• Since July 2016

Coming soon.

• Period 2011-2016

The work of H. Bommier (with H. Youssfi) focus on operators between Banach spaces of analytic functions, and the interaction between complex analysis and functional analysis. In particular, they studied the link between the spectral properties of an operator, the behavior of its symbol at the edge or at infinity, and the geometric properties of the domain under consideration. A. Borichev was interested in random integer functions. He obtained results on the distributions of the zeros of functions of the type ∑n≥0 ζ (n)zn /n!, where the sequence ζ = (ζ (n))n is random (stationary) or pseudo-random (ζ (n) = exp[2π iP(n)]P a polynomial with irrational coefficients). A. Borichev was also interested in Riesz bases of reproducing kernels in spaces of holomorphic functions. He showed that any rotation-invariant Branges space is equivalent to a ? 2 -φ(|z|) ? generalized radial Fock space Fφ = f ∈ Hol(C) : ? |f(z)| e dm(z) < ∞ , and therefore admits a C such base.A. Borichev also studied the following question. Let ψ be a regular even weight on [-1, 1], ψ ̸≍ 1. Is it true that L2 ([-1, 1], ψ ) has no Riesz basis of exponentials? His recent results indicate that this is the case when ψ (x) ≍ (1 – x2 )a with a > 0. Using the Fourier transform, this question translates into terms of Riesz bases of reproducing kernels in non-radial Fock spaces.
S. Charpentier’s work focused mainly on functional analysis and complex one-variable analysis. He was interested in universal series, integer series whose partial sums enjoy extreme approximation properties outside their disk of convergence.
In operator theory, universality is very wide-ranging, and comes under the notion of hypercyclicity, which must be clearly understood. S. Charpentier has obtained characterizations that generalize recent results.
H.’s research Daudé are at the interface of Mathematics, Computer Science and Physics. It develops techniques from random graph theory and percolation to describe the phase transitions associated with constraint satisfaction problems. Analytical combinatorics, via complex analysis, enabled H. Daudé to obtain a very precise description of certain transitions, such as those associated with certain linear systems over finite fields.
S. Rigat studied elliptic PDEs using techniques from complex analysis. With S. Chaabi, he studied axially symmetric potentials and obtained fundamental solutions of the operators ∆ + m∂x in the form of elliptic integrals. They obtained a decomposition theorem in ring-like domains. This enabled them to build a Riesz base of solutions in certain ring domains. With F. Wielonsky, they applied Fokas’ methods to obtain integral solutions of elliptic partial differential equations by solving Riemann-Hilbert problems on Riemann surfaces. Taking these techniques a step further, they studied the Dirichlet-Neumann application associated with certain elliptic operators.
C. Samuel’s recent work concerns the classification of spaces of continuous functions on a countable compact metric space. In particular, do isomorphisms of compact or nuclear operator spaces with values in these spaces imply isomorphisms between these same spaces? These questions lead to the study of injective and projective tensor products, one factor of which is a space of continuous functions C(α) on an interval ⟨1,α⟩. The latest results obtained on the subprojectivity of C(α)⊗π C(α) spaces lead us to study the relationships between these projective tensor products and C(β ) spaces.
F. Wielonsky’s research focuses on complex analysis, potential theory and approximation theory. In addition to the work already mentioned, he has obtained results of great deviation in the independent framework of potential theory, relating to sets of measurements derived from the theory of random matrices.
The problems studied by H. Youssfi stem from the theory of spaces of holomophic functions and their operators, as well as from the harmonic analysis associated with Dunkl operators. He was particularly interested in calculating and estimating Bergman kernels. He has given applications to the spectral theory of Hankel and Toeplitz operators.
He then studied the spectral properties of composition operators by means of the size of the level en- sembles associated with their symbol. In connection with the ∂ ̄ problem, he obtained an analytic extension of Toeplitz operators and studied Dunkl’s interleaving operator.
V. Zagrebnov obtained important results in semi-group theory and spectral theory: generalization of the von Neumann – Van Daele – Schmüdgen theorem, realization of unitary group approximations by the Trotter – Kato product formula. He established the Robinson-Link bound for irreversible quantum evolution. The Cayley transformation enabled him to construct out-of-equilibrium quantum states and analyze quantum transport. He solved inverse problems using the Dirichlet-Neumann operator for transporting the Laplacian.
In statistical quantum mechanics, V. Zagrebnov proposed stochastic bosonic point processes to establish anisotropic Bose-Einstein condensation. Finally, in quantum dynamics, he constructed the dynamical group for an open system and unbounded perturbations, as well as the W∗-dynamics of an infinite system.
R. Zarouf has worked on classical inequalities for rational functions. He simplified and generalized the proofs of Peller inequalities by dispensing with Hankel operators. He has also worked in Matrix Analysis on problems of conditioning and estimating the resolvent of a matrix. In particular, he answered a conjecture by N. Nikolski by combining the latter’s ideas with ancient techniques of matrix analysis (Egervary, 1928).

PR : A. Borichev, B. Coupet, C. Samuel (emeritus), H. Youssfi, V. Zagrebnov (emeritus)
MCF : S. Charpentier, H. Daudé, S. Rigat, F. Wielonsky (HDR), R. Zarouf (HDR)
Associate members: H. Bommier (PR CPGE, HDR), S. Damour (PR CPGE)
ATER: R. Ernst (2014-2015), R. Tytgat (2013-2014)
Doctoral students: S. Chaabi (2010-2013), N. Combe (2013-), A. Dumont (2010-2013), A. Hanine (2010-2013), V. Le (2015-), L. Merghni (2013-), R. Tygtat (2010-2013)
Movements. R. Zarouf arrived as an MCF (attached to IUFM) in 2011, and defended his HDR in 2015. S. Charpentier was recruited as MCF in 2012.
80 articles published or accepted (including Advances, CMP, Duke, JEMS, JFA); 8 theses, 2 in progress.

Since July 2016

Coming soon.

• Period 2011-2016

This theme has wide-ranging expertise and has worked on a variety of subjects, in connection with international conjectures. Its results are part of extensive, structured and ambitious research programs. The strong interaction between the algebraic geometry and analytic differential geometry problems studied is remarkable. This section presents some of the results obtained, while the list of publications provides a more comprehensive overview.
G. Dloussky is interested in the classification of compact holomorphic surfaces, special structures on non-Kählerian holomorphic varieties (locally Kählerian conformal structure, Bihermitian, G-structure) and invariant calculi such as Bott-Chern or Aeppli cohomology spaces. In this capacity, he collaborated actively with A. Teleman. With Apostolov (UQAM), he showed the existence of a locally conformal symplectic structure that dominates the holomorphic structure for all surfaces with odd Betti number b1. If we focus on special structures, this is, along with Lamari’s theorem, the only such general result for surfaces with b1 = 1 and b2 > 0.
A. Teleman continues to work on the classification of Class VII surfaces, having solved the cases b2 = 0, 1 and made very significant progress in the case b2 = 2 in a series of seminal papers. This extremely delicate work is based on the theory of instanton moduli spaces, Donaldson’s work and the Kobayashi-Hitchin correspondence for non-Kählerian surfaces. He seeks to extend these methods to demonstrate the existence of curve cycles in all generality, and has established several results in this direction.
With Okonek, he studied determinant fibers for a real family of Dirac operators, and theta divisors on the Picard group of a Klein surface (Riemann surface with anti-holomorphic involution). In particular, they have explicitly determined the Stiefel-Whitney classes of the associated real straight fibers, the first such calculation in the literature. The results obtained and the methods developed to achieve them will certainly have a major impact on real algebraic geometry.
J. Keller has undertaken a program of research into the geometric quantization of certain natural objects in complex analytic geometry. He presented a quantification of Calabi’s fundamental problem (fixing the volume form in a Kähler class), which led to another proof of the Aubin-Yau theorem, showing a different flow from the Kähler-Ricci flow. In the same way, he obtained a quantization of the Yang-Mills flow, the solutions of the Hitchin equations on Higgs fibers and the Kodaira Laplacian. For example, in the projective case, this gives an algorithm for approximating the eigenvalues of the Laplacian defined on a simple holomorphic vector fibered. The methods employed are based on symplectic constructions (moment application), heat kernel and Bergman asymptotics, and Mumford theory of geometric invariants. So there’s a connection between these problems and questions of stability of varieties and fibers. This led him to develop another line of research, the study of regulated varieties. For example, he established the existence of Kählerian metrics with constant scalar curvature and conic singularities on set varieties given as projectivizations of semistable fibers over a curve.
With Guedj and Kolev, N. Yeganefar has shown that on any strictly pseudoconvex domain of Cn there exists a Kähler-Einstein metric with strictly positive curvature, which induces on the edge a metric conforming to the Levi form. This is in complete contrast to the situation of closed (Fano) varieties, on which the existence of Kähler-Einstein metrics with strictly positive curvature is not always guaranteed. He is also interested in the multi-dimensional systems used by automation engineers. The questions asked are related to dynamical systems and PDEs: control, stability, Lyapunov theory, etc. The aim here is to provide a solid theoretical foundation for the field, using the rigor and concepts of Geometry.
J. Hubbard continued to work on hyperbolic geometry, Teichmueller theory and dynamics. In particular, he was interested in the quotient of the Teichmueller space augmented by the action of the mapping class group, which gives a compactification of the moduli space of curves. With Koch, J. Hubbard introduced an analytic structure on this compact quotient and proved that it was isomorphic, as an analytic space, to the Deligne-Mumford compactification.
P. Roesch is interested in the parameter space of families of rational fractions, with the aim of classifying possible dynamics. With Yongcheng, she showed that the edge of any bounded Fatou component for a polynomial is a Jordan curve as soon as it does not have a rotation domain. With Petersen, she proved a Milnor conjecture asserting that the locus of connectedness of rational fractions of degree 2 having a fixed point of multiplier 1 at infinity is homeomorphic to the Mandelbrot set.
E. Rousseau continued his research on hyperbolicity in complex geometry. With Diverio, he recently proved an all-or-nothing principle concerning the basic loci of algebraic differential operators on quotients of bounded symmetric domains. These basic locations are important because they contain all the entire curves drawn on an algebraic variety. The principle states that for the ball, this place is empty, while for the other domains, it covers the whole variety. With Touzet, E. Rousseau established the Green-Griffiths-Lang conjecture on the algebraic degeneracy of integer curves for modular Hilbert varieties.
K. Oeljeklaus, with Miebach and Gilligan, studied Hamiltonian actions of reductive Lie groups on Kählerian varieties, showing that orbital adhesions are analytic complex. This has enabled us to characterize reductive homogeneous Kählerian varieties in terms of their isotropy subgroups. These isotropy groups are algebraic if and only if the variety has a moment application associated with the action. K. Oeljeklaus also studied Schottky group actions on rational varieties, and found new examples of non-Kählerian compact complex varieties with free fundamental groups. In this way, he was able to construct new equivariant compactifications of SL(2,C)/Γ where Γ is a discrete loxodromic free subgroup of SL(2,C).
L. Manivel continues his quest for new compact holomorphic symplectic varieties constructed as moduli spaces of subvarieties in Fano varieties with special Hodge structure, or of objects in their derived categories. These special Fano varieties are often linked to homogeneous or quasi-homogeneous spaces under the action of algebraic groups. L. Manivel is also interested in questions of algorithmic complexity, in particular Valiant’s version of the P vs NP problem. The program recently proposed by Mulmuley to tackle this problem relies, among other things, on certain expected properties of Kronecker coefficients, the multiplicities of tensor products of irreducible complex representations of symmetric groups. L. Manivel has shown how algebraic geometry techniques give access to these coefficients, and in particular to their asymptotic behavior.

PR: G. Dloussky, J. Hubbard (emeritus), K. Oeljeklaus, P. Roesch, E. Rousseau, A. Teleman
DR: L. Manivel
MCF: J. Keller (HDR), N. Yeganefar
Post-docs: S.A. Filippini (2016-2017), F. Tanturri (2015-2017).
Doctoral students : I. Bachy (2008-2011), L. Battisti (2009-2012), A. Bazhdar (2013-), V. Benedetti (2015-), B. Cadorel (2015-), V. Plechinger (2015-), J.R. Velasquez (2015-), D. Veloso (2011-2014).
Movements. E. Rousseau was recruited as a professor in 2011, while V. Guedj was transferred to Toulouse. P. Roesch joined in 2012 and has just left the laboratory. J. Keller defended his HDR in 2014. L. Manivel arrived in January 2015 on a two-year A*MIDEX chair. G. Dloussky retires during 2016. E. Rousseau has just been named a junior member of the IUF.
80 published or accepted articles (including Advances, Ann. ENS, CMP, Crelle, Duke, JEMS, J.G.Phy., Math. Annalen) and 5 books; 8 theses, 5 in progress.

Source: i2m evaluation report 2011-2016 (hceres) p.64-66

• Since July 2016

Coming soon.

• Period 2011-2016

Topology and geometry in dimension 3 have undergone spectacular developments in recent years. Several major conjectures (Poincaré conjecture, geometrization of 3-dimensional varieties) have been resolved, opening up new fields of application. In addition, new methods from geometric group theory, as well as new homological theories (Khovanov and Heegaard-Floer homology) linked to symplectic geometry and quantum field theory, have appeared, raising new questions about intrinsic characterizations of 3-varieties. Infinite-dimensional differential geometry has played an important role in this progress. At the same time, infinite-dimensional Riemannian geometry is also a subject of active research, particularly on diffeomorphism groups and spaces of curves, because of its applications in shape recognition and image processing. A new theory, diffeology, has also developed, generalizing differential geometry to finite and infinite dimensions and incorporating, in particular, stability by quotient. In infinite dimension, Riemannian geometry on diffeomorphism groups has also seen new developments.
These various developments have strong repercussions outside mathematics, particularly in physics and mechanics, but also in quantum computing. It is in this context of interactions, both internal – between algebraic, geometric and topological properties of low-dimensional varieties – and external to related fields such as physics, mechanics and computer science, that the work of the Geometry-Topology theme is situated.
Research topics
The themes addressed can be broken down into two aspects.

The first, more geometrical and analytical, is made up of five members, and we now present some of their most striking results.
B. Kolev is interested in global analysis on diffeomorphism groups and its applications to mathematical physics equations. In particular, together with Escher and Bauer, he showed the local and global existence of geodesics on the group of C∞ diffeomorphisms of the torus Td or Rd for Hs metrics, when s is non-integer. This also includes the existence of solutions for the Constantin-Lax-Majda equation and the Euler-Weil-Petersson equation in the C∞ category. B. Kolev is also working on effective invariant theory and its application to mechanics. Together with Auffray and his student M. Olive, he determined a minimal basis of invariants of the elasticity tensor, thus solving an old open problem.
C. Pittet’s research concerns the asymptotic invariants of infinite groups. The application of techniques from Riemannian geometry and spectral analysis is a leitmotif. Together with Chatterji and Mislin, he gave, on the one hand, a necessary and sufficient criterion, in terms of the group derived from the radical of a connected Lie group provided with a complex structure G, for any principal G-fibered over a finite CW – complex to be virtually trivial; and on the other hand, necessary and sufficient criteria, in terms of the radical, for the primary characteristic classes of a (virtually) connected Lie group G to be all bounded. Concerning the spectral analysis of Laplacians on groups, he established, with Bendikov and Sauer, a formula for calculating the spectral distribution in the vicinity of zero of a Laplacian, in terms of the isospectral profile of domains whose volume tends to infinity.
P. Iglesias-Zemmour has written and published a reference work on diffeology. In addition, with P. Donato and J-P. Mohsen, two weekly working groups on the subject.
The second, more topological, aspect brings together the other four members on themes around dimension 3 and 4.
B. Audoux is interested in the interactions between welded diagrammatics and knotted surfaces in di- mension 4. With his co-authors from the ANR VasKho project, he has notably classified, in terms of automorphisms of the reduced free group, the “ribbon” type plungings of rings in B4 to within link-homotopy. This generalizes Habegger and Lin’s results for classical string-links to dimension 4. In collaboration with Couvreur and using topological tools such as Khovanov homology, B. Audoux has also constructed new families of CSS-type quantum codes with, asymptotically, good minimum distances.
M. Boileau’s work focuses on the geometry and topology of 3-dimensional varieties and the study of their fundamental groups. In collaboration with, among others, L. Paoluzzi, he has recently given an increase in the number of nodes in S3 with the same hyperbolic branched cyclic covering. M. Boileau is also interested in Heegaard-Floer homology and the topological characterization of full homology spheres that are L-spaces, giving with Boyer an answer in the case of graph varieties.
P. Derbez is interested in Chern-Simons theory, particularly in the hyperbolic case. Together with Wang, he gave a necessary and sufficient criterion for an irreducible closed variety to admit a finite covering of Seifert volume on the one hand, and hyperbolic volume on the other, which is non-zero. Based on these results, he determined the set of 3-varieties N such that the set of degrees of applications between M and N is finite for any 3-variety M, thus answering a question posed by Gromov.
D. Matignon continues to work on codimension one laminations. With his student S. Gibert-Caillat, he showed the existence of tense laminations on spheres of Seifert integer homology (apart from S3 and the Poincaré sphere), while there are an infinite number of spheres of rational homology that admit none.
Note that topological themes of dimension 3 and 4 are shared with certain members of the GDAC team (e.g. A. Garcia-Lecuona, L. Paoluzzi or H. Short). This broad porosity between the two teams has led to a joint working group on the Boyer-Gordon-Watson conjecture on L-spaces.

PR: M. Boileau, P. Donato (retired), C. Pittet
CR: P. Iglesias-Zemmour (PhD), B. Kolev (HDR)
MCF : B. Audoux, P. Derbez (HDR), D. Matignon (HDR), J.P. Mohsen
Doctoral students : A. Boyer (2011-2014), S. Gibert-Caillat (2008-2011), M. Olive (2011-2014), A. Pinochet-Lobos (2015-)
Movements. P. Derbez defended his HDR in 2012. Mr. Boileau was recruited in 2013. P. Donato retired in 2014.
63 published articles (including AJM, CMH, G& T, JDE, JDG, Math. Annalen) and 1 book; 8 theses, 2 in progress.

• Since 2016

Coming soon.

• Period 2011-2016

Topology of singular varieties. With B.Teissier, J.-P.Brasselet showed a relative Poincaré theorem. With S. Yokura and J. Schurmann, he gave a formula for the Hirzebruch χy-generus χy(X) and for the motivic Hirzebruch class Ty∗(X) for singular varieties X, using Vandermonde matrices. Motivated by the notions of secondary Euler characteristic class and “larger” Euler characteristic classes, they consider a similar notion for the motivic Hirzebruch class, called the derived motivic Hirzebruch class. With Nonato Araújo dos Santos, N. Dutertre proved a Milnor conjecture on the topology of the real Milnor fiber in a more general framework than that of applications with isolated singularities.
Geometry of sub-analytical sets. N. Dutertre has proved several singular versions of the Gauss-Bonnet formula (closed semi-algebraic sets, Milnor fiber of a sub-analytic function seed on a closed sub-analytic set, etc.) and a local singular version of the linear kinematic formula. He applied these results to complex analytic sets and obtained a characterization of the Euler obstruction of a complex analytic seed as a function of the Lipschitz-Killing curvatures of its regular part. He deduced a Fu conjecture on the Euler obstruction of a complex analytic germ and the Gauss-Bonnet curvature of the regular part of its link. He established a link between the Lipschitz-Killing curvatures of sub-analytic sets and the volumes of polar images of generic projections (a singular version of work by Langevin and Shifrin). Applying the theory of real equisingularity, he linked the densities of polar images to the polar invariants of Comte and Merle. Euler obstruction and generalizations. With de Góes Grulha Jr, N. Dutertre proved a Lê-Greuel-type formula for the Euler obstruction of a function. He deduced a multiplicity formula for this obstruction (a generalization of the Lê-Teissier formula), a curvature formula (a generalization of the Kennedy formula) and a relative version of the local index formula of Brykinski, Dubson and Kashiwara. Equisingularity and Bilipschitz geometry of complex singularities. Together with Birbrair and Neumann, A. Pichon established the complete classification of singularities of complex surfaces modulo bilipschitz equivalence for the intrinsic metric. Known results mainly concerned real analytic spaces, with the complex framework remaining virtually unexplored outside the case of curves. With W. Neumann, she demonstrated the equivalence between equisingularity in the Zariski sense and Bilipschitz equisingularity. She also established that a number of classical analytic invariants of singularities of normal surfaces are in fact bilipschtiz invariants for the external metric, notably multiplicity (which from a certain point of view, gives a positive answer to Zariski’s question on multiplicity with the (strong) Lipschitz hypothesis).
Still in the case of surfaces, A. Pichon has begun studying the links between singularity resolution and bilipschitz geometry, as well as the relations between bilipschitz geometry and the diving topology of a hypersurface singularity.
C. Plénat and D. Trotman studied a Teissier conjecture on families of complex hypersurfaces, and gave sufficient criteria for the multiplicity to remain constant. With K. Bekka, Trotman performed calculations for the example families of complex hypersurfaces introduced by Briançon and Speder, testing the weak Whitney condition. With D. van Straten, Trotman proved that the weak Whitney condition for a family of complex hypersurfaces implies equimultiplicity, a weak version of a Teissier conjecture.
M. Bilski, K. Kurdyka, A. Parusin ́ski and G. Rond proved that any analytic space germ is homeomorphic to an algebraic space germ (demonstrated by T. Mostowski in 1984) and that this homeomorphism can be assumed to have a prescribed order of tangency as large as desired with the identity. G. Rond, M. Bilski, and A. Parusinski have shown that every analytic function seed is homeomorphic to a polynomial function seed.
Nash problem. C. Plénat is interested in the Nash problem. In 2011/12 the problem was solved by J. Fernandez de Bobadilla and M. Pe Pereira in dimension 2 in the affirmative, and by a counterexample in dimension 3 by T. de Fernex and then J. Kollàr in dimension 4. C. Plénat, with M. Spivakovsky, has produced two surveys, one aimed at a wide audience, the other more detailed.
The second theme of C. Plénat’s research is a kind of inverse problem to Nash’s: how to characterize resolutions from the space of arcs and jets? A first result on toric plunging resolutions of simple singularities is the subject of a paper in collaboration with H. Mourtada.
Real filtration by weight. F. Priziac has constructed and studied a weight filtration on the equivariant homology of real algebraic varieties equipped with the action of a finite group. He also constructed equivariant zeta functions, equivariant analogues of Fichou’s zeta functions, themselves inspired by Denef and Loeser’s motivic zeta functions. He applied these tools to the problem of classifying simple Nash germs invariant with respect to equivariant Nash equivalence.
Commutative algebra. G.Rond is interested in the question of the algebraic quotient of formal series k[[x1, …,xn]]/I, with k any field. He shows that if I is generated by algebraic series, then the cancellation order in 0 modulo I of a polynomial p is increased by an affine function of the degree of p. More generally, the cancellation order in 0 modulo I of a formal algebraic series f is increased by an affine function of the height of f (the height being the maximum degree of the coefficients of the minimal polynomial). This allows us to show that certain solution series of functional equations, appearing for example in singularity theory or combinatorics and which are not algebraic series, are not too transcendental (in the sense that these series do not have too large gaps).
Whitney stratifications. With H. King, D. Trotman proved Poincaré-Hopf theorems for stratified vector fields on stratified spaces, extending the work of M.-H. Schwartz on analytic varieties. With S. Trivedi, he solved a conjecture concerning the extension to stratified morphisms of a theorem characterizing transversality stability by Thom’s condition. With N. Nguyen and S. Trivedi, he gave a geometric proof of the existence of Whitney stratifications of definable sets, correcting and generalizing a Kaloshin proof for semi-algebraics. In order to show the Whitney triangulation of a Whitney stratification, C. Murolo studied stratified cylinders of application.

PR: Lê Dung Tràng (emeritus), A. Pichon, D. Trotman
DR: J.P. Brasselet (emeritus)
MCF: N. Dutertre (HDR), C. Murolo, C. Plénat, F. Priziac, G. Rond (HDR)
ATER: J. Lapébie (2015-2016)
Doctoral students: J. Giacomoni (2012-), J. Lapébie (2012-2015), T.B.T. Nguyen (2010-2013), X.V.N. Nguyen (2012-2015), S. Trivedi (2010-2013)
Movements : A. Pichon was promoted to professor in 2012. F. Priziac was recruited as MCF in 2014.
49 published articles (including Acta, Advances, Crelle, J. Algebra, JAG, Proc. LMS), 11 theses (2 in progress).