Schedule for: 18w5011 - Lipschitz Geometry of Singularities

We investigate connections between Lipschitz geometry of real algebraic varieties and properties of their arc spaces. First we define a real motivic integral which admits a change of variable formula not only for the birational but also for generically one-to-one Nash maps. As a consequence we obtain an inverse mapping theorem which holds for generically arc-analytic maps. Then we characterize in terms of the motivic measure, germs of arc-analytic homeomorphisms between real algebraic varieties which are bi-Lipschitz for the inner metric. (Based on a joint paper with J.-B. Campesato, T. Fukui, and K. Kurdyka.)

I will speak about a common work with García Barroso and González P\'erez. Favre and Jonsson described in 2004 from various viewpoints a structure of real tree on the projectified space of semivaluations centered at a smooth point of a complex surface. We describe it from another viewpoint, as the projective limit of the Eggers-Wall trees of the reduced germs of curves at the given point. The Eggers-Wall tree measures the contacts between the various Newton-Puiseux series of a given curve singularity. Therefore, our viewpoint is adapted whenever one studies the local geometry of surfaces using such series.

I will introduce a non-archimedean version of the link of a singularity. This object will be a space of valuations, a close relative of non-archimedean analytic spaces (in the sense of Berkovich) over trivially valued fields. After describing the structure of these links, I will deduce information about the resolutions of surface singularities. If times allows, I will then characterize those normal surface singularities whose link satisfies a self-similarity property. The last part is a joint work with Charles Favre and Matteo Ruggiero.

Take polynomials $f,g\in k[X]$, where $k$ is the field of complex or real numbers. Under certain assumptions we show equivalence of the following conditions: (i) $(f,g)$ is radical (ii) for every polynomial $h$ if there exists a pointwise solution of $$A\cdot f + B\cdot g =h$$ then there exists its polynomial solution (iii) every continuous function $$F=\left\{\begin{array}{ll} \alpha & on\ \{f=0\}\\ \beta & on\ \{g=0\} \end{array}\right.$$ with $\alpha,\beta\in{k}[X]$, is a restriction of a polynomial. We will discuss relation of (i-iii) with Lipschitz normal embeddings. Work in progress.

We would like to present some bi-Lipschitz invariants that are used in our recent work on classification of Lipschitz simple germs. The invariants include rank, corank and algebraic tangent cone of non-quadratic part of function germs. 

In the first part we presented some bi-Lipschitz invariant. In this second part on the classification of Lipschitz simple germs we present techniques for checking bi-Lipschitz triviality of one-parameter deformations and give a list of smooth one-modal germs that are bi-Lipschitz trivial. Furthermore, we present the complete list of Lipschitz simple germs.

We consider the problem of Lipschitz classification of singularities of Real Surfaces definable in a polynomially bounded o-minimal structure (e.g., semialgebraic or subanalytic) with respect to the outer metric. The problem is closely related to the problem of classification of definable functions with respect to Lipschitz Contact equivalence. Invariants of bi-Lipschitz Contact equivalence presented in Birbrair et al. (2017) are used as building blocks for the complete invariant of bi-Lipschitz equivalence of definable surface singularities with respect to the outer metric.

We consider the problem of Lipschitz classification of singularities of Real Surfaces definable in a polynomially bounded o-minimal structure (e.g., semialgebraic or subanalytic) with respect to the outer metric. The problem is closely related to the problem of classification of definable functions with respect to Lipschitz Contact equivalence. Invariants of bi-Lipschitz Contact equivalence presented in Birbrair et al. (2017) are used as building blocks for the complete invariant of bi-Lipschitz equivalence of definable surface singularities with respect to the outer metric.

The motivic zeta function $Z_{mot}(f;s)$ is a geometrical invariant associated to a complex polynomial $f\in\mathbb{C}[x_1,\ldots,x_n]$ or germ map, introduced by Dener and Loeser in 1998 as a generalization of the topological zeta function $Z_{top}(f;s)$ by using Kontsevich's motivic integration theory. These zeta functions are related with the topology of $f^{-1}(0)$ via the Monodromy Conjecture, which affirms that any pole $s_0$ gives an eigenvalue of the monodromy on the cohomology of the Milnor fiber of $f^{-1}(0)$. They are classically computed in terms of an embedded resolution of singularities of $f^{-1}(0)$, where every exceptional divisor gives a ``pole candidate'' $s_0$ for $Z_{mot}(f;s)$ (or $Z_{top}(f;s)$), which could be not an actual pole when one gets the final expression. The conjecture is proved in some particular cases, but one of the main difficulties is to deal with resolution models with a lot of exceptional divisors, providing ``bad pole candidates''. In this work, we study the notion of motivic zeta functions in the context of $\mathbb{Q}$-divisors in a $\mathbb{Q}$-Gorenstein variety, where we obtain a useful version of the motivic change of variables formula depending on the relative canonical divisor. As an application, we obtain a closed formula for $Z_{mot}(f;s)$ in terms of weighted blow-ups and more generally the so-called \emph{embedded $\mathbf{Q}$-resolutions of singularities} of $f^{-1}(0)$, which are roughly embedded resolutions $\pi:X\to\mathbb{C}^n$ where the ambient space $X$ is allowed to contain abelian quotient singularities, providing a ``simpler'' model with less exceptional divisors and thus less ``bad pole candidates'' for $Z_{mot}(f;s)$.

We say that two germs of analytic sets $(M,0)$ and $(N,0)$ in $(\mathbb{C}^n,0)$ have the same embedded topological type or are topologically equivalent if there exists a germ of homeomorphism $\varphi : (\mathbb{C}^n,M,0) \rightarrow (\mathbb{C}^n,N,0)$. We consider a topologically trivial (flat) family of generically reduced curves $p : (X,0) \rightarrow (\mathbb{C},0)$ in $(\mathbb{C}^n \times \mathbb{C},0)$ with a section $\sigma: (\mathbb{C},0)\rightarrow (X,0)$ and fibers $(X_t,\sigma(t)):=p^{-1}(t)$. In this case, we know that the special curve $(X_0,\sigma(0))$ and the generic curve $(X_t,\sigma(t))$ are topologically equivalent, for all $t$. So we can ask in what conditions the tangent cone $C_{(X_t,\sigma(t))}$ of $(X_t,\sigma(t))$ does not change the topological type under topological trivial deformations of $(X_0,\sigma(0))$. In others words, we study the following question: if $p:(X,0)\rightarrow (\mathbb{C},0)$ is a topological trivial family of generically reduced curves, under what conditions the Zariski tangent cones $C_{(X_0,\sigma(0))}$ and $C_{(X_t,\sigma(t))}$ are homeomorphic? (Joint work with J. Snoussi and A. Giles Flores)

Let $f, g \in \Bbb C{x, y}$ be germs of functions defining plane curve singularities without common components in $(\Bbb C^2, 0)$ and let $\phi(x,y,z) = f(x,y) + zg(x,y)$. We give an explicit algorithm producing a plumbing graph for the boundary of the Milnor fiber of $\phi$ in terms of a common resolution for f and g.

Milnor fibers play a crucial role in the study of the topology of a singularity of surface. They correspond to the different possible smoothings of this singularity. A description of this fiber is known in some particular cases, but in general it is not, even for isolated singularities. However, the study of its boundary has been an active field of research in the last decades. In different settings, this boundary has been proven to be a graph manifold. (Mumford, 1961, for isolated singularities, Michel-Pichon, 2003, 2014, for a smoothing of a reduced surface with smooth total space, N\'emethi-Szilard, 2012, with the same hypothesis, Bobadilla-Menegon Neto, 2014, for a non-reduced surface and a total space with isolated singularity). I will explain how the constructive proof provided by N\'emethi and Szilard can be adapted to prove, constructively, the same result for a smoothing of a reduced surface with any total space. This allows the hope for a characterization of the manifolds bounding Milnor fibers of surface singularities. Furthermore, I provide a simple algorithm for computing the boundary of the Milnor fiber, in the case of a surface defined by a generic function on a toric germ.

In earlier work we used the double of an ideal to define in integral closure terms a notion of infinitesimal Lipschitz equisingularity for hypersurfaces. We describe different ways of extending this notion to general spaces, and show the most restrictive version is a generic condition. We then show that this condition is necessary for strong bi-Lipschitz equisingularity, a notion due to Fernandes and Ruas.  Hence our condition gives a necessary condition for a smooth subset of an analytic set to be a stratum in a Mostowski stratification of the set. 

We prove that multiplicity of complex analytic singularities of dimension $d$ is invariant under bi-Lipschitz homeomorphisms if, and only if, $d=1$ or $d=2$. This result was obtained in joint works with J. de Bobadilla, J. E. Sampaio. and L. Birbrair, J. E. Sampaio, M. Verbitsky.

I will describe work in progress on a homology theory for singularities that is able to capture outer metric phenomena. For example it is a complete invariant for the outer metric of complex plane curves. This is a joint work with S. Heinze, M. Pe pereira, and E. Sampaio.

Joint work with Guillaume Rond, based on the paper arXiv:1802.07083.

In this work in progress, jointly with A. Fernandes (UFC) and H. Soares (UFPi), I will discuss the problem of local (bi)-Lipschitz triviality of a complex polynomial mapping nearby a value, starting with the case of the functions.

It is proved by Pham and Teissier in [PT] (also Fernandes in [F]) that two irreducible complex plane curve singularities in $\mathbb{R}^4$ are outer bi-Lipschitz equivalent if and only if are topologically equivalent. This result was generalized by Pichon and Neumann in [NP]. The topological type, in this case, is equivalent to the knot type of your "link" (that is always an iterated knot). In this talk, we consider the similar approach for the general case, that is, singular real surfaces $X$ in $\mathbb{R}^4$ parametrized by polynomial map germs $f \colon (\mathbb{R}^2,0){\rightarrow }(\mathbb{R}^4,0)$ with isolated singularity. We show that, given $X{=}f(\mathbb{R}^2)$, the knot type of the link $X \cap \mathbb{S} ^3(0,\epsilon)$ determines completely the $C^0$-$\mathscr{A}$-class of $f$ and all parametrizations of this type are $C^0$-finitely determined. Moreover, we show that if $X$ is a bi-Lipschitz embedded parametrized surface, then $X$ is smooth. This is a joint work with Juan Jose Nuno Ballesteros. [BFLS] Birbrair L., Fernandes A., Lê, D. T., Sampaio J. E., {\it Lipschitz regular complex algebraic sets are smooth}, Proc. Amer. Math. Soc. 144 (2016), no. 3, 983--987. [F] Fernandes A., {\it Topological equivalence of complex curves and bi-Lipschitz homeomorphisms}, Michigan Math. J. 51 (2003), n. 3, 593--606. [NP] Neumann W. D., Pichon, A., {\it Lipschitz geometry of complex curves}, J. Singul. 10 (2014), 225--234. [PT] Teissier B., Pham, F, {\it Fractions lipschitziennes d'une alg\'ebre analytique complexe et saturation de Zariski}, Centre de Math\'ematiques de l'Ecole Polytechnique (Paris), June 1969.

Let R be a local ring, e.g. power series in several variables. Denote by Mat(m,n,R) the space of matrices with entries in R. Various groups act on this space. We study the corresponding Tjurina modules, the tangent spaces to the miniversal deformation. The first step is to check whether/ when these modules are finite dimensional. (This ensures the finite determinacy.) We compute/bound the support of these modules, achieving numerous geometric criteria of determinacy.

In this talk we will review the definition of local polar variety of a germ of singularity $(X,x)$ and discuss the fundamental role they play in describing the minimal Whitney stratification of the germ, and the set of limits of tangent spaces to $(X,x)$ as developed  mainly by Lê Dung Tràng and Bernard Teissier in the late 70's and 80's.

In a joint work with Claudio Murolo and Andrew du Plessis we proved the smooth Whitney fibering conjecture, in particular for every stratum X of a Whitney stratified set, locally near points of X the foliation defined by the Thom-Mather topological trivialization can be chosen, via suitable vector fields, so that the tangent spaces to the leaves are continuous at X. Moreover the associated wings have a similar property and are Whitney regular. As an application we describe a joint result with Claudio Murolo: every compact Whitney stratified set admits a Whitney cellulation, i.e. a cellulation such that the cells form a Whitney stratification. This resolves a homology problem of Mark Goresky.

In this talk I would like to present a "new class" of real singularities where one can associate a fibration structure as a generalisation of the called tube Milnor fibration. This is a joint work with Maico F. Ribeiro (UFES-Brazil) and M. Tibar (Université de Lille-France).

Any germ of a complex analytic space $(X,0) \subset (\mathbb R^n,0)$ is equipped with two metrics: the outer metric induced by the euclidian metric of the ambient space and the inner metric, which is the associated length metric on the germ. The two metrics are natural in the sense that up to local bilipschitz homeomorphisms, they do not depend on the choice of embedding in some $(\mathbb R^n,0)$. These two metrics are in general nonequivalent up to bilipschitz homeomorphism. We say that $(X,0)$ is Lipschitz normally embedded if it is the case. I will present a characterization of Lipschitz normally embedding among normal surface singularities. Joint work with Walter Neumann and Helge Pedersen.

I will present an infinite family of Lipschitz normally embedded singularities among superisolated hypersurface singularities in $(\Bbb C^3,0)$. The proof is based on the characterization of Lipschitz normal embedding presented in the previous talk. Joint work with Anne Pichon.

The 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. In this talk we prove Lipschitz normal embeddedness of some algebraic subsets of the space of matrices. These include the space of matrices, symmetric matrices and skew-symmetric matrices of rank equal to a given number and their closures, the upper triangular matrices with determinant $0$ and linear space transverse to the rank stratification away from the origin.