Neste trabalho, nos concentramos no estudo da topologia da fibração de Milnor associada a um germe de aplicação polinomial f : (Rn , 0) → (Rp , 0) com uma singularidade isolada na origem. O primeiro resultado é uma extensão da caracterização de germes de aplicações triviais nos pares de dimensões (n; p) quando n - p = 3: Uma caracterização inicial foi apresentada por Church e Lamotke em 1975. O segundo resultado é a caracterização de NS-pares (S5 , K2), usando a topologia de espaços de configuração. Como uma consequência desta caracterização, mostramos a existência de germe de aplicação polinomial real nos pares de dimensões (6; 3) com uma singularidade isolada na origem tal que sua fibra de Milnor não é difeomorfa a um disco. A existência desses exemplos coloca um fim ao problema da não-trivialidade proposto por Milnor em 1968 e além disso, nos permite apresentar um novo resultado sobre a topologia da fibra de Milnor real nos pares de dimensões (2n; n) e (2n + 1; n); n ≥ 3: Tal resultado garante a existência de germes de aplicações polinomiais (Rn , 0) → (Rp, 0); n ≥ p ≥ 2; com uma singularidade isolada na origem tais que suas fibras de Milnor têm o tipo de homotopia de um buquê de um número positivo de esferas. / In this work, we focus on the study of the topology of the Milnor fibration associated with a polynomial map germ f : (Rn , 0) → (Rp , 0) with an isolated singularity at the origin. The first result is an extension of the characterization of trivial map germs in the pairs of dimensions (n; p) when n - p = 3: An initial characterization was presented by Church and Lamotke in 1975. The second result is a characterization of NS-pairs (S5 , K2), using the topology of configuration spaces. As a consequence of this characterization, we show the existence of real polynomial map germs in the pairs of dimensions (6; 3) with an isolated singularity at the origin such that its Milnor fibers are not diffeomorphic to a disc. The existence of such examples ends a non-triviality problem posed by Milnor in 1968 and furthermore, it allows us to show a new result about the topology of the real Milnor fibers in the pairs of dimensions (2n; n) and (2n + 1; n); n ≥ 3. This result ensure the existence of polynomial map germs (Rn , 0) → (Rp, 0); n ≥ p ≥ 2; with an isolated singularity at the origin such that its Milnor fibers has the homotopy type of a bouquet of a positive number of spheres.

Ce mémoire de thèse présente les travaux effectués pour la réalisation de sources optiques fibrées d’impulsions picosecondes cadencées à 40 GHz dans la bande C des télécommunications. Dans une première partie, nous présentons des études numériques et expérimentales mises en place pour la génération d’un train d’impulsions cadencé à 40 GHz par la compression non-linéaire d’un battement sinusoïdal via un processus de mélanges à quatre ondes multiples. Afin d’obtenir des impulsions stables, le battement sinusoïdal initial est obtenu par la modulation en intensité d’un signal continu grâce à un modulateur Mach-Zehnder piloté au point nul de transmission. Nous démontrons également l’amélioration de la qualité des impulsions générées par la suppression de la diffusion Brillouin stimulée grâce à la mise en place d’isolateurs optiques dans la ligne fibrée de la source. Nous présentons ensuite la génération d’impulsions ultra-courtes grâce à un compresseur non-linéaire composé de quatre étages fibrés. Le train d’impulsions obtenu est alors codé puis multiplexé jusqu’à un débit optique de 160 Gbit/s. Dans la dernière partie, nous présentons les démarches mises en place en vue d’un transfert technologique, telles que la réalisation d’un prototype de la source, la recherche d’antériorité et l’étude de marché. / This thesis presents the work carried out on the realization of fibered 40-GHz picosecond optical pulse sources in the telecommunications C-band. In the first part, we present a numerical and experimental study of the generation of 40-GHz pulse trains thanks to the nonlinear compression of an initial beat-signal by multiple Four-Wave Mixing process. Enhanced temporal stability is achieved by generating the sinusoidal beating thanks to a Mach-Zehnder modulator driven at its zero-transmission working point. In order to improve the quality of the generated pulses, we also demonstrate the suppression of stimulated Brillouin back-scattering by inserting several optical isolators into the compression line. In the next part, we present the generation of low duty-cycle pulse trains by using a nonlinear compressor line based on 4 segments of fiber. The generated pulse trains have been encoded and then multiplexed to achieve a high bit rate signal (160 Gb/s). In the last part, we present the technology transfer steps of this optical source, such as creating a prototype of the source, prior art search and market research.

The 4-genus of a knot in S3 is an important measure of complexity, related to the unknotting number. A fundamental result used to study the 4-genus and related invariants of homology classes is the Thom conjecture, proved by Kronheimer-Mrowka, and its symplectic extension due to Ozsv´ath-Szab´o, which say that closed symplectic surfaces minimize genus. In this thesis, we prove a relative version of the symplectic capping theorem. More precisely, suppose (X, ω) is a symplectic 4-manifold with contact type bounday ∂X and Σ is a symplectic surface in X such that ∂Σ is a transverse knot in ∂X. We show that there is a closed symplectic 4-manifold Y with a closed symplectic submanifold S such that the pair (X, Σ) embeds symplectically into (Y, S). This gives a proof of the relative version of Symplectic Thom Conjecture. We use this to study 4-genus of fibered knots in S3 . We also prove a relative version of the sufficiency part of Giroux’s criterion for Stein fillability, namely, we show that a fibered knot whose mondoromy is a product of positive Dehn twists bounds a symplectic surface in a Stein filling. We use this to study 4-genus of fibered knots in S3 . Using this result, we give a criterion for quasipostive fibered knots to be strongly quasipositive. Symplectic convexity disc bundles is a useful tool in constructing symplectic fillings of contact manifolds. We show the symplectic convexity of the unit disc bundle in a Hermitian holomorphic line bundle over a Riemann surface.

It goes back to Ahlfors that a real algebraic curve admits a real-fibered morphism to the projective line if and only if the real part of the curve disconnects its complex part. Inspired by this result, we are interested in characterising real algebraic varieties of dimension n admitting real-fibered morphisms to the n-dimensional projective space. We present a criterion to classify real-fibered morphisms that arise as finite surjective linear projections from an embedded variety which relies on topological linking numbers. We address special attention to real algebraic surfaces. We classify all real-fibered morphisms from real del Pezzo surfaces to the projective plane and determine which such morphisms arise as the composition of a projective embedding with a linear projection. Furthermore, we give some insights in the case of real conic bundles.