# Lipschitz Geometry of Singularities (18w5011)

Arriving in Oaxaca, Mexico Sunday, October 21 and departing Friday October 26, 2018

## Organizers

Jawad Snoussi (Universidad Nacional Autonoma de México)

Walter Neumann (Columbia University)

Anne Pichon (Aix Marseille University)

## Objectives

Lipschitz geometry has potential promising applications in several different areas of singularity theory. The meeting will focus not only on the understanding of the Lipschitz geometry of singular spaces and maps but also on its interaction with other aspects of singularities: resolutions, abstract and embedded topology of hypersurfaces, equisingulity, deformations, smoothings, natural stratifications, arc and jet spaces, geodesics on singular spaces, curvature, etc. Part of the meeting will concentrate on the questions and methods which emerged recently in the complex setting, but it will leave a large place for questions arising in both complex and real settings, and it will promote exchanges between complex and real algebraic geometers and topologists.

There has been considerable recent progress in Lipschitz geometry of complex algebraic surfaces. In fact, despite much activity in the real semi-algebraic context, it was believed by some experts that complex germs are simply metrically conical (i.e., bi-Lipschitz equivalent to the metric cone on their links), so inner Lipschitz geometry would say nothing more than topology. But the first counter-example was found by Birbrair and Fernandes in 2006 in [BF}, and then Birbrair, Fernandes and Neumann discovered several interesting geometric phenomena, such as fast loops and separating sets \cite{BFN}, \cite{Fe2}, which soon made clear that metric conicalness is the exception rather than the rule. The subject then developed very rapidly, leading to the complete classification by Birbrair, Neumann and Pichon \cite{BNP} of the inner Lipschitz geometries of germs of normal complex surfaces, and building on it to several works on outer geometry of complex surfaces. In particular, using outer geometry Neumann and Pichon \cite{NP} proved the equivalence of Lipschitz equisingularity and Zariski equisingularity for germs of surfaces and that many analytic invariants such as multiplicity, topology of discriminants and polars, etc.\ are determined by outer geometry of surface germs. There has also been interesting work on ``normal embedding'' (bilipschitz equivalence of outer and inner geometry). For example in \cite{NPP] it is shown that minimal singularities are characterized among rational surface singularities by being normally embedded, a result which is a tantalizing step towards L\^e D\~ung Tr\'ang's suggested duality between resolution of surface singularities by repeated normalized blowing up and resolution by repeated normalized Nash transform.

There are some partial studies of Lipschitz geometry for families of singularities in higher dimensions, such as Birbrair, Fernandes, Gaffney and Grandjean [BFHS}, and other higher dimensional results were also obtained recently. For example, it was shown by Birbrair, Fernandes, L\^e and Sampaio \cite{BFLS} that Lipschitz regularity of complex analytic sets implies smoothness and Sampaio \cite{S] proved that bi-Lipschitz equivalent semialgebraic sets have bi-Lipschitz equivalent tangent cones (the topological equivalence of tangent cones was conjectured by Zariski but disproved by Bobadilla). Moreover, in a recent preprint of K.U.Katz, M.G. Katz, D.Kerner and Y. Liokumovich (arXiv:1602.01227) and more general results of Petersen and Ruas (arXiv:1607.07746), normal embedding is proved for various determinantal singularities. However, the general understanding of Lipschitz geometry in higher dimensions and for non isolated singularities is not yet well developed and will provide significant research projects well into the future.

An important long-standing question in singularity theory is Zariski's multiplicity conjecture (actually stated as a question), which states that the multiplicity of an algebraic hypersurface germ is determined by its embedded topological type. A Lipschitz version of Zariski multiplicity conjecture was proved by Fernandes and Sampaio [FS} for complex hypersurface germs (not necessarily with isolated singularities). The analogous result for normal surface germs (of any embedding dimension) was proved earlier in \cite{NP]. The higher dimensional case remains open, as well as the non-Lipschitz version in all generality, and is being very actively investigated.

There are also important new results in Lipschitz geometry of germs of functions. Birbrair, Fernandes, Gabrielov and Grandjean [BFGG} obtained finite Lipschitz classification of germs of definable functions on the real plane (Lipschitz K-equivalence). Notice that Lipschitz A-equivalence has so-called ``moduli'' and does not allow discrete classification (Henry and Parusinski \cite{HP]). In work in progress, Gaffney, Neumann, Pichon and Teissier extended the collection of Henry-Parusinski moduli for germs of functions $(\Bbb C^2,0)\to (\Bbb C,0)$ and are close to a full Lipschitz A-equivalence classification of such germs. Again, higher dimensions remain a challenge.

The topics planned to be covered by the meeting include:

There has been considerable recent progress in Lipschitz geometry of complex algebraic surfaces. In fact, despite much activity in the real semi-algebraic context, it was believed by some experts that complex germs are simply metrically conical (i.e., bi-Lipschitz equivalent to the metric cone on their links), so inner Lipschitz geometry would say nothing more than topology. But the first counter-example was found by Birbrair and Fernandes in 2006 in [BF}, and then Birbrair, Fernandes and Neumann discovered several interesting geometric phenomena, such as fast loops and separating sets \cite{BFN}, \cite{Fe2}, which soon made clear that metric conicalness is the exception rather than the rule. The subject then developed very rapidly, leading to the complete classification by Birbrair, Neumann and Pichon \cite{BNP} of the inner Lipschitz geometries of germs of normal complex surfaces, and building on it to several works on outer geometry of complex surfaces. In particular, using outer geometry Neumann and Pichon \cite{NP} proved the equivalence of Lipschitz equisingularity and Zariski equisingularity for germs of surfaces and that many analytic invariants such as multiplicity, topology of discriminants and polars, etc.\ are determined by outer geometry of surface germs. There has also been interesting work on ``normal embedding'' (bilipschitz equivalence of outer and inner geometry). For example in \cite{NPP] it is shown that minimal singularities are characterized among rational surface singularities by being normally embedded, a result which is a tantalizing step towards L\^e D\~ung Tr\'ang's suggested duality between resolution of surface singularities by repeated normalized blowing up and resolution by repeated normalized Nash transform.

There are some partial studies of Lipschitz geometry for families of singularities in higher dimensions, such as Birbrair, Fernandes, Gaffney and Grandjean [BFHS}, and other higher dimensional results were also obtained recently. For example, it was shown by Birbrair, Fernandes, L\^e and Sampaio \cite{BFLS} that Lipschitz regularity of complex analytic sets implies smoothness and Sampaio \cite{S] proved that bi-Lipschitz equivalent semialgebraic sets have bi-Lipschitz equivalent tangent cones (the topological equivalence of tangent cones was conjectured by Zariski but disproved by Bobadilla). Moreover, in a recent preprint of K.U.Katz, M.G. Katz, D.Kerner and Y. Liokumovich (arXiv:1602.01227) and more general results of Petersen and Ruas (arXiv:1607.07746), normal embedding is proved for various determinantal singularities. However, the general understanding of Lipschitz geometry in higher dimensions and for non isolated singularities is not yet well developed and will provide significant research projects well into the future.

An important long-standing question in singularity theory is Zariski's multiplicity conjecture (actually stated as a question), which states that the multiplicity of an algebraic hypersurface germ is determined by its embedded topological type. A Lipschitz version of Zariski multiplicity conjecture was proved by Fernandes and Sampaio [FS} for complex hypersurface germs (not necessarily with isolated singularities). The analogous result for normal surface germs (of any embedding dimension) was proved earlier in \cite{NP]. The higher dimensional case remains open, as well as the non-Lipschitz version in all generality, and is being very actively investigated.

There are also important new results in Lipschitz geometry of germs of functions. Birbrair, Fernandes, Gabrielov and Grandjean [BFGG} obtained finite Lipschitz classification of germs of definable functions on the real plane (Lipschitz K-equivalence). Notice that Lipschitz A-equivalence has so-called ``moduli'' and does not allow discrete classification (Henry and Parusinski \cite{HP]). In work in progress, Gaffney, Neumann, Pichon and Teissier extended the collection of Henry-Parusinski moduli for germs of functions $(\Bbb C^2,0)\to (\Bbb C,0)$ and are close to a full Lipschitz A-equivalence classification of such germs. Again, higher dimensions remain a challenge.

The topics planned to be covered by the meeting include:

- Lipschitz Geometry of germs complex surfaces. This will contain some introductory lectures for students and post-docs.
- Zariski Multiplicity Conjecture. This will include an elementary talk on regularity and invariance of tangent cones, and a review of recent results in this direction.
- Lipschitz geometry and resolution of singularities
- Equisingularity
- Lipschitz geometry of Determinantal manifolds
- Lipschitz geometry and arc spaces and jet schemes

- 2 or 3 mini-courses, each of three hours, for students, post-docs and researchers starting this area;
- 12 to 15 talks of one hour each.

#### Bibliography

\bibitem{BL} Andreas Bernig and Alexander Lytchak, Tangent spaces and Gromov-Hausdorff limits of subanalytic spaces, J. Reine Angew. Math. { 608} (2007), 1--15. \bibitem{B} Lev Birbrair, Local bi-Lipschitz classification of two-dimensional semialgebraic sets. Houston Journal of Mathematics, v. 25, n.3, p.453-472, 1999 \bibitem{BF} Lev Birbrair, Alexandre Fernandes, Inner metric geometry of complex algebraic surfaces with isolated singularities. Communications on Pure and Applied Mathematics, v. 61, p.1483-1494, 2008. \bibitem{BFGG} Lev Birbrair, Alexandre Fernandes, Andrei Gabrielov, Vincent Grandjean, Lipschitz contact equivalence of function germs in R2 (2014). arXiv:1406.2559 - to appear in Annali della Scuola Normale Superiore di Pisa. \bibitem{BFGaffG} Lev Birbrair, Alexandre Fernandes, Terry Gaffney, Vincent Grandjean, Blow-analytic equivalence versus contact bi-Lipschitz equivalence, arXiv:1601.06056. \bibitem{BFHS} Lev Birbrair, Alexandre Fernandes, Vincent Grandjean, Donal O'Shea, Choking horns in Lipschitz geometry of complex algebraic varieties. Appendix by Walter D. Neumann. Journal of Geometric Analysis, v. 24, p.1971-1981, 2013. \bibitem{BFLS} Lev Birbrair, Alexandre Fernandes,; L\^e, Dung Trang, Edson Sampaio, Lipschitz regular complex algebraic sets are smooth. Proceedings of the AMS, 2015. \bibitem{BFN} Lev Birbrair, Alexandre Fernandes, Walter D Neumann, Separating sets, metric tangent cone and applications for complex algebraic germs. Selecta Mathematica, New Series, v. 16, p.377-391, 2010. \bibitem{BNP} Lev Birbrair, Walter D Neumann, Anne Pichon, The thick-thin decomposition and the bilipschitz classification of normal surface singularities, Acta Math. (2014) 212, 199-256. \bibitem{Fe1} Alexandre Fernandes, Topological equivalence of complex curves and bi-Lipschitz maps, Michigan Math. J. { 51} (2003), 593--606. \bibitem{Fe2} Alexandre Fernandes, Separating sets on semi-weighted homogeneous hypersurface singularities. Results in Math. vol 60 (2011), n. 1--4, 361--367. \bibitem{FS} Alexandre Fernandes, Edson J. Sampaio, Multiplicity of analytic hypersurface singularities under bi-Lipschitz homeomorphisms (2015). arXiv:1508.06250. \bibitem{FR} Alexandre Fernandes, Maria Aparecida Ruas, Rigidity of bi-Lipschitz equivalence of weighted homogeneous function-germs in the plane, Proc. Amer. Math. Soc. 141 (2013), 1125-1133. \bibitem{GG} Vincent Grandjean, Daniel Grieser, The exponential map at a cuspidal singularity. Journal fur die Reine und Angewandte Mathematik, 2015. \bibitem{G} Daniel Grieser, Local geometry of singular real analytic surfaces. Transactions of the AMS, v.355 (2003), p.1559-1577. \bibitem{HP} Jean-Pierre Henry, Adam Parunsinski, Existence of moduli for bi-lipschitz equivalence of analytic functions, Compositio Mathematica 136; 217-235, 2003. \bibitem{M} Tadeusz Mostowski, Lipschitz equisingularity, Dissertationes Math. (Rozprawy Mat.) {\bf243} (1985), 46pp. \bibitem{M1} Tadeusz Mostowski, Tangent cones and Lipschitz stratifications.

*Singularities (Warsaw, 1985)}, Banach Center Publ., {20*(Warsaw, 1988), 303--322 \bibitem{NPP} Walter D Neumann, Helge M\o ller Pedersen and Anne Pichon, Minimal surface singularities are Lipschitz normally embedded (2015), 24 pages, arXiv:1503.03301. \bibitem{NP} Walter D Neumann and Anne Pichon, Lipschitz geometry of complex surfaces: analytic invariants and equisingularity, arXiv:1211.4897v1. \bibitem{NP2} Walter D Neumann and Anne Pichon, Lipschitz geometry of complex curves, Journal of Singularities {10} (2014), 225--234. \bibitem{parusinski1} Adam Parusi\'nski, Lipschitz properties of semi-analytic sets, Universit\'e de Grenoble, Annales de l'Institut Fourier, { 38} (1988) 189--213. \bibitem{parusinski} Adam Parusi\'nski, Lipschitz stratification of subanalytic sets, Ann. Sci. Ec. Norm. Sup. (4) { 27} (1994), 661--696. \bibitem{PT} Fr\'ed\'eric Pham and Bernard Teissier, Fractions Lipschitziennes d'une algebre analytique complexe et saturation de Zariski. Pr\'epublications Ecole Polytechnique No. M17.0669 (1969). Available at http://hal.archives-ouvertes.fr/hal-00384928/fr/ \bibitem{S} Edson J. Sampaio, Bi-Lipschitz homeomorphic subanalytic sets have bi-Lipschitz homeomorphic tangent cones. Selecta Mathematica, v. Online, p. 1-7, 2015 \bibitem{SS} Laurent Siebenmann, Dennis Sullivan, On complexes that are Lipschitz manifolds, Proceedings of the Georgia Topology Conference, Athens, Ga., 1977 (Academic Press, New York- London, 1979), 503--525. \bibitem{V} Guillaume Valette, The link of the germ of a semi-algebraic metric space. Proc. Amer. Math. Soc. {135} (2007), 3083--3090.