Conference: "SINGULAR LANDSCAPES"

22 juin 2015, 09:30 → 26 juin 2015, 20:00 Europe/Paris

Centre Paul-Langevin AUSSOIS

Charles Favre, Evelia Garcia Barroso (Universidad de la Laguna), Marine AMIER, Patrick Popescu-Pampu (Laboratoire Paul Painlevé), Pedro Gonzalez Pérez (ICMAT , Departamento de Álgebra)

This international conference is aimed at exploring various aspects of the geometry of analytic singular spaces. We shall especially emphasize the delicate interplay between commutative algebra and topology, and try to present new point of views on these problems that arose recently (valuations, non-archimedean geometry). This will be the opportunity to celebrate the accomplishments of B. Teissier in which all these aspects were always delicately interwoven. We shall also dedicate several talks to interactions between singularity theory and neuroscience.

Registration are now closed. Please contact the organizers by email if you want to attend.

Poster Aff-_Singular_Landscapes4.pdf

Program Programme_-_Singular_LandscapesBD.pdf

lundi 22 juin ¶

Welcome¶

Morning session: GAFFNEY: Families of isolated singularities and three inspirations from Bernard Teissier¶

1 Families of isolated singularities and three inspirations from Bernard Teissier¶

Part of Bernard Teisier¹s work is a substantial contribution to equisingularity theory. In this talk I will discuss three of the many inspirations his work has given me, and their role in my current approach to the equisingularity of isolated singularities. The talk will use determinantal singularities as an illustration of these ideas.

Orateur: M. Terence Gaffney (Northeastern University)

Slides 2015-Gaffney-Ausois.pdf

11:00 Coffee break

Morning session: CUTKOSKY: Local monomialization of analytic maps¶

2 Local monomialization of analytic maps¶

We prove that germs of analytic maps of complex analytic varieties can be made monomial by sequences of local blow ups of nonsingular analytic subvarieties in the domain and target along an arbitrary étoile. An étoile and the voûte étoilée is a generalization by Hironaka of valuations and the Zariski Riemann manifold to analytic spaces.

Orateur: M. Steven Dale Cutkosky (University of Missouri)

Slides 2015-Cutkosky-Aussois.pdf

12:30 Lunch break

Afternoon session: HUH: Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs¶

3 Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs¶

I will give an overview of a proof of a conjecture of Read that the coefficients of the chromatic polynomial of any graph form a unimodal sequence. There are two main ingredients in the proof, both coming from works of Bernard Teissier: The first is the idealistic Bertini for sectional Milnor numbers, and the second is the isoperimetric inequality for mixed multiplicities of ideals.

Orateur: M. June Huh (Princeton)

Slides 2015-Huh-Aussois.pdf

Afternoon session: KHOVANSKII: Algebraic geometry, theory of singularities, and convex geometry¶

4 Algebraic geometry, theory of singularities, and convex geometry¶

I will review some results which relate these areas of mathematics. Newton polyhedra connect algebraic geometry and the theory of singularities to the geometry of convex polyhedra. This connection is useful in both direc- tions. On the one hand, explicit answers are given to problems of algebra and the theory of singularities in terms of the geometry of polyhedra. On the other hand, algebraic theorems of general character (like the Hirzebruch–Riemann–Roch theorem) give significant information about the geometry of polyhedra. In this way one obtains, for example, a multidimensional generalization of the classical one-dimensional Euler–Mclaurin formula. Combinatorics related to the Newton polyhedra theory allows to prove that in hyperbolic space of high dimension there do not exist discrete groups generated by reflections with fundamental polyhedron of finite volume (it was a longstanding conjecture). The theory of Newton–Okounkov bodies relates algebra, singularities and geom- etry outside the framework of toric geometry. This relationship is useful in many directions. For algebraic geometry it provides elementary proofs of intersection- theoretic analogues of the geometric Alexandrov–Fenchel inequalities and far-reach- ing generalizations of the Fujita approximation theorem. The local version of the theory provides a new proof of the famous Teissier’s inequalities for the multiplici- ties of primary ideals in a local ring. In geometry it suggests a transparent analog of Alexandrov–Fenchel inequality for coconvex bodies.

Orateur: M. Askold Khovanskii (University of Toronto)

Slides 2015-Khovanskii-Aussois.pdf

16:30 Coffee break

Specific talk: POPESCU-PAMPU: How Teissier mixed multiplicities¶

mardi 23 juin ¶

Morning session: YU: Enumeration of curves via non-archimedean geometry¶

10:30 Coffee break

Morning session: NICAISE: Refined curve counting and Hrushovski-Kazhdan motivic integration¶

12:30 Lunch break

Afternoon session: ULRICH : Rees algebras of codimension three Gorenstein ideals¶

Afternoon session: YIN: Non-Archimedean Lipschitz stratification¶

16:15 Coffee break

Afternoon session: PE PEREIRA: About the arc space of C2 and adjacencies of plane curves.¶

mercredi 24 juin ¶

Neuroscience and Singularity theory: LORENCEAU: Continuities, discontinuities and singularities in Visual Perception¶

10:30 Coffee break

Neuroscience and Singularity theory: PETITOT: Geometry of some functional architectures of vision¶

13:30 Free afternoon

jeudi 25 juin ¶

Morning session: DRAISMA: Noetherianity up to symmetry¶

10:30 Coffee break

Morning session: DIMCA: Free divisors and rational cuspidal curves in the plane¶

12:30 Lunch break

Afternoon session: MUSTATA: On some questions about valuations and minimal log discrepancies¶

Afternoon session: McLEAN: Minimal Log Discrepancy of Isolated Singularities and Reeb Orbits.¶

16:15 Coffee break

Specific talk: SPIVAKOVSKY: An overview of Teissier's works on valuations¶

vendredi 26 juin ¶

Morning session: KURDYKA: Convexifying positive polynomials and a proximity algorithm¶

10:00 Coffee break

Morning session: TEMKIN: Resolution by alterations¶

Morning session: COMTE: Sets with few rational points¶