Inverse Problems for multiple invariant curves

Planar polynomial vector fields which admit invariant algebraic curves, Darboux integrating factors or Darboux first integrals are of special interest. In the present paper we solve the inverse problem for invariant algebraic curves with a given multiplicity and for integrating factors, under generic assumptions regarding the (multiple) invariant algebraic curves involved. In particular we prove, in this generic scenario, that the existence of a Darboux integrating factor implies Darboux integrability. Furthermore we construct examples where the genericity assumption does not hold and indicate that the situation is different for these.

Well-posedness and invariant measures for HJM models with deterministic volatility and Lévy noise

We give sufficient conditions for existence, uniqueness and ergodicity of invariant measures for Musiela's stochastic partial differential equation with deterministic volatility and a Hilbert space valued driving Lévy noise. Conditions for the absence of arbitrage and for the existence of mild solutions are also discussed.

Euclidean geometric invariant of framed knots in manifolds

We present an invariant of a three dimensional manifold with a framed knot in it based on the Reidemeister torsion of an acyclic complex of Euclidean geometric origin. To show its nontriviality, we calculate the invariant for some framed (un)knots in lens spaces. An important feature of our work is that we are not using any nontrivial representation of the manifold fundamental group or knot group.

Inverse problems for invariant algebraic curves: explicit computations

Given an algebraic curve in the complex affine plane, we describe how to determine all planar polynomial vector fields which leave this curve invariant. If all (finite) singular points of the curve are nondegenerate, we give an explicit expression for these vector fields. In the general setting we provide an algorithmic approach, and as an alternative we discuss sigma processes.

Fast numerical algorithms for the computation of invariant tori in Hamiltonian systems

In this paper, we develop numerical algorithms that use small requirements of storage and operations for the computation of invariant tori in Hamiltonian systems (exact symplectic maps and Hamiltonian vector fields). The algorithms are based on the parameterization method and follow closely the proof of the KAM theorem given in [LGJV05] and [FLS07]. They essentially consist in solving a functional equation satisfied by the invariant tori by using a Newton method. Using some geometric identities, it is possible to perform a Newton step using little storage and few operations. In this paper we focus on the numerical issues of the algorithms (speed, storage and stability) and we refer to the mentioned papers for the rigorous results. We show how to compute efficiently both maximal invariant tori and whiskered tori, together with the associated invariant stable and unstable manifolds of whiskered tori. Moreover, we present fast algorithms for the iteration of the quasi-periodic cocycles and the computation of the invariant bundles, which is a preliminary step for the computation of invariant whiskered tori. Since quasi-periodic cocycles appear in other contexts, this section may be of independent interest. The numerical methods presented here allow to compute in a unified way primary and secondary invariant KAM tori. Secondary tori are invariant tori which can be contracted to a periodic orbit. We present some preliminary results that ensure that the methods are indeed implementable and fast. We postpone to a future paper optimized implementations and results on the breakdown of invariant tori.

Destruction of invariant curves in the restricted circular planar three body problem using comparison of action

"Vegeu el resum a l'inici del document del fitxer adjunt."

Generic super-exponential stability of invariant tori in Hamiltonian systems

In this article, we consider solutions starting close to some linearly stable invariant tori in an analytic Hamiltonian system and we prove results of stability for a super-exponentially long interval of time, under generic conditions. The proof combines classical Birkhoff normal forms and a new method to obtain generic Nekhoroshev estimates developed by the author and L. Niederman in another paper. We will mainly focus on the neighbourhood of elliptic fixed points, the other cases being completely similar.

Sobolev mappings: Lipschitz density is not an isometric invariant of the target

"Vegeu el resum a l'inici del document del fitxer adjunt."

The eta invariant and equivariant index of transversally elliptic operators

We prove a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a G-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications, we obtain an index formula for basic Dirac operators on Riemannian foliations, a problem that was open for many years.

The eta invariant and equivariant index of transversally elliptic operators

We prove a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a G-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications, we obtain an index formula for basic Dirac operators on Riemannian foliations, a problem that was open for many years.

Recovering the Elliott invariant from the Cuntz semigroup

Let A be a simple, separable C*-algebra of stable rank one. We prove that the Cuntz semigroup of C (T, A) is determined by its Murray-von Neumann semigroup of projections and a certain semigroup of lower semicontinuous functions (with values in the Cuntz semigroup of A). This result has two consequences. First, specializing to the case that A is simple, finite, separable and Z-stable, this yields a description of the Cuntz semigroup of C (T, A) in terms of the Elliott invariant of A. Second, suitably interpreted, it shows that the Elliott functor and the functor defined by the Cuntz semigroup of the tensor product with the algebra of continuous functions on the circle are naturally equivalent.

Human invariant V alpha 24-J alpha Q TCR supports the development of CD1d-dependent NK1.1+ and NK1.1- T cells in transgenic mice.

A sizable fraction of T cells expressing the NK cell marker NK1.1 (NKT cells) bear a very conserved TCR, characterized by homologous invariant (inv.) TCR V alpha 24-J alpha Q and V alpha 14-J alpha 18 rearrangements in humans and mice, respectively, and are thus defined as inv. NKT cells. Because human inv. NKT cells recognize mouse CD1d in vitro, we wondered whether a human inv. V alpha 24 TCR could be selected in vivo by mouse ligands presented by CD1d, thereby supporting the development of inv. NKT cells in mice. Therefore, we generated transgenic (Tg) mice expressing the human inv. V alpha 24-J alpha Q TCR chain in all T cells. The expression of the human inv. V alpha 24 TCR in TCR C alpha(-/-) mice indeed rescues the development of inv. NKT cells, which home preferentially to the liver and respond to the CD1d-restricted ligand alpha-galactosylceramide (alpha-GalCer). However, unlike inv. NKT cells from non-Tg mice, the majority of NKT cells in V alpha 24 Tg mice display a double-negative phenotype, as well as a significant increase in TCR V beta 7 and a corresponding decrease in TCR V beta 8.2 use. Despite the forced expression of the human CD1d-restricted TCR in C alpha(-/-) mice, staining with mCD1d-alpha-GalCer tetramers reveals that the absolute numbers of peripheral CD1d-dependent T lymphocytes increase at most by 2-fold. This increase is accounted for mainly by an increased fraction of NK1.1(-) T cells that bind CD1d-alpha-GalCer tetramers. These findings indicate that human inv. V alpha 24 TCR supports the development of CD1d-dependent lymphocytes in mice, and argue for a tight homeostatic control on the total number of inv. NKT cells. Thus, human inv. V alpha 24 TCR-expressing mice are a valuable model to study different aspects of the inv. NKT cell subset.

Cutting edge: influence of the TCR Vbeta domain on the selection of semi-invariant NKT cells by endogenous ligands.

Invariant Valpha14 (Valpha14i) NKT cells are a murine CD1d-dependent regulatory T cell subset characterized by a Valpha14-Jalpha18 rearrangement and expression of mostly Vbeta8.2 and Vbeta7. Whereas the TCR Vbeta domain influences the binding avidity of the Valpha14i TCR for CD1d-alpha-galactosylceramide complexes, with Vbeta8.2 conferring higher avidity binding than Vbeta7, a possible impact of the TCR Vbeta domain on Valpha14i NKT cell selection by endogenous ligands has not been studied. In this study, we show that thymic selection of Vbeta7(+), but not Vbeta8.2(+), Valpha14i NKT cells is favored in situations where endogenous ligand concentration or TCRalpha-chain avidity are suboptimal. Furthermore, thymic Vbeta7(+) Valpha14i NKT cells were preferentially selected in vitro in response to CD1d-dependent presentation of endogenous ligands or exogenously added self ligand isoglobotrihexosylceramide. Collectively, our data demonstrate that the TCR Vbeta domain influences the selection of Valpha14i NKT cells by endogenous ligands, presumably because Vbeta7 confers higher avidity binding.

Cutting edge: influence of the TCR V beta domain on the avidity of CD1d:alpha-galactosylceramide binding by invariant V alpha 14 NKT cells.

CD1d tetramers loaded with alpha-galactosylceramide (alpha-GalCer) bind selectively to mouse invariant Valpha14 (Valpha14i) NKT cells and their human counterparts. Whereas tetramer binding strictly depends on the expression of a Valpha14-Jalpha18 chain in murine NKT cells, the associated beta-chain (typically expressing Vbeta8.2 or Vbeta7) appears not to influence tetramer binding. In this study, we describe novel alpha-GalCer-loaded mouse and human CD1d-IgG1 dimers, which revealed an unexpected influence of the TCR-beta chain on the avidity of CD1d:alpha-GalCer binding. A subset of Valpha14i NKT cells clearly discriminated alpha-GalCer bound to mouse or human CD1d on the basis of avidity differences conferred by the Vbeta domain of the TCR-beta chain, with Vbeta8.2 conferring higher avidity binding than Vbeta7.

Characterization of tumor antigen specific CD8+ T cells and semi-invariant Vα24/Vβ11 NKT cells in tumor invaded liver

Summary: Detailed knowledge on tumor antigen expression and specific immune cells is required for a rational design of immunotherapy for patients with tumor invaded liver. In this study, we confirmed that Cancer/Testis (CT) tumor-associated antigens are frequently expressed in hepatocellular carcinoma (HCC) and searched for the presence of CD8+ T cells specific for these antigens. In 2/10 HLA-A2+ patients with HCC, we found that MAGE-A10 and/or SSX-2 specific CD8+ T cells naturally responded to the disease, since they were enriched in tumor lesions but not in non-tumoral liver. Isolated T cells specifically and strongly killed tumor cells in vitro, suggesting that these CTL were selected in vivo for high avidity antigen recognition, providing the rational for specific immunotherapy of HCC, based on immunization with CT antigens such as MAGE-Al 0 and SSX-2. Type 1 NKT cells express an invariant TCR α chain (Vα24.1α18, paired with Vβ11 in human) and share a specific reactivity to αGalactosylceramide (αGC) presented by CD1d. These cells can display paradoxical immuno-regulatory properties including strong anti-tumor effects upon αGC administration in murine models. To understand why NKT cells were not sufficiently protective against tumor development in patients with tumor invaded liver, we characterized the diversity of Vα24/Vβ11 NKT cells in healthy donors (HD) and cancer patients: NKT cells from HD and patients were generally diverse in terms of TCR β chain (Vβ11) variability and NKT cells from HD showed a variable recognition of αGC loaded CD 1 d multimers. Vα24/ Vβ11 NKT cells can be divided in 3 populations, the CD4, DN (CD4-/CD8-) and CD8 NKT cell subsets that show distinct ability of cytokine production. In addition, our functional analysis revealed that DN and CD8 subsets displayed a higher cytolytic potential and a weaker IFNγ release than the CD4 NKT cell subset. NKT cell subsets were variably represented in the blood of HD and cancer patients. However, HD with high NKT cell frequencies displayed an enrichment of the DN and CD8 subsets, and few of them were suggestive of an oligoclonal expansion in vivo. Comparable NKT cell frequencies were found between blood, non-tumoral liver and tumor of patients. In contrast, we identified a gradual enrichment of CD4 NKT cells from blood to the liver and to the tumor, together with a decrease of DN and CD8 NKT cell subsets. Most patient derived NKT cells were unresponsive upon αGalactosylceramide stimulation ex vivo; NKT cells from few patients displayed a weak responsiveness with different cytokine polarization. The NKT cell repertoire was thus different in tumor tissue, suggesting that CD4 NKT cells infiltrating tumors may be detrimental for protection against tumors and instead may favour the tumor growth/recurrence as recently reported in mice. Résumé en français scientifique : Afin de développer le traitement des patients porteurs d'une tumeur dans le foie par immunothérapie, de nouvelles connaissances sont requises concernant l'expression d'antigènes par les tumeurs et les cellules immunitaires spécifiques de ces antigènes. Nous avons vérifié que des antigènes associés aux tumeurs, tels que les antigènes « Cancer-Testis » (CT), sont fréquemment exprimés par le carcinome hepatocéllulaire (CHC). La recherche de lymphocytes T CD8+ spécifiques (CTL) de ces antigènes a révélé que des CTL spécifiques de MAGE-A10 et/ou SSX-2 ont répondu naturellement à la tumeur chez 2/10 patients étudiés. Ces cellules étaient présentes dans les lésions tumorales mais pas dans le foie adjacent. De plus, ces CTL ont démontré une activité cytolytique forte et spécifique contre les cellules tumorales in vitro, ce qui suggère que ces CTL ont été sélectionnés pour une haute avidité de reconnaissance de l'antigène in vivo. Ces données fournissent une base pour l'immunothérapie spécifique du CHC, en proposant de cibler les antigènes CT tels que MAGE-A10 ou SSX-2. Les cellules NKT de type 1 ont une chaîne α de TCR qui est invariante (chez l'homme, Vα24Jα18, apparié avec Vβ11) et reconnaissent spécifiquement l'αGalactosylceramide (αGC) présenté par CD1d. Ces cellules ont des propriétés immuno¬régulatrices qui peuvent être parfois contradictoires et leur activation par l'αGC induit une forte protection anti-tumorale chez la souris: Afin de comprendre pourquoi ces cellules ne sont pas assez protectrices contre le développement des tumeurs dans le foie chez l'homme, nous avons étudié la diversité des cellules NKT Vα24/Vβ11 d'individus sains (IS) et de patients cancéreux. Les cellules NKT peuvent être sous-divisées en 3 populations : Les CD4, DN (CD4- /CD8-) ou CDS, qui ont la capacité de produire des cytokines différentes. Nos analyses fonctionnelles ont aussi révélé que les sous-populations DN et CD8 ont un potentiel cytolytique plus élevé et une production d'IFNγ plus faible que la sous-population CD4. Ces sous-populations sont représentées de manière variable dans le sang des IS ou des patients. Cependant, les IS avec un taux élevé de cellules NKT ont un enrichissement des sous- populations DN ou CDS, et certains suggèrent qu'il s'agit d'une expansion oligo-clonale in vivo. Les patients avaient des fréquences comparables de cellules NKT entre le sang, le foie et la tumeur. Par contre, la sous-population CD4 était progressivement enrichie du sang vers le foie et la tumeur, tandis que les sous-populations DN ou CD8 était perdues. La plupart des cellules NKT des patients ne réagissaient pas lors de stimulation avec l'αGC ex vivo et les cellules NKT de quelques patients répondaient faiblement et avec des polarisations de cytokines différentes. Ces données suggèrent que les cellules NKT CD4, prédominantes dans les tumeurs, sont inefficaces pour la lutte anti-tumorale et pourraient même favoriser la croissance ou la récurrence tumorale. Donc, une mobilisation spécifique des cellules NKT CD4 négatives par immunothérapie pourrait favoriser l'immunité contre des tumeurs chez l'homme. Résumé en français pour un large public Au sein des globules blancs, les lymphocytes T expriment un récepteur (le TCR), qui est propre à chacun d'entre eux et leur permet d'accrocher de manière très spécifique une molécule appelée antigène. Ce TCR est employé par les lymphocytes pour inspecter les antigènes associés avec des molécules présentatrices à la surface des autres cellules. Les lymphocytes T CD8 reconnaissent un fragment de protéine (ou peptide), qui est présenté par une des molécules du Complexe Majeur d'Histocompatibilité de classe I et tuent la cellule qui présente ce peptide. Ils sont ainsi bien adaptés pour éliminer les cellules qui présentent un peptide issu d'un virus quand la cellule est infectée. D'autres cellules T CD8 reconnaissent des peptides comme les antigènes CT, qui sont produits anormalement par les cellules cancéreuses. Nous avons confirmé que les antigènes CT sont fréquemment exprimés par le cancer du foie. Nous avons également identifié des cellules T CD8 spécifiques d'antigènes CT dans la tumeur, mais pas dans le foie normal de 2 patients sur 10. Cela signifie que ces lymphocytes peuvent être naturellement activés contre la tumeur et sont capables de la trouver. De plus les lymphocytes issus d'un patient ont démontré une forte sensibilité pour reconnaître l'antigène et tuent spécifiquement les cellules tumorales. Les antigènes CT représentent donc des cibles intéressantes qui pourront être intégrés dans des vaccins thérapeutiques du cancer du foie. De cette manière, les cellules T CD8 du patient lui-même pourront être induites à détruire de manière spécifique les cellules cancéreuses. Un nouveau type de lymphocytes T a été récemment découvert: les lymphocytes NKT. Quand ils reconnaissent un glycolipide présenté par la molécule CD1d, ils sont capables, de manière encore incomprise, d'initier, d'augmenter, ou à l'inverse d'inhiber la défense immunitaire. Ces cellules NKT ont démontré qu'elles jouent un rôle important dans la défense contre les tumeurs et particulièrement dans le foie des souris. Nous avons étudié les cellules NKT de patients atteints d'une tumeur dans le foie, afin de comprendre pourquoi elles ne sont pas assez protectrice chez l'homme. Les lymphocytes NKT peuvent être sous-divisés en 3 populations: Les CD4, les DN (CD4-/CD8-) et les CD8. Ces 3 classes de NKT peuvent produire différents signaux chimiques appelés cytokines. Contrairement aux cellules NKT DN ou CDS, seules les cellules NKT CD4 sont capables de produire des cytokines qui sont défavorables pour la défense anti-tumorale. Par ailleurs nous avons trouvé que les cellules NKT CD4 tuent moins bien les cellules cancéreuses que les cellules NKT DN ou CD8. L'analyse des cellules NKT, fraîchement extraites du sang, du foie et de la tumeur de patients a révélé que les cellules NKT CD4 sont progressivement enrichies du sang vers le foie et la tumeur. La large prédominance des NKT CD4 à l'intérieur des tumeurs suggère que, chez l'homme, ces cellules sont inappropriées pour la lutte anti-tumorale. Par ailleurs, la plupart des cellules NKT de patients n'étaient pas capables de produire des cytokines après stimulation avec un antigène. Cela explique également pourquoi ces cellules ne protègent pas contre les tumeurs dans le foie.