The theory of toric varieties plays a prominent role in various domains of mathematics, providing explicit relations between combinatorial geometry and algebraic geometry. It gives an important field of examples and models. According to Fulton ``Toric varieties provide... a way to see many examples and phenomena in algebraic geometry. For example, they are rational, and, although they may be singular, the singularities are rational. Nevertheless, toric varieties provided a remarkably fertile ground for general theories."

The course will be an elementary one, using and abusing examples. After general definitions and examples, the course will deal with torus action, orbits, divisors and homology of toric varieties. Special interest will be given to resolution of singularities and recent developments on Betti numbers, characteristic classes, intersection homology. One of the main points of the course will be applications of toric varieties in various domains, in particular Sommerville relations, lattice points, magic squares and sudoku. Other applications will be mentioned in order to help interested students.

There is no need of special prerequisites to follow the course, accessible to all students. According to the audience, language will be english or (approximation of) spanish.

Graph manifolds fibred over the circle appear in many classical situations involving complex surface singularities : Milnor fibration associated with a plane curve ; more generally, Milnor fibration of a holomorphic germ or a meromorphic germ from a normal complex surface singularity ; degenerating family of complex curves, etc.

The aim of this course is to introduce, through many examples, the topological tools to understand the topology of these situations : seifert and graph manifolds, plumbing, fibred links, open-book fibrations, quasi-periodic diffeomorphisms and their Nielsen invariants.

The course will first study the link of a germ of plane curve, and more generally of a germ of holomorphic function on a complex surface singularity through the topology of the resolution.

We will also give some applications in other singular situations such as germs of meromorphic functions or degenerating families of curves.

It will be explained how most of the invariants of the Milnor fibration (topology of the fibre, Nielsen invariants of the quasi-periodic monodromy, etc.) can be readen in the combinatoric data of the resolution.

We will also present several realization theorems on these invariants (Grauert, Winter, Montesinos-Matsumoto, etc.), and the relations between them.

The course will conclude by an application to a less classical situation : it will be explained on an example how the boundary of the Milnor fibre of a germ with a non-isolated singularity can be explicitely described by computing the Nielsen invariants of its vertical monodromies.

This course will consist of four lectures, each of them being somehow independent of the others, but all of them related. The aim is to give some basic facts about the topology of isolated singularities in analytic spaces, which will play a key role in several of the courses and lectures in this “School and Workshop”. The first lecture will be about the local conical structure of analytic spaces, looking particularly at the cases of complex plane curves (and their relation with knot theory) and surfaces (and their relation with plumbing). The second lecture will focus on the case of isolated singularities of complex surfaces and look at some additional structures one may have away from the singular point, specially spin, spin-c and contact structures; we will begin by recalling what these structures are. The third lecture will be about Milnor fibrations for real and complex singularities, and the fourth lecture will be about the topology of the Milnor fibers.

In these lectures we will review what we do and do not know about the existence and uniqueness of smooth and symplectic structures on closed, simply connected 4-manifolds. We will provide an introduction to the study of 4-manifolds and introduce those surgical techniques that have been effective in altering their smooth and symplectic structures and the Seiberg-Witten invariants that are used to distinguish them.

1. Stratifications.

It has been known for 40 years that analytic varieties and semianalytic sets admit Whitney stratifications, since 30 years ago that the same is true for subanalytic sets, and since 10 years ago for definable sets with respect to o-minimal structures. For all these classes of sets the strengthened regularity conditions introduced by T.-C. Kuo and Verdier can also be imposed. The even stronger conditions proposed by Mostowski which ensure local Lipschitz triviality along strata may be also obtained for subanalytic sets but not in general for definable sets with respect to non polynomially bounded o-minimal structures. We define all these regularity conditions, giving examples to confirm they are distinct, and state the corresponding existence theorems. As it is often useful to know that the transversal intersection of two regularly stratified sets possesses the same regularity, we describe a general theorem of this type.

2. Equisingularity.

The fundamental theorem about Whitney's -regular stratifications which makes them so useful is the Thom-Mather isotopy theorem. This says that every stratum Y of a Whitney stratified set has a neighbourhood which is the total space of a locally trivial topological fibre bundle with base space the stratum. One talks of the stratified set being locally topologically trivial along strata. The proof of Mather uses controlled stratified vector fields, which are shown to be integrable, and whose resulting controlled stratified isotopies yield the local topological trivialisations. The same proof works for the weaker -regular stratifications of K. Bekka; this is important when studying topological stability or classification up to homeomorphism. A natural weakening of Whitney regularity still implies -regularity. The stronger regularity conditions of Verdier and Mostowski give more information about the resulting stratified isotopies. Theorems for manifolds giving transversality after isotopy allow stratified generalisations. Classical Morse theory statements have stratified counterparts due to Goresky and MacPherson, by using systematically the Thom-Mather theorem, at least for sufficiently general functions. For constructible sets the isotopy theorem has constructible counterparts, due to M. Coste and M. Shiota.

3. Stratifications and determinacy.

A very weak regularity condition introduced by Thom in 1964 says that submanifolds transverse to a stratum remain transverse to nearby strata locally. Specifying that the submanifolds be of class gives . It turns out that the conditions and the Kuo-Verdier condition are related in a striking fashion. Generalising the blow-up of a point by considering planes of given dimension through a point rather than just lines defines what is called the Grassmann blow-up (Kuo-Trotman, 1987). The induced canonical projection has the property that the pullback of is , while the pullback of is . Moreover the pushforward of is . This shows at once that implies topological uniqueness of germs of transversal intersections at a point and generalisations of this provide precise characterisations of jet sufficiency (Trotman-Wilson 1999), recovering early theorems of Kuo. Incidentally this context was precisely that which led Thom to the introduction of stratification theory in his 1964 Bombay paper.

4. Metric and tangential properties.

To generalise the tangent space of a differentiable manifold to a stratified space one has various notions of tangent cone at a point and normal cone to a stratum. Whitney and Hironaka proved several properties of analytic Whitney stratifications, for example equimultiplicity in the complex case and the openness of the projection to a stratum associated to its normal cone. These can be extended to differentiable Kuo-Verdier stratifications (Orro-Trotman, 2002), and provide criteria for approximation of a subanalytic set by a normal cone (Ferrarotti-Fortuna-Wilson, 2004). Hironaka's equimultiplicity has a real counterpart in the continuity of the density (generalising to subanalytic sets the Lelong number), proved by G. Comte for Verdier stratifications (2000) and G. Valette for Whitney stratifications (2004). Another generalisation of the complex multiplicity to real functions is the Fukui invariant (1995), introduced as a way of testing for blow-analytic equivalence (a useful way of classifying real analytic functions due to Kuo). Finally we note that compact (weakly) Whitney stratified sets have finite geodesic diameter but not necessarily finite volume.

K. Bekka, C-régularité et trivialité topologique, Singularity theory and its applications, Warwick 1989, Part I, Lecture Notes in Math. 1462, Springer, Berlin, 1991, 42-62.

J. Bochnak, M. Coste, M.-F. Roy, Real algebraic geometry, Springer-Verlag, 1998.

C. G. Gibson, K. Wirthmüller, A. A. du Plessis and E. J. N. Looijenga, Topological stability of smooth mappings, Lecture Notes in Math. 552, Springer-Verlag, 1976.

M. Goresky, Whitney stratified chains and cochains, Trans. Amer. Math. Soc. 267 (1981), 175-196.

M. Goresky and R. MacPherson, Stratified Morse theory, Springer-Verlag, 1988.

J. Mather, Notes on topological stability, Mimeographed notes, Harvard University, 1970.

M. Pflaum, Analytic and geometric study of stratified spaces, Lecture Notes in Mathematics 1768, Springer, 2001.

A. A. du Plessis and C. T. C. Wall, The Geometry of Topological Stability, Oxford University Press, Oxford, 1995.

M. Shiota, Geometry of subanalytic and semialgebraic sets, Progress in Mathematics 150, Birkhauser, 1997.

R. Thom, Ensembles et morphismes stratifiés, Bull.A.M.S. 75 (1969), 240-284.

J.-L. Verdier, Stratifications de Whitney et théorème de Bertini-Sard, Inventiones Math. 36 (1976), 295-312.

H. Whitney, Local properties of analytic varieties, Differential and Combinatorial Topology, Princeton Univ. Press, (1965), 205-244.

The geometry of curves and surfaces (more generally, the geometry of submanifolds) is a very classic subject which is related to physics, geometrical optics, thermodynamics and to several branches of mathematics as well as to their applications. So there are always mathematicians attracted by the geometry of curves and surfaces. Following the idea that one should always start by the simple things, the submanifolds that we will consider in this lectures will be only curves and surfaces (although most of the facts and methods extend to genaral submanifolds). Usually, the generic submanifolds have special points (at which the submanifold have some special property) whose existence is due to to the (stable) generic singularities of some differentialble maps whose construction depends on the geometric properties we are studying (and, of course, on the considered submanifold). Very often such maps are the so-called Legendrian maps and Lagrangian maps, whose singularities are called, respectively, Legendrian and Lagrangian singularities. In these lectures, we will introduce those Legendrian and Lagrangian objects together with the notions of wave front and caustic, which one can use to study, to understand and to discover several interesting properties of curves and surfaces. Wave fronts, Legendrian maps and Legendrian singularities belong to contact geometry. Caustics, Lagrangian maps and Lagrangian singularities belong to symplectic geometry. Last years, the ideas and technics of singularity theory of wave fronts and caustics revealed to be a powerful tool to discover new theorems on the differential geometry of curves and surfaces. Indeed, already in 1988 V.I. Arnold pointed out that ``most of the facts of the differential geometry of submanifolds of Euclidean or of Riemannian space may be translated into the language of contact (or symplectic) geometry, whose applications to the problem of ordinary differential geometry provide new information in this classical domain''. We will introduce the basic notions of contact and symplectic geometry in order to show their relations with several interesting problems and properties of elementary differential geometry of curves and surfaces. We will see that some very natural (and new!) theorems on curves and surfaces are obtained (or discovered) by using Legendrian or Lagrangian singularities.

We will discuss the Seiberg-Witten invariants of isolated complex surface singularities, provided that they are rational homology spheres. We will follow two different descriptions: the first is given by Turaev's torsion normalized by the Casson-Walker invariant, the second by the Heegaard-Floer homology of Ozsvath and Szabo. We will provide combinatorial formulas extracted from the dual resolution graph. The main examples will include rational and almost-rational graph-manifolds and Seifert 3-manifolds.

These lectures will describe classification schemes for graph manifolds with applications to the topology of singularities. A tentative schedule is as follows:

Lectures 1 and 2: Classification in the context of general 3-manifolds; sketch proof that the resolution of a surface singularity is determined by its topology.

Lecture 3: Other classification schemes for graph manifolds and applications.

Lectures 4 and 5: Singularities of splice type.

I stress that this schedule is very tentative.

References:

"Graph 3-Manifolds, splice diagrams, singularities. (Notes for a short course in Trieste August 2005.)" http://www.math.columbia.edu/~neumann/preprints/graphmans0.1.pdf

"A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves", Trans. Amer. Math. Soc. 268 (1981), 299--343.

Contact topology is the odd-dimensional twin of symplectic geometry. Giroux discovered around 2000 that there is a tight relation between contact structures on a given manifold and open book decompositions of it.  The aim of the course is to explain this relation and to show how to use it in order to study the natural contact structure on the link of an isolated complex analytic singularity.

We characterize Hardy fields by intrinsic properties of their valuation and their derivative, in order to give a differential version of Kaplansky’s theorem

Let denote a germ of a normal complex analytic surface singularity. In this talk I will outline an explicit topological construction which gives the Milnor open books associated to germs of holomorphic functions which define isolated singularities at .

I will also discuss the following joint result with Selma Altinok (Adnan Menderes University, Turkey): Fix a representative of a germ of a rational surface singularity. The page genus of the associated Milnor open books is minimized when is taken to be the restriction of a ``generic' linear form on to .

For a theorem of Stein says that the set of values such that is reducible is either equal to or is a finite set of cardinality .

In a first time we state such a result for rational fractions .

In a second time we answer the following question: Given , which monomials could be added in order that becomes irreducible for generic ?

This second part is a joint work with P. Dèbes and S. Najib.

The minimal singularities of surface are the rational singularities with reduced fundamental cycle (or equivalently reduced tangent cone). In a 2004 paper, I gave an explicit description of the critical locus of a generic projection of such a singularity S onto C^2, also called the polar curve of S. The first proof of this result rested on computation done by M. Spivakovsky for these surfaces. I will explain how one can circumvent these computations by geometric arguments, and if time permits tell more on the geometry of these surfaces, which are a good source of examples and counterexample in the study of normal surfaces singularities.

Daremos un panorama de los resultados conocidos de reducción de singularidades en sistemas dinámicos, alguna de sus aplicaciones y potenciales avances.

The Weierstrass-Hironaka division theorem for formal and for convergent power series can be extended to the ring of linear differential operators. We will give some appliations of this division theorem to the so called computational D-module theory.

When there are monodromy phenomena, as for instance in the sections of a variety by a pencil of hyperplanes, the difference between a cycle and its image by monodromy is a vanishing cycle. It is natural to call this difference a variation. Similarly for homotopy -cells. In the van Kampen theorem for plane curves, the generators of the fundamental group of the complement can be taken in the section by a generic line and the relations are their variations by monodromies. Such a point of view is insufficient in higher dimensional analogues of this theorem. One needs to add the effect by another operator or to consider a more sophisticated kind of homotopical variation. We shall follow this second way in this talk and construct such a homotopical variation.

We shall take our inspiration from a special homological variation which comes in when generalizing the so-called second Lefschetz theorem. The well known first (or hyperplane section) Lefschetz theorem states that the homology groups of a variety are, up to a certain rank, the same as those of a generic hyperplane section of it and that, at the immediately next rank, all the generators are in the hyperplane section. The less known second Lefschetz theorem gives the relations between these generators with the help of a pencil containing the hyperplane. In its classical form, which applies to compact non-singular varieties, the theorem uses thimbles obtained by crashing frontally on the exceptional hyperplanes of the pencil. This no more works for non-closed varieties, even if non-singular. Nor does, in either cases, the mere consideration of ordinary variations. A way which goes through in both cases is to use a more sophisticated homological variation as follows.

Consider monodromies which leave the axis of the pencil pointwise fixed and take a relative cycle of the hyperplane modulo the axis. Its image by monodromy has the same boundary so that the difference between the cycle and its image is actually an absolute cycle. This absolute cycle is the variation we consider. These new variations (which include the ordinary ones as a special case) give all the relations.

This kind of more general homological variation was already considered in the context of holomorphic functions with isolated critical points. In this talk, we shall look at an homotopical analogue of it. We shall indicate how it can be used in a higher dimensional analogue of the van Kampen theorem which was formerly stated with ordinary variations plus degeneration operators. We shall also indicate a conjectural homotopy version of the second Lefschetz theorem, a statement which in fact unifies the van Kampen and second Lefschetz theorems.

Given a holomorphic map-germ with a critical value at , there are two equivalent ways of defining its Milnor fibration. The first is:

(1)

(2)

We show that there is a canonical decomposition of the whole ball into real analytic hypersurfaces that spin around their ''axis'' forming a kind of ''open-book'' with singular binding. Using this canonical decomposition we give a slight refinement of Milnor Fibration Theorem.

Joint work with J. Seade and J. Snoussi.

Let be a finite dimensional algebra over an algebraically closed field . The Auslander-Reiten transformation associates an indecomposable module to each indecomposable non-projective module. We consider the graded algebra with n-homogeneous space . If is the path algebra of a quiver whose underlying graph is an extended Dynkin diagram and are the complex numbers, then can be chosen such that is the ring of a Kleinian singularity. For a canonical algebra, it is possible to get all the singularities associated to Fuchsian groups. We present joint work with Helmut Lenzing describing properties of and such that is an ICIS (=isolated complete intersection singularity) of Fuchsian type. Our results yield alternative description of the Poincare series of the singularities.

In this talk I will show that if we consider a real isolated singularities defined by map-germs , and a convenient one parameter family associated to , using some condition of transversality in this family, will be possible to guarantee that the map-germ satisfy the strong Milnor condition. Motivations and appropriated definition will be given during the talk.

The theory of Hodge structures plays a central role in Algebraic Geometry.

Delgne introduced mixed Hodge structure to extend the theory to cohomology of singular varieties and gave a proof of Hard Lefschetz theorem based on such theory.

Constructible sheaves reflects the special properties of cohomology of algebraic families of varities.

The discovery of the Intersection complex opened the way to the generalization of Lefschetz theorems.

The conference will consist in an orientation guide with references to help the students to find the link between the various courses and fundamental research articles.

If is a scheme of finite type over a perfect field , and is a geometric point, Grothendieck defined the arithmetic fundamental group as the automorphism group of the fiber functor from étale coverings to finite sets. If is separably closed in , then surjects to where is the separable closure of in the residue field of . An open question is to understand geometrically the sections of this surjection. Grothendieck conjectured that if is of finite type over and is an an abelian curve, then sections come from rational points of or of its compactification. More precisely he conjectured that the universal covering based at , or its compactification, should have a -form with a -rational pro-proint.

We show the existence of a -linear abelian rigid tensor category which underlines . It allows us to define nice geometric fiber functors. Then Deligne's theory of groupoid schemes allows us to see that a section of exists if and only if our category is neutral. We define a general universal covering associated to the choice of a fiber functor. This construction allows us to see the easier part of Grothendieck's prediction: if a section comes from a -rational point, it defines automocally a -rational pro-proint.

(Joint with Phùng Hô Hai)

We produce examples of complex algebraic surfaces with isolated singularities such that these singularities are not metrically conic, i.e. the germs of the surfaces near singular points are not bi-Lipschitz equivalent, with respect to inner metric, to cones.

Topological equisingularity is a subtle notion, weaker than Whtney equisingularity and more difficult to characterise in terms of invariants. I will report on recent progress towards the understanding of topological equisingularity of germs of hypersurfaces. I will introduce new notions such as equisingularity at the critical set or equisingularity at the normalisation and show how they can be used to grasp topological equisingularity.

The Milnor fibration is a well known tool for studying hypersurface singularities. The proof of the existence of the Milnor fibration depends on the existence of a stratification of the function satisfying the condition. If is a mapping which defines a complete intersection with an isolated singularity, then an stratification exists, hence so does a Milnor fibration. Motivated by a question and example of Lê, in this talk we discuss the case of complete intersections with non-isolated singularities. It turns out that the existence of an stratification puts stringent conditions on . The principle tool used is the theory of the integral closure of modules.

For an absolutely isolated singularity in real n-dimensional space we show how to construct a flag of ideals in the Milnor Algebra of the singularity and a family of bilinear forms in the flag. We will give a topological interpretation of these numbers.

A survey on some topics of the geometry and algebra of the Nash blow-up, which replaces singular points with limiting positions of tangent spaces to nearby smooth points. In particular we consider singularities of orbifolds, with some new results in higher dimension.

The famous 5-points Theorem of Nevanlinna asserts that if and are nonconstant meromorphic functions, and if there exist distinct points ( ) such that , then . If instead of preimages of single points we consider the preimage of a set, then we have the notion of unique range sets as follows. Let be a morphic function on the complex plane, and be a subset of points in . Define

a with multiplicity

The problem of unique range sets is closely related to the problem of decompositions of meromorphic functions, i.e, presentation of a meromorphic function in the form ..

The mail tools of studying the above problems are Nevanlinna theory and singularity theory. In this talk we give a survey of recent results on unique range sets and decompositions of meromorphic functions.

The aim of theorems of Lefschetz type is to compare invariants for a complex projective or quasi-projective variety and those for some hyperplane section. Such theorems have been developed in joint work with Le first in the setting of algebraic topology. Later we have shifted to the Picard group, the talk will primarily focus upon this subject. It is known that it is useful to consider a neighbourhood of the hyperplane section, too, instead of the hyperplane section itself, in this case we speak of Zariski-Lefschetz theorems. This makes it possible to pass to other objects related to the Picard group, like meromorphic functions and divisors.

Given a function , integers and non-zero complex numbers we consider the following families of functions from :

For the second family, the functions are quasi-homogeneous only over the real numbers and the coefficients cannot all be taken to be 1. We characterize those for which the corresponding singularity is isolated and we extend some results of Seade who studied the case where is bijective. We are still in the process of defining and computing invariants for these singularities and of determining the topology of their zero set in some cases.

This is joint work with Vinicio Gómez Gutiérrez and Luis Hernández de la Cruz.

I will report on a joint work with E. Artal, J. Fernandez de Bobadilla and A. Melle. We will consider Weighted-Iomdine singularities defined in by a series , where is weighted-homogeneous such that the singularities of in the weighted-projective plane do not intersect the curve defined by . We will give a formula for the Milnor number and relate it with the geometry and resolution of the singularity to give interesting examples of such singularities. We give a negative answer to Question B of Zariski for surface singularities.

We will discuss the work of Lê on hypersurface singularities, beginning with his results from 1973 and ending with our recent joint work from 2006.

In this talk I will discuss some open problems related with Igusa Zeta functions (topological zeta functions, motivic zeta functions,...) associated to a complex hypersurface. These zeta functions are, in some sense, rational functions and the main conjectures are about their poles. The conjectures provides an interesting bridge between some arithmetic and the complex topology of the hypersurface. We will mainly focusses on poles of maximal order and their properties. This is a work in progress jointly with Tristan Torrelli and Wim Veys.

Consider two smooth cubic curves which are tangent at one point with intersection number 9. There are two different geometry for such casse by Artal Bartolo (Zariski pairs). We generalize this observation and explain this phenomenon from Taylor expansions of the polynomial at the intersection point. Let be the space of smooth curves of degree with and the tangent line . We consider the space of Taylor expansion of one variable up to degree . By implicit function theorem, we have a canonical mapping . We analyze this mapping.

This talk is concerned with a problem of affine global differential geometry of the parabolic curve of an algebraic smooth surface in . A hessian curve in the real plane is a projection of the set of parabolic points of the graph of some function in two variables. In particular, it will be discussed the question: Given the graph of a real polynomial on the plane, is it possible that some usual configurations of plane algebraic curves are forbidden for the parabolic curve, due to some obstruction associated to the condition "hessian of equal to zero"? This is a question on "hessian topology", where hessian topology means the study of discrete invariants of smooth objects associated to the restriction Hessian equal to zero.

We will introduce the notion of alternate product of Frobenius manifolds and we will give an interpretation of the Frobenius manifold structure canonically attached to the quantum cohomology of the Grassmannian in terms of alternate products. We also investigate the relationship with the alternate Thom-Sebastiani sum of Laurent polynomials. This is a joint work with Bumsig Kim (Seoul) and relies on a joint work with I. Ciocan-Fontanine and B. Kim.

(a joint work with H. Kajiura and A. Takahashi)

We explain about the program to construct primitive forms associated to a hypersurface isolated singularity by a use of infinite dimensional Lie algebras. We discuss about some recent results on certain infinite root systems obtained by a use of certain triangulated category associated to exceptional singularities.

We give a method for characterizing base points of polar curves on a normal surface of type , after a resolution of singularities by Jung's method.

I will ilustrate by examples the philosophy that sometimes the understanding of a singular object can be vastly improved by viewing it as a deformation of an apparently more complicated singular object. For example, singularities of plane analytic branches are better understood by viewing them as deformations of non-plane monomial curves. I will discuss other examples as well.

Simple elliptic singularities of normal surfaces were defined by Saito and several special types were named as , , and . Beyond Grothendieck-Brieskorn theory on the relation between simple singularities of surfaces and simple Lie algebras, many mathematicians tried to discover some similar relations between simple elliptic singularities and Lie algebras or related objects.

Here we construct the simple elliptic singularities of type and their semi-universal deformation spaces by using .

We study germs of pairs of codimension one regular foliations in . We show that the discriminant of the pair determines the topological type of the pair. We also consider various classifications of the singularities of the discriminant.

In this work we study some properties of the bi-Lipschitz contact equivalence. Our main result is a finiteness theorem with respect to bi-Lipschitz contact equivalence classes of real polynomial map-germs.

Our objective is to present a generalization for the local Euler obstruction, that was defined for holomorphic function with isolated singularity at the origin, to the case of a holomorphic map , where (V, 0) is a germ of complex analytic variety, equidimensional of dimension . The main result is a formula which computes the local Euler obstruction, defined for -frames in terms of the local Euler obstruction of .

We study the local topology and topological triviality of ruled surfaces in . In this work we compare the singularities of germs from to with the singularities appearing in the set of ruled surfaces, doing a local topology classification of the ruled surface and study the topological triviality of families of ruled surfaces.