)$ is a regular system of parameters of $R$.

The need for a noncommutative algebraic geometry is apparent in classical invariant and moduli theory. It is, in general, impossible to find commuting parameters parametrizing all orbits of a Lie group acting on a scheme. When one orbit is contained in the closure of another, the orbit space cannot, in a natural way, be given a scheme structure. In this paper we shall show that one may overcome these difficulties by introducing a noncommutative algebraic geometry, where affine "schemes" are modeled on associative algebras. The points of such an affine scheme are the simple modules of the algebra, and the local structure of the scheme at a finite family of points, is expressed in terms of a noncommutative deformation theory proposed by the author in \cite{Laudal2002}. More generally, the geometry in the theory is represented by a {\it swarm}, i.e. a diagram (finite or infinite) of objects (and if one wants, arrows) in a given $k$-linear Abelian category ($k$ a field), satisfying some reasonable conditions. The noncommutative deformation theory refered to above, permits the construction of a presheaf of associative $k$-algebras, locally {\it parametrizing} the diagram. It is shown that this theory, in a natural way, generalizes the classical scheme theory. Moreover it provides a promising framework for treating problems of invariant theory and moduli problems. In particular it is shown that many moduli spaces in classical algebraic geometry are commutativizations of noncommutative schemes containing additional information.

Let $H$ denote the set of formal arcs going through a singular point of an algebraic variety $V$ defined over an algebraically closed field $k$ of characteristic zero. In the late sixties, J. Nash has observed that for any nonnegative integer $s$, the set $j^s(H)$ of $s$-jets of arcs in $H$ is a constructible subset of some affine space. Recently (1999), J. Denef and F. Loeser have proved that the Poincar\'{e} series associated with the image of $j^s(H)$ in some suitable localization of the Grothendieck ring of algebraic varieties over $k$ is a rational function. We compute this function for normal toric surface singularities.

We investigate different concepts of modular deformations of germs of isolated singularities (infinitesimal, Artinian, formal). An obstruction calculus based on the graded Lie algebra structure of the tangent cohomology for modular deformations is introduced. As the main result the characterisation of the maximal infinitesimally modular subgerm of the miniversal family as flattening stratum of the relative Tjurina module is extended from ICIS to space curve singularities.

In this lecture we introduce the reader to the proof of the p-adic monodromy theorem linking the p-adic differential equations theory and the local Galois p-adic representations theory.

Let $K$ be a finite field of $2^{\ell}$ elements. Let $\phi_4,\phi_3, \phi_2,\phi_1$ be tame mappings of the $n\!+r$-dimensional affine space $K^{n+r}$. Let the composition $\phi_4\phi_3\phi_2\phi_1$ be $\pi$. The mapping $\pi$ and the $\phi_i$'s will be hidden. Let the component expression of $\pi$ be $(\pi_1(x_1,\dots,x_{n+r}),\dots \pi_{n+r}(x_1,\dots,x_{n+r}))$. Let the restriction of $\pi$ to a subspace be $\hat\pi$ as $\hat\pi=(\pi_1(x_1,\dots,x_n,0,\dots,0),\dots,\pi_{n+r}(x_1,\dots, x_n,0,\dots,0))=(f_1,\dots,f_{n+r}) : K^n\ mapsto K^{n+r}$. The field $K$ and the polynomial map ($f_1,\dots,f_{n+r}$) will be announced as the public key. Given a plaintext $(x'_1,\dots,x'_n)\in K^n$, let $y'_i=f_i(x'_1,\dots,x'_n)$, then the ciphertext will be $(y'_1,\dots,y'_{n+r})\in K^{n+r}$. Given $\phi_i$ and ($y'_1,\dots,y'_{n+r}$), it is easy to find $\phi_i^{-1}(y'_1,\dots,y'_{n+r})$. Therefore the plaintext can be recovered by $(x'_1,\dots,x'_n,0,\dots,0) = \phi_1^{-1}\phi_2^{-1} \phi_3^{-1}\phi_4^{-1}\,\hat\pi\,(x'_1,\dots,x'_n)=\phi_1^{-1} \phi_2^{-1}\phi_3^{-1}\phi_4^{-1}(y'_1,\dots, y'_{n+r})$. The private key will be the set of maps $\{\phi_1,\phi_2,\phi_3,\phi_4\}$. The security of the system rests in part on the difficulty of finding the map $\pi$ from the partial informations provided by the map $\hat\pi$ and the factorization of the map $\pi$ into a product (i.e., composition) of tame transformations $\phi_i$'s.

In this paper we review the moduli theory of polarized CY manifolds. We briefly sketched Kodaira-Spencer-Kuranishi local deformation theory developed by the author and G. Tian. We also construct the Teichm\"{u}ller space of polarized CY manifolds following the ideas of I. R. Shafarevich and I. I. Piatetski-Shapiro. We review the fundamental result of E. Viehweg about the existence of the course moduli space of polarized CY manifolds as a quasi-projective variety. Recently S. Donaldson computed the moment map for the action of the group of symplectic diffeomorphisms on the space of K\"{a}hler metrics with fixed class of cohomology. Combining this results with the solution of Calabi conjecture by Yau one obtain a very conceptual proof of the existence of the coarse moduli space for a large class of varieties. We follow the approach developed in \cite{LTYZI} to study the global properties of the moduli of polarized CY manifolds. We discuss the latest results connecting the discriminant locus in the moduli space of polarized odd dimensional CY manifolds with the Bismut-Gillet-Soule-Quillen-Donaldson Theory of Determinant line bundles.

This paper is part of the author's thesis, recently presented, where the following problem is treated: Characterizing the tangent cone and the equimultiple locus of a Puiseux surface (that is, an algebroid embedded surface admitting an equation whose roots are Puiseux power series), using a set of exponents appearing in a root of an equation. The aim is knowing to which extent the well-known results for the quasi-ordinary case can be extended to this much wider family.

If $B$ is a branch at $O\in\mathbb{C}^2$ of a holomorphic curve, a Puiseux parametrisation $y=\psi(x)$ of $B$ determines "pro-branches" defined over a sector $|\mathrm{arg} x-\alpha| < \varepsilon $. The exponent of contact of two pro-branches is the (fractional) exponent of the first power of $x$ where they differ. We first show how to use exponents of contact to give simple proofs of several well known results. For $C$ the germ at $O$ of a curve in $\mathbb{C}^2$, the Eggers tree $T_C$ of $C$ is defined. We also introduce combinatorial invariants (particularly, a certain 1-chain) on $T_C$. Any other germ $\Gamma$ at $O$ has contact with $C$ measured by a unique point $X_{\Gamma}\in T_C$, and this determines the set of exponents of contact with $C$ of any pro-branch of $\Gamma$. A simple formula establishes the converse, and this leads to a short proof of the theorem on decomposition of a transverse polar of $C$ into parts $P_i$, where both the multiplicity of $P_i$, and the order of contact with $C$ of each branch $Q$ of $P_i$ are explicitly given.