This is the second annual meeting of the ANR group Lipschitz Geometry of Singularities (LISA).
There will be several talks discussing open problems, questions, and techniques broadly related to the Lipschitz geometry of real and complex singular spaces.

The workshop will take place at the FRUMAM, in the Campus Saint-Charles of the Aix-Marseille University, from Monday, June 3rd to Thursday, June 6th, 2019.

**Organizing committee:**
Lorenzo Fantini, Anne Pichon.

- Javier Fernandez de Bobadilla
- Sonja Lea Heinze
- Maria Pe Pereira

- André Belotto da Silva
- Lorenzo Fantini
- Alexandre Fernandes
- Françoise Michel
- Hussein Mourtada
- Adam Parusinski
- Jose Edson Sampaio
- Bernard Teissier
- David Trotman

Monday | Tuesday | Wednesday | Thursday | |
---|---|---|---|---|

9:30-10:30 | HEINZE | BOBADILLA | FANTINI | |

10:30-11:00 | Coffee | Coffee | Coffee | |

11:00-12:00 | SAMPAIO | BELOTTO | TROTMAN | |

12:00-14:30 | Lunch | Lunch | Social Lunch | Lunch |

14:30-15:30 | TEISSIER | PARUSINSKI | MOURTADA | |

15:30-16:00 | Coffee | Coffee | Coffee | |

16:00-17:00 | PE | FERNANDES | MICHEL | |

17:00-18:00 | Meeting of the ANR members |

- Monday: Maria Pe Pereira (Universidad Complutense, Madrid)

*"Part I: Definition and first properties (relative homology, moderately discontinuous functoriality, homotopy invariance)"* - Tuesday: Sonja Lea Heinze (Instituto de Ciencias Matemáticas, Madrid)

*"Part II: the Mayer-Vietoris sequence, Excision, and the computation for plane curves"* - Wednesday: Javier Fernandez de Bobadilla (Basque Center for Applied Mathematics, Bilbao)

*"Part III: Computations in interesting cases and applications"*

Joint work by: J. Fernandez de Bobadilla, S. Heinze, M. Pe Pereira, E. Sampaio.

Moderately Discontinuous Homology is a new algebraic topology invariant specially suited to study the Lipschitz Geometry of subanalytic sets. It is a Homology Theory having values in diagrams of groups, each of them called the MD Homology group associated with a given "discontinuity parameter". It satisfyies adequate forms of all axioms and computational tools of an ordinary homology theory: relative homology sequence, the value at a "point", Lipschitz homotopy invariance, Mayer-Vietoris, excision and the spectral sequences associated with filtrations and Cech coverings. A distinguishing feature of the theory is the functoriality for maps allowing a controlled type of discontinuities in the Lipschitz sense.

Besides being quite computable, it is able to capture the outer metric Lipschitz phenomena, and can be used as an invariant obstructing Lipschitz normally embedded in certain cases. As an example we will fully compute the invariant for complex plane curve germs for the outer metric, and show, in the irreducible case that it recovers all Puiseux pairs (we also compute the the non irreducible case in terms of the Eggers tree).

In the case of arbitrary dimension germs with the outer metric it recovers the number of irreducible components of the tangent cone, the relative multiplicities, and that it is powerful enough to characterize the smooth germ.

Moderately Discontinuous Homology is a new algebraic topology invariant specially suited to study the Lipschitz Geometry of subanalytic sets. It is a Homology Theory having values in diagrams of groups, each of them called the MD Homology group associated with a given "discontinuity parameter". It satisfyies adequate forms of all axioms and computational tools of an ordinary homology theory: relative homology sequence, the value at a "point", Lipschitz homotopy invariance, Mayer-Vietoris, excision and the spectral sequences associated with filtrations and Cech coverings. A distinguishing feature of the theory is the functoriality for maps allowing a controlled type of discontinuities in the Lipschitz sense.

Besides being quite computable, it is able to capture the outer metric Lipschitz phenomena, and can be used as an invariant obstructing Lipschitz normally embedded in certain cases. As an example we will fully compute the invariant for complex plane curve germs for the outer metric, and show, in the irreducible case that it recovers all Puiseux pairs (we also compute the the non irreducible case in terms of the Eggers tree).

In the case of arbitrary dimension germs with the outer metric it recovers the number of irreducible components of the tangent cone, the relative multiplicities, and that it is powerful enough to characterize the smooth germ.

- André Belotto da Silva (Université Aix-Marseille)

*"Hsiang–Pati coordinates and reduction of singularities of the inner metric"*

Coming soon! -
Lorenzo Fantini (Université Aix-Marseille)

*"A valuative approach to the inner geometry of complex surfaces"*

I will discuss a new approach to the study of the inner geometry of complex surfaces, and in particular of the classical invariants called inner rates, based on the combinatorics of a space of valuations associated with the singularity. More precisely, I will describe completely the inner metric structure of a complex surface germ with isolated singularities by showing that its inner rates both determine and are determined by global geometric data: the topology of the germ, its generic hyperplane sections, and its generic polar curves. This is a joint work with André Belotto and Anne Pichon (arXiv:1905.01677). - Alexandre Fernandes (Universidade Federal do Ceará, Fortaleza)

*"Lipschitz contact equivalence of two variable function-germs"*

The aim of this talk is to delivery a complete discrete invariant that describes two real variable function-germs up to Lipschitz contact equivalence. Time permitting, I will mention what happens with two complex variable funtion-germs up to Lipschitz contact.equivalence. (joint works with Lev Bibrair, Andrei Gabrielov and Vincent Grandjean) - Françoise Michel (Université de Toulouse)

*"Jacobian curves for normal complex surfaces"*

Let \((X , p)\) be a germ of complex surface, let \( \psi : (X , p) \rightarrow ( \bf {C} ^2 , 0) \) be a finite holomorphic germ. The talk is devoted to the study the contact zone of the strict transform of the Jacobian curve \( \Gamma \) of \( \psi \) in the minimal good resolution of \(\psi\). We define "bunches" of branches of \(\Gamma\) which have the same contact and we evaluate the "multiplicity" of every bunch.

(The results presented in this talk are summarized in this short text, which also contains several references.) - Hussein Mourtada (Sorbonne Université, Paris)

*"The embedding dimension of the completion of the space of arcs at a point associated with a divisorial valuation"*

To a divisorial valuation centered on a variety X, one can associate a schematic point (equivalently a family of arcs) in the space of arcs L(X) of X.

Let us call such a point divisorial. We will explain a formula for the embedding dimension of the completion of L(X) at a divisorial point in terms of the Mather discrepancy.

All the terms in the abstract and the title will be introduced.

This is a joint work with Ana Reguera. - Adam Parusinski (Université Nice Sophia Antipolis)

*"Bi-Lipschitz Right Equivalence of Real Analytic Function Germs"*

It is known that bi-Lipschitz right equivalence of analytic function germs admits continuous moduli. In this talk we discuss bi-Lipschitz classification and bi-Lipschitz determinacy of real analytic two variable quasi-homogeneous function germs. - Jose Edson Sampaio (Basque Center for Applied Mathematics, Bilbao)

*"A version of the Mumford's Theorem on regularity of normal complex surfaces in high dimension"*

**(A pdf abstract including references and the author's affiliation can be downloaded here.)**

In 1961, D. Mumford proved that a normal complex analytic surface \(X\) with simply connected link at \(0\) need to be smooth at \(0\). In the case \(X\subset \mathbb C^3\) is a complex surface with an isolated singularity at \(0\), Mumford's result is equivalent to say that \(X\) is smooth at \(0\) if and only if \(X\) is a topological manifold at \(0\). However, this result does not hold true if \(\dim X>2\). In this talk, we prove a version of Mumford's result in high dimension. More precisely, if \(X\subset \mathbb C^n\) is a LNE complex analytic set, we prove that the following statements are equivalent:- \(X\) is a topological manifold at \(0\);
- \(X\) is smooth at \(0\).

- Bernard Teissier (Sorbonne Université, Paris)

*"Around the valuative Cohen theorem"*

The valuative Cohen theorem is a tool to provide geometric meaning to valuations in algebraic geometry. I shall explain its origin, statement, the main ideas of the proof, and applications.

The prerequisites are basic knowledge in commutative algebra, and a little toric geometry. - David Trotman (Université Aix-Marseille)

*"Whitney cellularisation and Goresky’s homology conjecture"*

(Joint work with Claudio Murolo) In 1981 Mark Goresky conjectured that the homology of a Whitney stratified set can be represented by Whitney stratified cycles. More precisely he defined a Whitney homology theory using Whitney stratified chains and conjectured the bijection of the resulting homology groups with those of the usual homology. He proved such a bijection for cohomology, and in the special case of a Whitney stratified manifold proved the bijection for homology. We prove Goresky’s conjecture by showing that every Whitney stratified set admits a refinement which is a Whitney regular cell decomposition. Our proof depends on results obtained in our recent proof of the smooth Whitney fibering conjecture (2016), in particular on a horizontally \(C^1\) improvement of the Thom-Mather isotopy theorem and the existence of a local Whitney regular wing structure in a neighbourhood of each stratum.

If you are interested in attending the workshop you can get in touch with Lorenzo Fantini and/or Anne Pichon.

The talks will take place on the Conference Room at the second floor of the FRUMAM, in the St. Charles campus, just north of the main station. Here's a maps of the campus. The entrance of the campus is in Place Victor Hugo, in front of the station's north exit (on the right if you are coming from the platforms), at GPS coordinates 43°18'16.7"N,5°22'42.6"E (the link will take you to Google maps); to enter tell the security guards there that you are coming for the LISA Workshop, they should have a list of all the participants. The entrance of the FRUMAM is on the map linked above, but it can be a bit hard to find, its coordinates are 43°18'21.1"N,5°22'48.2"E. Go up to the second floor and walk through the door on the right, the Conference Room is at the end of the corridor.

For your accommodation, here are a couple of nice and reasonably priced hotels close to the Vieux Port, about 10/15 minutes walking from the FRUMAM:

The participants are free to explore the city during the lunch breaks. Here are a couple of simple places within reasonable walking distance from the FRUMAM that we recommend:

- Café l'Écomotive (vegetarian, 2 Place des Marseillaises)
- Sur le Pouce (couscous, 2 Rue des Convalescents)
- Zafer (Turkish, 12 Rue de Jemmapes)

This workshop is founded by the following institutions: ANR LISA, FRUMAM, GDR Singularités et Applications, Institut de Mathématiques de Marseille, Labex Archimède.