Analyse Complexe, Géométrie Complexe et Applications

11 déc. 2017, 08:30 → 12 déc. 2017, 18:00 Europe/Paris

Amphi Herpin & Amphi Astier (Université Pierre et Marie Curie Paris 6)

Analyse Complexe, Géométrie Complexe et Applications

Complex Analysis, Complex Geometry and Applications
Colloque en mémoire de G. Henkin

Organisateurs :

• T .-C. Dinh (National University of Singapour & Université Pierre et Marie Curie Paris 6) courriel: matdtc@nus.edu.sg
• A. Iordan (Université Pierre et Marie Curie Paris 6) courriel: andrei.iordan@imj-prg.fr

Conférenciers :

B. Berndtsson (Göteborg),  J. -M.Bismut (Orsay), M. Kontsevich (IHES), G. Marinescu (Köln), N. Mok (Hong Kong) , R. Novikov (École Polytechnique), D. H. Phong (Columbia) , N. Sibony (Orsay), Y-T. Siu (Harvard).

Affiche Affiche_Hommage_Henkin_V3.pdf

lundi 11 décembre

1 Acceuil des participants / Reception of participants

2 Ouverture / Opening

Orateur: Prof. Jean Chambaz (Président de l'Université Pierre et Marie Curie)

3 NEW CURVATURE FLOWS IN NON-KÄHLER GEOMETRY

Orateur: Prof. D. H. Phong (University of Columbia)

4 Pause café

5 Twistor families of categories

I will give a definition of a twistor family (Cζ), ζ belonging to the Riemann sphere, of triangulated categories. The propotypical example is the family of derived categories of coherent sheaves on compact hyperkähler manifold, endowed with complex structures parametrized by twistor parameter ζ. Another basic example comes from Simpson's non-abelian Hodge theory. In a joint work (in progress) with Y.Soibelman we propose a general approach to twistor families using Fukaya categories associated with holomorphic symplectic manifolds. The most clean case is the product of an elliptic curve and C*. For ζ≠0,∞ the corresponding category has a decription in terms of elliptic difference equations. Harmonic objects are solutions of Bogomolony equations on 3-dimensional torus with isolated singularities. The universal family of categories in this example is parametrized by the non-Hausdorff quotient (CP2 -RP2)/GL(3; Z).

Orateur: Prof. M. Kontsevich (IHES)

6 Geometric substructures, uniruled projective subvarieties, and applications to Kähler geometry

In a series of articles with Jun-Muk Hwang starting from the late 1990s, we introduced a geometric theory of uniruled projective manifolds based on the variety of minimal rational tangents (VMRT), i.e., the collection of tangents to minimal rational curves on a uniruled projective manifold (X;K) equipped with a minimal rational component. This theory provides differential-geometric tools for the study of uniruled projective manifolds, especially Fano manifolds of Picard number 1. Associated to (X;K) is the fibered space π:C(X)→X of VMRTs called the VMRT structure on (X;K). I will discuss germs of complex submanifolds S on (X;K) inheriting geometric substructures, to be called sub-VMRT structures, obtained from intersections of VMRTs with tangent subspaces, i.e., from ϖ : C(S) →S, C(S) := C(X) \ PT(S). Central to our study is the characterization of certain classical Fano manifolds of Picard number 1 or special uniruled projective subvarieties on them in terms of VMRTs and sub-VMRTs. As applications I will relate the theory to the existence and uniqueness of certain classes of holomorphic isometries into bounded symmetric domains. For uniqueness results parallel transport (holonomy), a notion of fundamental importance both in Kähler geometry and in the study of sub-VMRT structures, will play an important role.

Orateur: Prof. N. Mok (University of Hong Kong)

7 EQUIVARIANT KODAIRA EMBEDDING FOR CR MANIFOLDS WITH CIRCLE ACTION

Orateur: Prof. G. Marinescu (Universität zu Köln)

8 Pause café

9 Unique ergodicity for foliations

Consider the polynomial differential equation in C² dz/dt = P(z;w); dw/dt = Q(z;w): The polynomials P and Q are holomorphic, the time is complex. In order to study the global behavior of the solutions, it is convenient to consider the extension as a foliation in the projective plane P². I will discuss some recent results around the following questions. What is the ergodic theory of such systems? How do the leaves distribute in a generic case? What is the topology of generic leaves?

Orateur: Prof. N. Sibony (Université d'Orsay)

mardi 12 décembre

10 Hypoelliptic deformation, self-adjointness, and analytic torsion

The purpose of the talk is to explain the construction of non self-adjoint Hodge Laplacians, which naturally deform classical Hodge theory. If X is a compact Riemannian manifold, let X be the total space of its tangent bundle. The deformed Hodge Laplacian is constructed over X. It is a hypoelliptic operator on X, which is essentially the sum of a harmonic oscillator and of the generator of the geodesic flow. In the real case, the symplectic form of X is used in its construction. Applications to analytic torsion, real and holomorphic, will be given. Time permitting, connections with Selberg's trace formula will be explained.

Orateur: Prof. J.-M. Bismut (Université d'Orsay)

11 Pause café

12 Pluri-Hodge Decomposition and Associated Jacobian

Orateur: Prof. Y-T. Siu (Univesity of Harvard)

13 Multidimensional inverse scattering problem

We give a review of old and recent results on the multidimensional inverse scattering problem related with works of G.M. Henkin. This talk is based, in particular, on the following references: ▶ G.M. Henkin, R.G. Novikov, The dbar-equation in the multidimensional inverse scat- tering problem, Russ. Math. Surv. 42(3), 109-180, 1987; ▶ G.M. Henkin, N.N. Novikova, The reconstruction of the attracting potential in the Sturm-Liouville equation through characteristics of negative discrete spectrum, Stud. Appl. Math. 97, 17-52, 1996; ▶ R.G. Novikov, The dbar-approach to monochromatic inverse scattering in three dimen- sions, J. Geom. Anal. 18, 612-631, 2008; ▶ R.G. Novikov, Formulas for phase recovering from phaseless scattering data at fixed frequency, Bull. Sci. Math. 139, 923-936, 2015.

Orateur: Prof. R. Novikov (École Polytechnique)

14 Pause café

15 Remarks on superforms and supercurrents

This is basically a survey of Lagerberg's work on superforms and supercurrents, with some additions. We will illustrate the formalism with a proof of Weyl's tube formula and state a conjecture related to the Alexandrov-Fenchel inequality.

Orateur: Prof. B. Berndtsson (Göteborgs Universitet)