Abstracts

Javier Fernandez de Bobadilla: An asymptotic Lefschetz Fixed Point Theorem.

Using the recently developed theory of Moderately Discontinuous Homology the following asymptotic fixed point theorem can be shown:

Let X_t be a subanalytic family of bounded subanalytic sets t\in (0,\delta). Let \phi_t:X_t\ to X_t be a subanalytic family of bi-Lipschitz maps with uniform Lipschitz constant, t\in (0,\delta). The b-Moderately Discontinuous Lefschetz number can be defined, and when it is different from 0 there exists the existence of a family of points x_t \in X_t such that \lim_{t\to0} d_t(x_t,\phi_t(x_t)(t^b=0. This should be regarded an asymptotic fixed point at speed b.

It is worth to mention that asymptotic fixed points need to exist even in situations in which each of the mappings $\phi_t$ does not have fixed points, like in the monodromy of Brieskorn-Pham hypersurface singularities.

This is a joint work with Maria Pe Pereira.

Maria Pe Pereira: Moderately Discontinuous Algebraic Topology

In the works [1] and [2] we develope a new metric algebraic topology,

called the Moderately Discontinuous Homology and Homotopy, in the

context of subanalytic germs in R^n (with a supplementary metric

structure) and more generally of (degenerating) subanalytic families.

This theory captures bilipschitz information, or in other words, quasi

isometric invariants, and aims to codify, in an algebraic way, part of

the bilipschitz geometry.

A subanalytic germ is topologically a cone over its link and the

moderately discontinuous theory captures the different speeds, with

respect to the distance to the origin, in which the topology of the

link collapses towards the origin. Similarly, in a degenerating

subanalytic family, it captures the different speeds of collapsing

with respect to the family parameter.

The MD algebraic topology satisfies all the analogues of the usual

theorems in Algebraic Topology: long exact sequences for the relative

case, Mayer Vietoris and Seifert van Kampen for special coverings...

In this talk, I will present the most important concepts in the theory

and some results or applications that we got until the present.

[1] (with J. Fernández de Bobadilla, S. Heinze, E. Sampaio)

Moderately discontinuous homology.

To appear in Comm. Pure App. Math.. Available in arXiv: 1910.12552

[2] (with J. Fernández de Bobadilla, S. Heinze) Moderately

discontinuous homotopy.

Submitted. Available in ArXiv:2007.01538

Metric geometry of complex surfaces, Lipschitz Normal Embeddings, and polar explorations

Lipschitz geometry is a branch of singularity theory that studies a complex analytic germ (X,0) \subset (C^n,0) by equipping it with either one of two metrics: its outer metric, induced by the euclidean metric of the ambient space, and its inner metric, given by measuring the length of arcs on (X,0).

Whenever those two metrics are equivalent up to a bi-Lipschitz homeomorphism, the germ is said to be Lipschitz normally embedded (LNE).

We will discuss a formula describing the inner metric structure of a normal complex surface germ, as well as several geometric properties of LNE surface germs, criteria to prove that a surface germ is LNE, and an application of this theory to the so-called problem of polar exploration, that is the quest to determine the combinatorics of the generic polar curve of a complex surface from its topology.

This minicourse will consist of three parts:

Part 1: Inner geometry and the Laplacian formula (by Lorenzo Fantini)

Part 2: Lipschitz Normally Embedded singularities (by Anne Pichon)

Part 3: Polar explorations (by André Belotto da Silva)

The course will discuss results from several papers of some subsets of the three authors, both published and unpublished, some of which are written jointly with other people including András Némethi, Walter Neumann, Helge Pedersen, and Bernd Schober.

Brian Hepler: Enhanced ind-sheaves and vanishing cycles at a fixed angle

We give an alternative construction of the enhanced nearby cycles and vanishing cycles functors recently introduced by D’Agnolo-Kashiwara in 2020 for the category of enhanced ind-sheaves in dimension one. Our approach is to extend the classical construction of nearby and vanishing cycles functors at a fixed angle to the framework of enhanced ind-sheaves on bordered subanalytic spaces, which has the benefit of simplifying many computations previously involving enhancements of the specialization and microlocalization functors, as well as allowing us to prove the existence of an enhanced version of the natural distinguished triangle relating the nearby and vanishing cycles. Time permitting, we will also describe this picture from the perspective of Stokes-filtered local systems, and show strictness of the canonical morphism.

Nicolas Dutertre: Principal kinematic formulas for germs of closed definable sets

We prove two principal kinematic formulas for germs of closed

definable sets in R^n, that generalize the Cauchy-Crofton

formula for the density due to Comte and the infinitesimal linear

kinematic formula due to the author. In this setting, we do not

integrate on the space of euclidian motions SO(n) \ltimes

R^n, but on the manifold SO(n) \times S^{n-1}.

Terence Gaffney: Infinitesimal Lipschitz Equisingularity: Genericity and Necessity

Joint with Thiago Filipe da Silva

In earlier work Gaffney used the Jacobian module and the theory of Integral closure of modules to construct an infinitesimal theory of Whitney equisingularity. Here we use the double of a the Jacobian module to define in integral closure terms a notion of infinitesimal Lipschitz equisingularity for analytic sets, and show that resulting condition is generic.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.

Adam Parusiński: Lipschitz stratification for real singularities

In this talk we discuss the classical constructions of Lipschitz stratification in the sense of Mostowski for real singularities. The original construction for subanalytic sets, as well as a later version for the sets definable in polynomially bounded o-minimal structures given by Nguyen and Valette, is based on the preparation theorem and the L-regular decomposition. Another proof for power bounded o-minimal structures due to Halupczok and Yin, that uses non-Archimedean field approach, is in essence fairly similar. Both tools, the preparation theorem and the L-regular decomposition, provide by themselves non-trivial information on the metric geometry of singular spaces.

Thierry De Pauw: Linear isoperimetric inequality for normal and integral currents in compact subanalytic sets

The isoperimetric inequality for a smooth compact Riemannian manifold $A$ provides a positive ${\bf c}(A)$, so that for any $(k+1)$-dimensional integral current $S_0$ in $A$ there exists an integral current $ S$ in $A$ with $\partial S=\partial S_0$ and $\mathbf{M}(S)\leq {\bc}(A)\mathbf{M}(\partial S)^{(k+1)/k}$. Although such an inequality still holds for any compact Lipschitz neighborhood retract $A$, it may fail in case $A$ contains a single polynomial singularity. Here, replacing $(k+1)/k$ by $1$, we find that a linear inequality $\mathbf{M}(S)\leq {\bc}(A)\mathbf{M}(\partial S)$ is valid for any compact algebraic, semi-algebraic, or even subanalytic set $A$. In such a set, this linear inequality holds not only for integral currents, which have $\mathbb{Z}$ coefficients, but also for normal currents having $\mathbb{R}$ coefficients and generally for normal flat chains with coefficients in any complete normed abelian group. A relative version for a subanalytic pair $B\subset A$ is also true, and there are applications to variational and metric properties of subanalytic sets.

Maria Ruas: Density of Lipschitz stable mappings

The main open question on density of stable mappings is to determine

the pairs $(n,p)$ for which Lipschitz stable mappings are dense. We discuss recent results

by Nguyen, Ruas and Trivedi on this subject, formulating conjectures for the density of Lipschitz stable

mappings in the boundary of the nice dimensions.

Antoni Rangachev: Algebraic theory of continuity for meromorphic functions on a complex analytic variety.

In the first lecture I will describe the algebra of Holder continuous meromorphic functions on a complex analytic variety

using rational powers of ideals and the work of Pham and Teissier on the Lipschitz saturation. In the second lecture

I will discuss how the Lipschitz saturation can be used to establish the bi-Lipschitz equivalence of quasi-ordinary singularities having the same characteristic exponents.

Thiago Filipe da Silva: The projective analytic spectrum of the double of a module.

Gaffney first introduced the double of an ideal to investigate bi-Lipschitz equisingularity of complex analytic hypersurfaces, using Lipschitz saturation of ideals and a notion of infinitesimal Lipschitz equisingularity (iL). Its genericity motivates to look for some relation with strongly bi-Lipschitz equisingularity (SL).

After we generalize the results about iL condition for modules and complex analytic varieties, and we prove that SL implies iL.

In this talk we show some preliminary results on the study about the fibers of Projan of the Rees algebra of the double of the jacobian module associated to a complex analytic variety and how it is related to Lipschitz problems.