# Holomorphic Foliations and Singularities of Mappings and Spaces

Organizers:

- Lev Birbrair (Univ. Federal do Ceará)
- Fernando Cukierman (Univ. de Buenos Aires)
- Dale Cutkosky (Univ. of Missouri Columbia)
- Marcelo Saia (Univ. de São Paulo/São Carlos)
- Jawad Snoussi (UNAM)
- Marcio G. Soares (University Federal de Minas Gerais)

Schedule:

- Tuesday, Jul 25 [McGill U., Burnside Hall, Room 1214]
- 11:45 Edward Bierstone (Univ. of Toronto - Canada), Global smoothing of a subanalytic set
- 12:15 Felipe Cano (Univ. Valladolid - Spain), Local Uniformization of Singular Codimension One Foliations
- 14:15 Fuensanta Aroca (Univ. Nac. Aut. Mexico - Mexico), Parametrizations with exponents in cones and the algebraic closure of the field of rational functions in several variables.
- 14:45 Alicia Dickenstein (Univ. Buenos Aires - Argentina), Sparse mixed discriminants, toric jacobians, and iterated discriminants
- 15:45 André Belotto da Silva (Univ. Paul Sabatier Toulouse III - France), Local monomialization of a system of first integrals of Darboux type
- 16:15 Alexandre Fernandes (Univ. Fed. Ceará - Brazil), On Lipschitz regularity at infinity of complex algebraic sets
- Wednesday, Jul 26 [McGill U., Burnside Hall, Room 1214]
- 13:45 Bruna Oréfice Okamoto (Univ. Fed. São Carlos - Brazil), Equisingularity of map germs from a surface to the plane.
- 14:15 Mauricio Corrêa Jr. (Univ. Fed. Minas Gerais - Brazil), Čech-Bott-Chern cohomology and refined residue theory.
- 14:45 Jose Seade (Univ. Nac. Aut. Mexico - Mexico), On the Chern classes of singular varieties
- 15:15 Jorge Vitorio Pereira (IMPA - Brazil), Effective algebraic integration in bounded genus
- 16:15 Maria Aparecida Ruas (Univ. São Paulo - Brazil), Lipschitz Normal Embeddings in the Space of Matrices

- Edward Bierstone

Univ. of Toronto - CanadaGlobal smoothing of a subanalytic setSemialgebraic and subanalytic sets have become ubiquitous in mathematics since their introduction by {\L}ojasiewicz in the 1960s, following the celebrated Tarski-Seidenberg theorem on quantifier elimination. I will discuss two long-standing questions in real-analytic geometry, on global smoothing of a subanalytic set (an analogue of resolution of singularities), and on transformation of a proper real-analytic mapping to a mapping with equidimensional fibres by global blowings-up of the target (a classical result of Hironaka in the complex-analytic case). These questions are related: a positive answer to the second can be used to reduce the first to the simpler semianalytic case. It turns out that the second question has a negative answer, in general, and that the first problem nevertheless has a positive solution. (Work in collaboration with Adam Parusi\'nski.) - Felipe Cano

Univ. Valladolid - SpainLocal Uniformization of Singular Codimension One FoliationsThis is a work in collaboration with Miguel Fernández-Duque. We prove the existence of Local Uniformization, in the classical sense of Zariski, for codimension one singular foliations in projective varieties of any dimension. The precise statement is as follows: Let K/k be the field of rational functions of a projective variety over a base field k of characteristic zero. A singular foliation F of codimension one is defined from a birational viewpoint as a 1-dimensional K-vector subspace of the Kähler differentials of K over k, satisfying Frobenius integrability condition. We take a k-valuation ring R of K. We show the existence of a projective model M of K such that F is simple at the center Y of the valuation in M. The definitions of simple points for F is compatible with the known ones in the holomorphic case, in particular with the definitions appearing in the statement of Cano's global reduction of singularities in dimension three. - Fuensanta Aroca

Univ. Nac. Aut. Mexico - MexicoParametrizations with exponents in cones and the algebraic closure of the field of rational functions in several variables.When working with an algebraically closed field of characteristic zero. Newton-Puiseux theorem asserts that, at a singular point, a plane curve may be parametrized by a Puiseux series. In arbitrary dimension, singularities of hypersurfaces cannot, in general, be parametrized, anyhow, we can assure the existence of parametrizations in "wedges". These parametrizations are represented as Laurent Puiseux series whose support is contained in the translation of a rational strongly convex cone. With the above result, we may construct different algebraically closed fields containing the field of power series in several variables. Which of these elements are algebraic over the field of power series? We will give some characterizations in terms of the support set. Joint work with G. Ilardy, V. Saavedra and G. Rond. - Alicia Dickenstein

Univ. Buenos Aires - ArgentinaSparse mixed discriminants, toric jacobians, and iterated discriminantsSparse mixed discriminants characterize those polynomial systems with fixed supports, which define a singular variety. We will discuss the relation between sparse mixed discriminants and the resultant of the given polynomials and their toric jacobian, which yields a multiplicativity formula for sparse mixed discriminants. We will also discuss conditions for mixed discriminants to be computed via iterated discriminants, with possible extra factors coming from the singularities of the discriminantal locus. - André Belotto da Silva

Univ. Paul Sabatier Toulouse III - FranceLocal monomialization of a system of first integrals of Darboux typeGiven a real- or complex-analytic singular foliation $\mathcal{F}$ with $n$ first integrals of Darboux type $(f_1,\dots,f_n)$, we prove that there exists a local monomialization of the first integrals. In particular, if $\mathcal{F}$ is generated by the $n$ first integrals, the foliation $\mathcal{F}$ is reduced to monomial singularities. - Alexandre Fernandes

Univ. Fed. Ceará - BrazilOn Lipschitz regularity at infinity of complex algebraic setsWe prove that any complex algebraic subset of $\mathbb{C}^n$ which is Lipschitz regular at infinity is an affine linear subspace of $\mathbb{C}^n$. No restrictions on dimension and codimension are needed. This is a joint work with J. Edson Sampaio. - Bruna Oréfice Okamoto

Univ. Fed. São Carlos - BrazilEquisingularity of map germs from a surface to the plane.Let $(X,0)$ be an ICIS of dimension 2 and let $f:(X,0)\to (\mathbb C^2,0)$ be a map germ with an isolated instability. We look at the invariants that appear when $X_s$ is a smoothing of $(X,0)$ and $f_s:X_s\to B_\epsilon$ is a stabilization of $f$. We find relations between these invariants and also give necessary and sufficient conditions for a $1$-parameter family to be Whitney equisingular. Joint work with J.J. Nuño-Ballesteros and J. N. Tomazella. - Mauricio Corrêa Jr.

Univ. Fed. Minas Gerais - BrazilČech-Bott-Chern cohomology and refined residue theory.The Bott-Chern cohomology of a complex manifold refines both the de Rham and Dolbeault cohomologies. In this talk, we present a Čech-Bott-Chern cohomology theory. In particular, we present a relative Bott-Chern cohomology and discuss applications in the refined residue theory. A joint work with Tatsuo Suwa. - Jose Seade

Univ. Nac. Aut. Mexico - MexicoOn the Chern classes of singular varietiesThere are several different extensions of the notion of Chern classes to the case of singular varieties. This gives rise to the notion of Milnor classes, which restrict to the usual Milnor number when the variety has only one singular point which is a local complete intersection. In this talk we will speak about new results on Milnor classes obtained with R. Callejas-Bedregal and M. Morgado. - Maria Aparecida Ruas

Univ. São Paulo - BrazilLipschitz Normal Embeddings in the Space of MatricesThe germ of an algebraic variety is naturally equipped with two different metrics up to bilipschitz equivalence. The inner metric and the outer metric. One calls a germ of a variety Lipschitz normally embedded if the two metrics are bilipschitz equivalent. We prove Lipschitz normal embeddedness of some algebraic subsets of the space of matrices. These include the space $m\times n$ matrices, symmetric matrices and skew-symmetric matrices of rank equal to a given number and their closures, and the upper triangular matrices with determinant $0$. We also make a short discussion about generalizing these results to determinantal varieties in real and complex spaces. This is a joint work with Dmitry Kerner and Helge Moeller Pedersen.