arXiv research

Extends boundary estimates for Monge-Ampère equations in polygonal domains.

Existence and boundary regularity away from the corners are established for two-dimensional Monge-Ampère equations on convex polytopes with Guillemin boundary conditions. An important step is to derive an expansion in terms of functions $y^n$ and $y^n\log y$ for solutions to equations of the form $\det D^2u(x,y) = y^{-…

Guillemin trace formula adapted for group actions.

We show that any compact convex simple lattice polytope is the moment polytope of a Kähler-Einstein orbifold, unique up to orbifold covering and homothety. We extend the Wang-Zhu Theorem \cite{WZ} giving the existence of a Kähler-Ricci soliton on any toric monotone manifold on any compact convex simple labelled polytop…

Study on determining metrics from minimal surface areas, extending earlier work.

Adapts a short argument to derive a stability theorem for smooth maps.

We construct Zollfrei Lorentzian metrics on every nontrivial orientable circle bundle over a orientable closed surface. Further we prove a weaker version of Guillemin's conjecture assuming global hyperbolicity of the universal cover.

The Guillemin-Uribe trace formula is a semiclassical version of the Selberg trace formula and more general Duistermaat-Guillemin formula for elliptic operators on compact manifolds, which reflects the dynamics of magnetic geodesic flows in terms of eigenvalues of a natural differential operator (the magnetic Laplacian)…

We discuss the construction of toric Kaehler metrics on symplectic 2n-manifolds with a hamiltonian n-torus action and present a simple derivation of the Guillemin formula for a distinguished Kaehler metric on any such manifold. The results also apply to orbifolds.

We develop a theory of "quasi"-Hamiltonian G-spaces for which the moment map takes values in the group G itself rather than in the dual of the Lie algebra. The theory includes counterparts of Hamiltonian reductions, the Guillemin-Sternberg symplectic cross-section theorem and of convexity properties of the moment map. …

Recently Guillemin gave an explicit combinatorial way of constructing "toric" Kahler metrics on (symplectic) toric varieties, using only data on the moment polytope. In this paper, differential geometric properties of these metrics are investigated using Guillemin's construction. In particular, a nice combinatorial for…

We study the asymptotic of the spectrum of the \spin Dirac operator on high tensor powers of a line bundle. As application, we get a simple proof of the main result of Guillemin-Uribe, which was originally proved by using the analysis of Toeplitz operators of Boutet de Monvel and Guillemin.

This expository monograph cuts a short path from the common, elementary background in geometry (linear algebra, vector bundles, and algebraic ideals) to the most advanced theorems about involutive exterior differential systems: (1) The incidence correspondence of the characteristic variety, (2) Guillemin normal form an…

Study of symplectic and Poisson reduction, proposing Poisson implosion.

The purpose of this note is to extend the results of V. Guillemin on elliptic self-adjoint pseudodifferential operators of order one, from operators defined on smooth functions on a closed manifold to operators defined on smooth sections in a vector bundle of Hilbert modules of finite type over a finite von Neumann alg…

This note is a step towards demonstrating the benefits of a symplectic approach to studying equivariant Kähler geometry. We apply a local differential geometric framework from Kähler toric geometry due to Guillemin & Abreu to the case of the standard linear $\SU(n)$ action on $\bbC^n\setminus\{0\}$. Using this framewor…

Study geodesic X-ray transforms on hyperbolic surfaces, proposing new reconstruction methods.

The abstract explores analogues of Hodge theory in Lie algebroids.

We extend Guillemin's formula for Kaehler potentials on toric manifolds to singular quotients of C^N and CP^N.

We prove a Duistermaat-Guillemin trace formula for transversally elliptic operators on a compact foliated manifold.

We define a moment map associated to a smooth torus action on a smooth manifold, without a two-form. We define cobordisms of such structures, allowing non compact manifolds as long as the moment maps are proper. We prove that a compact manifold with a torus action and a moment map is cobordant to the disjoint union of …

In this paper, we prove that the "quantization commutes with reduction" phenomenon of Guillemin-Sternberg applies in the context of the metaplectic correction.

If $(M,g)$ is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes of order $o(λ)$ saturating sup-norm estimates. In particular, it gives optimal conditions for existence of eigenfunctions satisfying maximal sup norm bounds. The condition is …

The elaboration of new quantization methods has recently developed the interest in the study of subalgebras of the Lie algebra of polynomial vector fields over a Euclidean space. In this framework, these subalgebras define maximal equivariance conditions that one can impose on a linear bijection between observables tha…

The classical Weyl Law says that if $N_M(λ)$ denotes the number of eigenvalues of the Laplace operator on a $d$-dimensional compact manifold $M$ without a boundary that are less than or equal to $λ$, then $N_M(λ)=cλ^d+O(λ^{d-1}).$ In this paper, we show Duistermaat and Guillemin's result allows us to replace the $O(…

The notion of a local line bundle on a manifold, classified by 2-cohomology with real coefficients, is introduced. The twisting of pseudodifferential operators by such a line bundle leads to an algebroid with elliptic elements with real-valued index, given by a twisted variant of the Atiyah-Singer index formula. Using …

For operators on a compact manifold $X$ with boundary $\partial X$, the basic zeta coefficient $C_0(B, P_{1,T})$ is the regular value at $s=0$ of the zeta function $\Tr(B P_{1,T}^{-s})$, where $B=P_++G$ is a pseudodifferential boundary operator (in the Boutet de Monvel calculus) -- for example the solution operator of …

In this paper we show that the `quantization commutes with reduction' principle of Guillemin-Sternberg holds for the coadjoint orbits that parametrize the discrete series of a real connected semi-simple Lie group.

We discuss Witten's formulas for the symplectic volumes of moduli spaces of flat connections on 2-manifolds from the viewpoint of Hamiltonian cobordism as introduced by Ginzburg-Guillemin-Karshon.

We show that the boundaries of thin strongly pseudoconvex Grauert tubes, with respect to the Guillemin-Stenzel Kähler metric canonically associated with the Poincaré metric on closed hyperbolic real-analytic surfaces, has nowhere vanishing Cartan CR-curvature. This result provides a wealth of examples of compact $3$-di…

We develop the theory of Berezin-Toeplitz operator on any compact symplectic prequantizable manifold from scratch. Our main inspiration is the Boutet de Monvel-Guillemin theory, that we simplify in several ways to obtain a concise exposition. A comparison with the spin-c Dirac quantization is also included.

Defines an equivariant Ruelle dynamical zeta function for flows on manifolds.

New proof of wave trace formula for 3D-contact manifolds.

We introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties. We develop K{ä}hler geometry on these varieties, with applications to canonical metrics in mind, as a generalization of the Guillemin-Abreu-D…

The purpose of this paper is to present the construction of a canonical determinant functional on elliptic pseudodifferential operators associated to the Guillemin-Wodzicki residue trace. The resulting functional is multiplicative, a local invariant, and not defined by a regularization procedure. The residue determinan…

This is my Habilitation a Diriger des Recherches (French thing). In this paper I summarize the work I have done to realize the program of Witten called -non abelian localization-. This work deals first with problems of localization in equivariant cohomology. The second part of this paper concerns the Guillemin-Sternber…

This note contains some observations on abelian convexity theorems. Convexity along an orbit is established in a very general setting using Kempf-Ness functions. This is applied to give short proofs of the Atiyah-Guillemin-Sternberg theorem and of abelian convexity for the gradient map in the case of a real analytic su…

We construct a counterexample to a conjectured inequality L<2D, relating the diameter D and the least length L of a nontrivial closed geodesic, for a Riemannian metric on the 2-sphere. The construction relies on Guillemin's theorem concerning the existence of Zoll surfaces integrating an arbitrary infinitesimal odd def…

We apply the Guillemin-Lerman-Sternberg theorem to reprove a formula of Heckman for the Duistermaat-Heckman measure associated to the coadjoint action of $T$, a maximal torus of a compact semisimple Lie group $G$, on a regular coadjoint $G$-orbit in the dual space of the Lie algebra of $G$. This formula is, in an appro…

We extend our earlier work in [TZ1], where an analytic approach to the Guillemin-Sternberg conjecture [GS] was developed, to cases where the Spin$^c$-complex under consideration is allowed to be further twisted by certain natural exterior power bundles. The main result is a weighted quantization formula in the presence…

Let $X$ be a compact connected orientable CR manifold with the action of a connected compact Lie group $G$. Under natural pseudoconvexity assumptions we show that the CR Guillemin-Strernberg map is Fredholm at the level of Sobolev spaces of CR functions. As application we study this map for holomorphic line bundles whi…

We present a K-theoritic approach to the Guillemin-Sternberg conjecture, about the commutativity of geometric quantization and symplectic reduction, which was proved by Meinrenken and Tian-Zhang. Besides providing a new proof of this conjecture for the full non-abelian group action case, our methods lead to a generalis…

We extend our earlier work in [TZ1], where an analytic approach to the Guillemin-Sternberg conjecture [GS] was developed, to cases where the Spin^c-complex under consideration is allowed to be further twisted by certain exterior power bundles of the cotangent bundle. The main result is a weighted quantization formula i…

Let $G$ be a real reductive Lie group and $τ:G \longrightarrow GL(V)$ be a real reductive representation of $G$ with (restricted) moment map $m_{\ggo}: V-{0} \longrightarrow \ggo$. In this work, we introduce the notion of "nice space" of a real reductive representation to study the problem of how to determine if a $G$-…

We extend a recent result of Burns, Guillemin and Uribe on the asymptotics of the spectral measure for the reduction metric on a toric variety to any toric metric on a toric variety. We show how this extended result together with the Tian-Yau-Zelditch asymptotic expansion can be used to deduce Abreu's formula for the s…

For a symplectic manifold with quantizing line bundle, a choice of almost complex structure determines a Laplacian acting on tensor powers of the bundle. For high tensor powers Guillemin-Uribe showed that there is a well-defined cluster of low-lying eigenvalues, whose distribution is described by a spectral density fun…

Classifies local boundary conditions for Dirac-type operators on manifolds.

Proof of local well-posedness for a specific boundary condition in general relativity.