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Colin de Verdière|An inverse semi-classical problem for the Schrödinger operator in dimension one| The main motivation comes from the method of passive imaging in seismology, developped by Michel Campillo and his group (LGIT, Grenoble). The potential $V(x)$ in the Schrödinger operator $\hat{H}= -\hbar^2 \frac{d^2}{dx^2} + V(x)$ can be recovered, under some weak genericity assumptions, from the spectrum of $\hat{H}$ modulo $o(\hbar^3)$. The proof uses Abel's integral transform and several semi-classical trace formulae. Details can be found in the papers arXiv:0802.1605 and arXiv:0802.1643. |Thomas Delzant|Volume des variétés de dimension 3 et complexité de leurs groupes fondamentaux| On compare le volume de certaines variétés de dimension trois avec la complexité de leurs groupes fondamentaux (travail en commun avec L. Potyagailo). |Carolyn Gordon|Spectral Isolation of Bi-invariant Metrics on Compact Lie Groups| We show that a bi-invariant metric on a compact connected Lie group $G$ is spectrally isolated within the class of left-invariant metrics. In fact, we prove that given a bi-invariant metric $g_0$ on $G$ there is a positive integer $N$ such that, within a neighborhood of $g_0$ in the class of left-invariant metrics of at most the same volume, $g_0$ is uniquely determined by the first $N$ distinct non-zero eigenvalues of its Laplacian (ignoring multiplicities). In the case where $G$ is simple, $N$ can be chosen to be two. (This is joint work with Dorothee Schueth and Craig Sutton.). |Laurent Guillopé|Determinants and zeta functions| On a Riemann surface Selberg trace formula can be expressed through the expression of the Laplacian determinant and Selberg zeta function. The lecture will consider the determinant for the Dirichlet to Neuman map. |Matthew Gursky|Obstructions and constructions of metrics with prescribed curvature conditions.| I will begin by summarizing some well known topological obstructions to admitting metrics with various curvature conditions. I will then specialize to the problem of prescribing curvature conditions for conformal metrics, and describe some results for closed manifolds and manifolds with boundary. |Pierre Jammes|On multiple eigenvalues of the Hodge Laplacian| I will survey some classical results about the multiplicity of eigenvalues of the standard Laplacian on compact manifolds, and present new results and open questions about multiple eigenvalues of the Hodge Laplacian. |Dan Jane|The effect of the Ricci flow on magnetic topological entropy| Suppose, on a given closed 2-manifold, we have a family of negatively curved metrics that satisfy the Ricci flow. Anthony Manning showed in 2004 that the topological entropy associated to the geodesic flow of a metric is decreasing as we move along the path. We extend this result to a magnetic setting, where the Ricci Yang-Mills flow is a more appropriate geometric evolution equation. |François Labourie|Proper actions of free groups on the affine space| In this talk, I will explain how to characterise using ergodic theory actions of free groups on the affine 3-dimensional space whose linear part is a fuchsian group. (joint work with Goldman and Margulis) |Harold Rosenberg|The geometry of surfaces in 3-dimensional homogeneous spaces.| I will discuss surfaces in the 3-dimensional homogeneous spaces $S \times R, H \times R$, Berger spheres, Heisenberg space, $PSL(2,R)$-tilda, and $Sol(3)$. Here, $S$ and $H$ are the sphere and hyperbolic plane of curvature one and minus one respectively. I describe some examples and theorems depending on the mean, intrinsic or extrinsic curvature of the surface. |Julie Marie Rowlett|The Laplace and length spectra of asymptotically hyperbolic manifolds| Asymptotically hyperbolic manifolds are a natural generalization of infinite volume hyperbolic manifolds and enjoy similar features. In this talk, I will present results for the Laplace and length spectra of $(n+1)$ dimension asymptotically hyperbolic manifolds with negative (but not necessarily constant) sectional curvatures. These results include: a dynamical wave trace formula relating the Laplace and length spectra, a prime orbit theorem for the geodesic flow based on the dynamical zeta function, and a relationship between the pure point spectrum of the Laplacian and the topological entropy of the geodesic flow. Key techniques and ideas from the proofs will be summarized, concluding with a discussion of open problems. |Andrea Sambusetti|On the growth of quotients of Klenian groups.| We present some results about the growth of quotients of a general Kleinian group $G$ (i.e. a discrete, torsionless group of isometries of a Cartan-Hadamard manifold with curvature pinched between two negative constants). Namely, we give general criteria ensuring the divergence of a quotient $\overline{G}$ of $G$ and the critical gap property $\delta_{\overline{G < \delta_G$. As a corollary, we prove growth tightness of geometrically finite Kleinian groups satisfying the parabolic gap condition (this means that $\delta_P < \delta_G$ for every parabolic subgroup $P$ of $G$). Notice that, as these quotient groups naturally act on non-simply connected quotients of a Cartan-Hadamard manifold, the classical arguments of Patterson-Sullivan's theory (topology of the boundary, shadows, quasi-conformal densities etc.) are not available here. This forces us to a more elementary approach for counting points in the orbit of a quotient (which gives a new elementary proof of the classical results of divergence of geometrically finite groups in the simply connected case). We also notice that, contrary to the simply connected case, there is large freedom for the behaviour of the growth function of quotients of Kleinian groups (even convex-cocompact): as a way of example, we will exhibit quotients of convex cocompact Kleinian groups with mixed polynomial-exponential growth. (joint work with F.Dal'Bo, M.Peigne, J.C.Picaud) |Walcy Santos|Curvature integral estimates for complete hypersurfaces| We consider the integrals of $r$-mean curvatures $S_r$ of a complete hypersurface $M$ in space forms $\mathcal{Q}_c^{n+1}$ which generalize volume $(r=0)$, total mean curvature $(r=1)$, total scalar curvature $(r=2)$ and total curvature $(r=n)$. Among other results we prove that a complete properly immersed hypersurface of a space form with $S_r\geq 0$, $S_r\not\equiv 0$ and $S_{r+1}\equiv 0$ for some $r\le n-1$ has $\int_MS_rdM=\infty.$ This is a joint work with H. Alencar and D. Zhou. |Harish Seshadri|Positive isotropic curvature and Einstein metrics| The first half of the talk will be a brief survey of the geometry and topology of manifolds with positive isotropic curvature. The second half will deal with recent results about metric and smooth rigidity of Einstein metrics with positive isotropic curvature. |Ricardo Sà Earp|Lindelöf's theorem revisited| In the second volume of the Mathematische Annalen (1874), in which he determines the maximal domains of stability of the catenoid in R^ 3 .
We will introduce a geometric approach to extend Lindelöf results to certain geometric situations in several ambient spaces.
This is a joint work with Pierre Bérard (Institut Fourier, Grenoble). |Peter Storm|Infinitesimal rigidity of hyperbolic manifolds with totally geodesic boundary| Using the Bochner technique, Steve Kerckhoff and I recently proved the following theorem. Let $M$ be a compact hyperbolic manifold with totally geodesic boundary. If $M$ has dimension at least four, then the holonomy representation of $M$ is infinitesimally rigid. This is an infinite volume analog of the Calabi-Weil rigidity theorem. I will explain some of the background and ideas used in the proof. |Jeff Viaclovsky|Limits of constant scalar curvature anti-self-dual metrics.| I will discuss some theorems regarding sequences of anti-self-dual metrics limiting to orbifold metrics, and give various examples and connections with the orbifold Yamabe problem.
Spectral Theory and Geometry
Lundi, 1 Juin, 2009 (Toute la journée)
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Journées organisées par l'Institut Fourier
