Roughly speaking, valuations and derivations are measures of contact, derivations are algebraic differential equations and after Kaplansty’s theorem valuations may be considered as generalized arcs. Then there are two problems:

• Relation between valuation and derivation corresponding to "generalized arc solution of a differential equation"
• Structure of the space of arcs solution of a differential equation.

Given a simplicial graph, its associated right-angled Artin group is a group generated by the vertices and such that two vertices commute if they are joined by an edge; the kernel of an epimorphism onto the infinite cyclic group (non vanishing on the vertices) is called an Artin kernel and its homology is in a natural way a module over a ring of Laurent polynomials. We study the module structure of this homology which can be interpreted as the homology of the Milnor fiber of a monomial function on a highly singular space for which a version of the Monodromy Theorem can be stated. It is joint work in progress with J.I. Cogolludo and D. Matei.

We study the topological triviality and the Whitney equisingularity of a family of isolated determinantal singularities. On one hand, we give a Lê-Ramanujam type theorem for this kind of singularities by using the vanishing Euler characteristic. On the other hand, we extend the results of Teissier and Gaffney about the Whitney equisingularity of hypersurfaces and complete intersections, respectively, in terms of the constancy of the polar multiplicities.

(Joint work with B. Oréfice-Okamoto and J.N. Tomazella)

In this talk we study handle decompositions of 3-manifolds in relation with an invariant defined by Eiji Ogasa. This invariant takes into account the complexity of regular level surfaces of the handle decompositions. We will discuss the relationship of the Ogasa invariant with the topology of the 3-manifold. Comparison with the Heegaard genus will give some informations about the geometry of the 3-manifold.
This is a joint work with Gioia Vago.

The Milnor number is one of the most important invariant associated to an isolated hypersurface singularity. Since its introduction by John Milnor in 1968 many generalizations have appeared in the literature, both local and global, even allowing arbitrary singularities. In this talk I will address some of these generalizations, giving special attention to the global aspects of these.

In this talk we give a review on the development on this area

For a complex analytic variety with an action of a finite group and for an invariant 1-form on it, we give equivariant versions (with values in the Burnside ring of the group) of the radial index, the GSV-index and of the local Euler obstruction of the 1-form and describe their relations. In particular, this leads to equivariant versions of the local Euler obstruction of a complex analytic space and of the global Euler obstruction. This is joint work with Sabir Gusein-Zade.

The front invariants under consideration are those whose increments in generic homotopies are determined entirely by diffeomorphism types of local bifurcations of the fronts.

Such invariants are dual to trivial codimension 1 cycles supported on the discriminant in the space of corresponding Legendrian maps. We describe the spaces of the discriminantal cycles (possibly non-trivial) for framed fronts in an arbitrary oriented 3-manifold, both for the integer and mod2 coefficients. For the majority of these cycles we find homotopy-independent interpretations which guarantee the triviality required. In particular, we show that all integer local invariants of Legendrian maps without corank 2 points are essentially exhausted by the numbers of points of isolated singularity types of the fronts.

I give a survey on results about the geometry of equisingular families V(s1,...,sr) of algebraic curves C in a linear system |D| on a fixed smooth projective surface S with prescribed singularities of given singularity type s1,...,sr (the types can be analytic or topological types or given by the Newton diagram, ...). A classical example is the family of plane curves of given degree in P^2 with nodes and cusps. We are interested in conditions guaranteeing the existence (non-emptiness) reps. smoothness resp. irreducibility of V(s1,...,sr). These conditions should be

- numerical: relating numerical invariants of the surface S, the linear system |D|, and s1,...,sr, - universal: applicable to any C, D, S, and an arbitrary number and type of singularities, - asymptotically proper: asymptotically of the same growth as the known necessary conditions

Substantial progress, even for curves in P^2, has been made only in the last decade. We show asymptotically proper conditions for existence in general (which are even optimal for plane curves with nodes and cusps). We discuss further proper and in some cases optimal asymptotic bounds for the T-smoothness and irreducibility of equisingular families and report on open problems and conjectures. Most of the results are joint work with Christoph Lossen and Eugenii Shustin.

In this talk we discuss the influence of the analytic type of a plane branch belonging to a given equisinsingularity class in the topology of its polar curve. This goes beyond the known results about the topology of a general member of an equisingularity class of plane branches.

A new fractal set of "complexified Arnold's tongues" will be discribed.
It occurs in the following way. Consider an analytic diffeomorphism f of a unit circle S1 into itself. Let fλ = λf , and |λ| ≤ 1. If |λ| = 1, then fλ is still an analytic diffeomorphism of a circle into itself; let ρ(λ) be its rotation number. If |λ| < 1, then an elliptic curve occurs as a factor space of an action of fλ . Let μ(λ) be the "multiplicative modulus" of this elliptic curve. A "moduli map" of unit discs λ → μ(λ) occurs. The problem is to describe its limit values.
It appears that the boundary values of the moduli map form a fractal set: a union of S1 and a countable number of "bubbles" adjacent to all the roots of unity from inside S 1 . Relations of these limit values and rotation numbers ρ(λ) will be described.
These results are motivated by problems stated by Arnold and Yoccoz, and are due to Risler, Moldavskis, Buff, Goncharuk and the speaker. Some open problems will be stated.

I will describe contruction of invariant of infinite cyclic covers taking values in the Grothendieck group of complex algebraic varieties. This will generalize Denef-Loeser motivic Milnor fiber as well as other versions of motivic Milnor fibers which appeared recently (joint work with M. Gonzalez-Villa and L. Maxim).

Let $F:R^n \rightarrow R^k$ be a homogeneous quadratic mapping, and consider its zero set $V=F^{-1}(0)$ and the intersection of $V$ with the unit sphere $Z=V\cap S^{n-1}$. The topology of these varieties has been studied for decades, but in the last five years many old and new topological problems about them have been solved: the topological description of the generic case for $k=2$, of many infinite families for $k>2$ (where complete results look out of reach) and of some of their variants (open-book structures on $Z$, semi-algebraic versions, smoothings of $V,\dots$), under some restrictions that should be removed soon by the same methods.

After reviewing briefly those results I will turn to some work in progress on questions that seem to require new methods: the topological description of their projective versions and of some interesting families that admit contact structures. The construction of another interesting family with dihedral symmetry depends on certain conjectures about the Vandermonde matrix on roots of unity.

(Joint work with Yadira Barreto, Samuel Gitler, Vinicio Gómez Gutiérrez and Alberto Verjovsky).

The goal is to describe the set on the topological classes of holomorphic foliations germs at the origin of C^2. We introduce notions of marking; We prove, under some assumptions, that when the Camacho-Sad indexes and the holonomy representations (which are topological invariants) are fixed, then the classes of marked foliations constitute an abelian group which is a finite extension of a quotient of C^N by a finitely generated subgroup. We compare this situation with the space of strongly marked foliation whose classification is given by the monodromy representation.

This talk will describe ongoing joint work with Paul Cadman and Duco van Straten, based on the PhD thesis of the former.

Givental and Varchenko used the period mapping to pull back the intersection form on the Milnor fibre of an irreducible plane curve singularity $C$, and thereby define a symplectic structure on the base space of a miniversal deformation. We show how to combine this with a symmetric basis for the module of vector fields tangent to the discriminant, to produce a family of involutive ideals which define the strata of parameter values $u$ such that $\delta(C_u)\leq k$. In the process we find an unexpected Lie algebra and a still mysterious canonical deformation of the module structure of the critical space over the discriminant. Much of this work is experimental - a crucial gap in understanding still needs bridging.

The local topology of isolated complex surface singularites is long understood, as cones on closed 3-manifolds obtained by negative definite plumbing. On the other hand a full understanding of the analytic types is out of reach, motivating Zariski's efforts into the 1980's to give a good concept of "equisingularity" for families of singularities.

The significance of Lipschitz geometry as a tool in singularity theory is a recent insight, starting (in complex dimension 2) with examples of Birbrair and Fernandes published in 2008. I will describe work with Anne Pichon proving that Zariski equisingularity in complex dimension two (and lower) is equivalent to constant Lipschitz geometry. This builds on earlier joint work with Birbrair and Pichon on classifying the inner geometry in terms of discrete data associated with a refined JSJ decomposition of the associated 3-manifold link.

Convenient mixed functions with strongly non-degenerate Newton boundaries have a Milnor fibration, as the isolatedness of the singularity follows from the convenience. In this paper, we consider the Milnor fibration for non-convenient mixed functions.
We also study geometric properties such as Thom's $a_f$ condition, the transversality of the nearby fibers and stable boundary property of the Milnor fibration and their relations.

T. Mostowski proved that every (real or complex) germ of an analytic set is homeomorphic to the germ of an algebraic set. We show that every (real or complex) analytic function germ, defined on a possibly singular analytic space, is topologically equivalent to a polynomial function germ defined on an affine algebraic variety. The main tools are: Artin approximation and Zariski equisingularity. (This is a joint work with Marcin Bilski and Guillaume Rond)

Minimal surface singularities, introduced by J. Kollár in 1985, play a key role in resolution theory of surface singularities since they appear as central objects in the two main resolution algorithms : the resolution obtained by a sequence of normalized Nash transformations (Spivakovky, 1990) and the one obtained by successive normalized blow-ups of points (Zariski 1939) as showed by Bondil and Lê in 2002. The question of the existence of a duality between these two algorithms, asserted by D. T. Lê, remains open, and the fact that minimal singularities seem to be the common denominator between them suggests the need of a better understanding of this class of surface germs.

I will present a recent joint work with Walter Neumann and Helge Moller Perdersen in which we show that minimal surface singularities can be characterized by a remarkable metric property among rational surface singularities.

I will present joint work with Burak Ozbagci. We propose a generalization of the classical notions of plumbing and Murasugi summing operations to smooth manifolds of arbitrary dimensions, so that in this general context Gabai's credo "the Murasugi sum is a natural geometric operation" holds. In particular, we prove that the sum of the pages of two open books is again a page of an open book and that there is an associated summing operation of Morse maps. I will conclude with several open questions relating our work with singularity theory or contact topology.

We review basic results on determinantal varieties and show how to apply methods of singularity theory of matrices to study their invariants and geometry.

In 1928, Brauner became the first to apply knot theory to the understanding of singular points of complex plane curves. In 1968, Milnor considerably extended the subject with his pioneering study of the topology of isolated singular points of complex hypersurfaces in all dimensions. Since then much progress has been made on this local theory; but little has been published on related larger-scale phenomena, except in complex dimension 2 where various tools and techniques (quasipositivity, the enhanced Milnor number, contact and symplectic geometry and topology, etc) have led to deeper though far from complete understanding. I will describe some old and many new (or not previously published) results in higher dimensions, and pose a number of open questions--both general and specific--for future work.

We discuss the existence of transversal projective foliations for regular foliations which are generically transverse to a curve.

Let $X$ and $Y$ be closed complex subvarieties in an ambient complex manifold $M$. We will explain the intersection formula $$c(X) \cdot c(Y)= c(TM)\cap c(X\cap Y)$$ for suitable notions of Chern classes and transversality for singular spaces. If $X$ and $Y$ intersect transversal in a Whitney stratified sense, this is true for the MacPherson Chern classes (of adopted constructible functions). If $X$ and $Y$ are "splayed" in the sense of Aluffi-Faber, then this formula holds for the Fulton-(Johnson-) Chern classes, and is conjectured for the MacPherson Chern classes.

We explain, that the version for the MacPherson Chern classes is true under a micro-local "non-characteristic" condition for the diagonal embedding of $M$ with respect to $X\times Y$. This notion of non-characteristic is weaker than the Whitney stratified transversality as well as the splayedness assumption.

The homology of projective hypersurfaces is classically known for smooth hypersurfaces. Due to results of Dimca the homology of a singular hypersurface with isolated singularities is related to the homology of the smooth case as follows: the difference is concentrated in one dimension and related to the direct sum of the Milnor lattices of the singular points.

In the talk we will treat 1- dimensional singularities. By using a one parameter smoothing of an n- dimensional hypersurface we can compare with a smooth hypersurface. We call this the vanishing homology of the smoothing. We will show that this (relative) homology is concentrated in two dimensions only: n+1 and n+2.

Moreover we will give precise information and bounds for the Betti numbers of the vanishing homology in terms of properties of the singular set, the generic transversal singularities, the 'special' non-isolated singularities and (if they occur) the isolated singularities.

As an example: the n+2 Betti number is bounded by the sum of (generic) transversal Betti numbers on each irreducible component of the 1-dimensional singular set. In several cases this Betti number is zero.

We discuss several examples.

This is joint work with Mihai Tibar, and a preprint can be found on http://arxiv.org/abs/1411.2640

We present a residual formula for vector fields on compact complex orbifolds. This is analogous to, and inspired by, the well known Bott residue formula for vector fields on complex manifolds. We then derive some applications, related to holomorphic foliations on weighted complex projective spaces and conclude by exploring Hirzebruch surfaces as resolutions of weighted projective planes of a simple type.

We define local invariants of three kinds of characteristic points on generic surfaces in projective 3-space or in $\mathbb{R}^3$. Our formulas for global counting of these invariants on the surface (and on domains in it) involve the Euler characteristic of the surface (and of those domains) and impose restrictions to the possible numbers of coexisting characteristic points of different types.

There is an intrinsic (fundamental) cubic form on the surface. The zeros of this form define some fields of lines tangent to the surface; these foelds degenerate at the caracteristic points. Tourning around a caracteristic point, the monodromy of these fields is fractional, but the sum of these fractions over all caracteristic points of the surface provides the Euler characteristic of the surface.

In this talk I will describe new examples of compact contact manifolds in arbitrarily large dimensions. These manifolds are modifications of certain so-called moment-angle manifolds which were obtained several years ago by Santiago López de Medrano and myself and later generalized by Laurent Meersseman. This is joint work with Yadira Barreto.

The paper on the subject has appeared very recently: "Moment-Angle Manifolds, Intersection of Quadrics and Higher Dimensional Contact Manifolds." Yadira Barreto and Alberto Verjovsky. Moscow Math. Journal. Volume 14, Issue 4 (October–December), 2014, pp. 669–696