Analytic Geometry

Real Analytic Geometry

Complex Geometry

Ergodic Geometry

Topology in low dimensions

Olivier Le Gal generalizes in his thesis, advised by Jean-Marie Lion, the explicit version of Gabrielov's Complement Theorem to certain families of smooth functions.

Jean-Marie Lion is studying analytic foliationswith methods from real analytic geometry. He proves that without spiraling phenomenon (kind of wandering assumption) leaves of codimension one foliations have similar properties with semi-analytic subsets. In particular, he just proved in collaboration with Patrick Speissegger the completion-model for the family of projections of nested sub-pfaffian subsets answering positively by the way to a question of Khovanskii from the 80's.

Rémy Soufflet (previous student of J.-M. Lion) proved, under some diophantine conditions, non oscillation properties for generalised abelian integrals.

In a paper published in Mémoires de la S.M.F., Dominique Cerveau and Julie Déserti study foliations associated to group actions on projective spaces. J. V. Pereira and F. Cuckierman recently deduced from this work new linerly stable foliations on those spaces.

In her thesis, co-advised by D. Cerveau and S. Cantat, J. Déserti describes automorphisms of some transformations groups, like the group of Cremona. She also studies representations of arithmetic groups in the group of Cremona. Technics involved are classical algebraic geometry as well as dynamics of rational mappings (works of Cantat, Diller,Favre,...).

In his thesis advised by D. Cerveau, Javier Ribon classified germs of holomorphic diffeomorphisms of the complex plane preserving the radial fibration. Among other tools, Écalle and Martinet-Ramis theories in dimension one are used. Having now a position at IMPA (Rio), he works in collaboration with D. Cerveau and D. Garba-Belko on a complex approach of some real foliations. C.R.-geometry plays a fundamental role in understanding normalization phenomena.

In his thesis advised by Frank Loray, Loic Jean-Dit-Teyssier provided the complete analytic classification of holomorphic vector fields at the neighborhood of a saddle-node type singular point. Infinitely many invariants are necessary beyond Martinet-Ramis' ones for the subjacent foliation.

D. Cerveau classifies, in collaboration with P. Sad, generic singularities of real foliations whose leaves are Levi-flat. He keep on working on the description of varieties of foliations on a given algebraic manifold, partially with A. Lins Neto. Recently, he obtained with A. Lins Neto a singular Frobenius Theorem "à la Malgrange" on singular varieties.

During his CNRS postdoc position in our group, Jorge Vitório Pereira uses flat divisors and asociated foliations to study the geometry of complex complact manifolds, in particular the existence of fibrations. He also introduced with F. Loray the notion of minimal model for a transversely projective foliations and prove the existence and unicity of such a model when the ambient manifold is a projective surface.

Frédéric Touzet is studying complex compact manifolds whose tangent bundle splits. For Kähler manifolds, he proves with M. Brunella and J. V. Pereira that total splitting of the tangent bundle implies that the universal cover is a product of curves.

D. Cerveau, A. Lins Neto, F. Loray, J. V. Pereira and F. Touzet studied codimension one singular foliations on compact complex manifolds. They prove among other results that, under some assumptions, such a foliation is transversely projective provided that it is not the pull-back of a foliation on the algebraic reduction of the manifold.

In two recent papers, Frank Loray constructs normal forms for singularities of analytic foliations by means of complex geometry (glueing foliated manifolds and uniformizing). In some cases, he obtains (infinite dimensional) versal deformations. He also wrote a book on the pseudo-group of a holomorphic foliation at the neighborhood of a singular point. He is now working on isomonodromic deformations of connexions on curves.

In his thesis, Luc Pirio studied planar webs, that is configuration of foliations in general position. His principal interest was the determination of exceptional webs. He now works on several directions: functional equations (in particular polylogarithms), differential equations, differential systems, projective geometry, differential invariants, geometric structures and holomorphic dynamics.

L. Pirio now works with J.-M. Trépreau on algebrization of higher dimensional webs and with D. Marín et J. V. Pereira on the study of planar and exceptional webs.

In collaboration with A. Verjovsky (Mexico), Laurent Meersseman constructed and studied a foliation of the 5-dimensional sphere by complex surfaces, i.e. a codimension one CR Levi-flat structure. Such a structure may be viewed as an analogous of a complex structure on an odd-dimensional manifold.

With Frédéric Bosio (Poitiers), L. Meersseman describes the topology of a wide class of non Käler complex compact manifolds. They are diffeomorphic to moment-angle manifolds of Buchstaber and Panov, and this work has consequences in algebraic topology.

In his thesis, advised by L. Meersseman, Guillaume Deschamps constructed non standart complex structures on real 6-manifolds by means of twistor space theory. He also precised the classification of parallelizable (as a real manifold) and spin complex surfaces.

Maximilien Bauer characterized the unique ergodicity of an interval exchange transformation in terms of Jacobi-Perron algorithm, and by means of properties of multidimensional continued fractions. With A. Lopez, he studied hyperbolic billiards associated to even continued fractions with an asymptotic estimate of the correlation of the first return map.

Corentin Boissy his in thesis, advised by A. Zorich. Studying configurations of rigid families of saddle connexions on half-translation surfaces, he describes the principal boundary of some stratas of the moduli space of quadratic differentials (genus 0, hyperelliptic and exceptional stratas).

In his thesis, advised by F. Dal'Bo, Damien Ferté studied the topological closure of an homogeneous group action. He characterised denses and closed orbits under certain actions.

Françoise Dal'Bo is working on the action of discrete isometry group G on a simply connected and negatively curved Riemannian manifold X. The main motivation is to relate some properties of this action with dynamics of geodesic flow or horospheric foliation on the unitary bundle of the quotient manifold G/X. Recently, F. Dal'Bo wrote a long introductory survey to the subject. Many results obtained in the area use asymptotic properties of the growth function of G and its Poincaré series. In a work in collaboration with M. Peigné, J.-C. Picaud and A. Sambusetti, F. Dal'Bo constructed an unexpected example of a non uniform lattice whose Poincaré series has critical exponent strictly less to the volumic entropy of X. Now, in the context of quotient groups, and thus of non simply connected manifolds, F. Dal'Bo and A. Sambusetti obtained results on the growth of G/N, where N is a normal subgroup of G.

In his PhD thesis, advised A. Zorich, Samuel Lelièvre studied "square-tiled surfaces" which, being the integer points of moduli spaces of abelian differentials, enable to tackle combinatorially the geometry and dynamics of these spaces. In joint work with P. Hubert, he studied Teichmüller discs of square-tiled surfaces of genus two with a cone-type singularity of angle 6 pi (abelian differentials with a double zero), as well as their stabilisers for the action of SL(2,R). He also studied the quadratic growth rates of cylinders of cylinders of closed geodesics for the flat metric on these surfaces.

Anton Zorich relates interval exchange transformations (ergodic point of view) with dynamics of Teichmüller geodesic flow. Improving renormalization procedure (Rauzy induction) and Veech approach, he obtains quantitative informations on the deviation of ergodic means for interval exchange transformations. In collaboration with J. Athreya and A. Eskin, he is studying billiards in "rectangular" polygons.

In collaboration with M. Kontsevich, A. Zorich described connected components of stratas in the moduli space of abelian quadratic differentials. They compute Lyapunov exponents of Teichmüller geodesic flow as well as volume of stratas.

In collaboration with A. Eskin and H. Masur, A. Zorich studies geometry and topology of moduli spaces: computation of volume, description of the main boundary, enumeration of closed geodesics on flat surfaces, computation of Siegel-Veech constants.

Let M be an irreducible compact closed manifold of dimension 3 with infinite fundamental group. If the boundary of M is a torus, there are finitely many closed surfaces properly embedded in M (Hatcher). Mark Baker proved that there can be infinitely many imerged closed surfaces with embedded boundary; in particular, he proved that this so occurs for the complement of the eight knot.

M. Baker proved Waldhausen conjecture for threefolds obtained by Dehn filling of torii bundles over the circle. Now, he is working on analogous problems for fibre bundles over the circle when fibres are higher genus surfaces. This approach is motivated by the fact that any orientable closed threefold can be obtained by this way.

In his thesis, advised by M. Baker, Franck Harou studied 3-dimensional manifolds from the point of view of Heegard splitting. He characterizes homeomorphisms used in Heegard splitting for which $M$ admits an hyperbolic structure.

M. Baker studied the action of arithmetic groups on the hyperbolic 3-space; one of the goal is to determine under which condition the quotient space is the complement of a link.

Bertold Wiest works in geometric group theory, especially in braid groups and their generalizations: Artin and Garside groups from the algebraic point of view and homotopy groups (mapping class groups) from the geometric point of view. He works on the geometry of spaces on which these groups act: Cayley graph, train tracks, Teichmüller spaces, complexes of curves, etc... B. Wiest also works on algorithmic questions (word problem, conjugacy classes,...) for such classes of groups.