Considérons une variété régulière analytique M sur le corps réel ou complexe, un faisceau d'idéaux J défini sur M, un diviseur à croisement normaux simples E et une distribution singulière involutive Θ tangent à E.L'objectif principal de ce travail est d'obtenir une résolution des singularités du faisceau d'idéaux J qui préserve certaines ``bonnes" propriétés de la distribution singulière Θ. Plus précisément, la propriété de R-monomialité : l'existence d'intégrales premières monomiales. Ce problème est naturel dans le contexte où on doit étudier l'interaction d'une variété et d'un feuilletage et, donc, est aussi reliée au problème de la monomilisation des applications et de résolution ``quasi-lisse" des familles d'idéaux.- Le premier résultat donne une résolution globale si le faisceau d'idéaux J est invariant par la distribution singulière;- Le deuxième résultat donne une résolution globale si la distribution singulière Θ est de dimension 1 ;- Le troisième résultat donne une uniformisation locale si la distribution singulière Θ est de dimension 2.On présente aussi deux utilisations des résultats précédents. La première application concerne la résolution des singularités en famille analytique, soit pour une famille d'idéaux, soit pour une famille de champs de vecteurs. Pour la deuxième, on applique les résultats à un problème de système dynamique, motivé par une question de Mattei. / Let M be an analytic manifold over the real or complex field, J be a coherent and everywhere non-zero ideal sheaf over M, E be a reduced SNC divisor and Θ an involutive singular distribution everywhere tangent to E. The main objective of this work is to obtain a resolution of singularities for the ideal sheaf J that preserves some ``good" properties of the singular distribution Θ. More precisely, the R-monomial property : the existence of local monomial first integrals. This problem arises naturally when we study the ``interaction" between a variety and a foliation and, thus, is also related with the problem of monomialization of maps and of ``quasi-smooth" resolution of families of ideal sheaves.- The first result is a global resolution if the ideal sheaf J is invariant by the singular distribution Θ;- The second result is a global resolution if the the singular distribution Θ has leaf dimension 1;- The third result is a local uniformization if the the singular distribution Θ has leaf dimension 2;We also present two applications of the previous results. The first application concerns the resolution of singularities in families, either of ideal sheaves or vector fields. For the second application, we apply the results to a dynamical system problem motivated by a question of Mattei.

We study the Hitchin component in the space of representations of the fundamental group of a Riemann surface into a split real Lie group in the rank 2 case. We prove that such representations are described by a conformal structure and class of Higgs bundle we call cyclic and we show cyclic Higgs bundles correspond to a form of the affine Toda equations. We also relate various real forms of the Toda equations to minimal surfaces in quadrics of arbitrary signature. In the case of the Hitchin component for PSL(3,R) we provide a new proof of the relation to convex RP²-structures and hyperbolic affine spheres. For PSp(4,R) we prove such representations are the monodromy for a special class of projective structure on the unit tangent bundle of the surface. We prove these are isomorphic to the convex-foliated projective structures of Guichard and Wienhard. We elucidate the geometry of generic 2-plane distributions in 5 dimensions, work which traces back to Cartan. Nurowski showed that there is an associated signature (2,3) conformal structure. We clarify this as a relationship between a parabolic geometry associated to the split real form of G₂ and a conformal geometry with holonomy in G₂. Moreover in terms of the conformal geometry we prove this distribution is the bundle of maximal isotropics corresponding to the annihilator of a spinor satisfying the twistor-spinor equation. The moduli space of deformations of a compact coassociative submanifold L in a G₂ manifold is shown to have a natural local embedding as a submanifold of H2(L,R). We consider G2-manifolds with a T^4-action of isomorphisms such that the orbits are coassociative tori and prove a local equivalence to minimal 3-manifolds in R^{3,3} = H²(T⁴,R) with positive induced metric. By studying minimal surfaces in quadrics we show how to construct minimal 3-manifold cones in R^{3,3} and hence G₂-metrics from equations that are a set of affine Toda equations. The relation to semi-flat special Lagrangian fibrations and the Monge-Ampere equation is explained.

The thesis deals with definability of certain Zariski geometries, introduced by Zilber, in the theory of algebraically closed fields. I axiomatise a class of structures, called 'abstract linear spaces', which are a common reduct of these Zariski geometries. I then describe what an interpretation of an abstract linear space in an algebraically closed field looks like. I give a new proof that the structure "quantum harmonic oscillator", introduced by Zilber and Solanki, is not interpretable in an algebraically closed field. I prove that a similar structure from an unpublished note of Solanki is not definable in an algebraically closed field and explain the non-definability of both structures in terms of geometric interpretation of the group law on a Galois cohomology group H¹(k(x), μ_n). I further consider quantum Zariski geometries introduced by Zilber and give necessary and sufficient conditions that a quantum Zariski geometry be definable in an algebraically closed field. Finally, I take an attempt at extending the results described above to complex-analytic setting. I define what it means for quantum Zariski geometry to have a complex analytic model, an give a necessary and sufficient conditions for a smooth quantum Zariski geometry to have one. I then prove a theorem giving a partial description of an interpretation of an abstract linear space in the structure of compact complex spaces and discuss the difficulties that present themselves when one tries to understand interpretations of abstract linear spaces and quantum Zariski geometries in the compact complex spaces structure.

No description available.

The Selmer varieties of a hyperbolic curve X over ℚ are refinements of the Selmer group arising from replacing the Tate module of the Jacobian with higher quotients of the unipotent étale fundamental group. It is hoped that these refinements carry extra arithmetic information. In particular the nonabelian Chabauty method developed by Kim uses the Selmer variety to give a new method to find the set X(ℚ). This thesis studies certain local and global properties of the Selmer varieties associated to finite dimensional quotients of the unipotent fundamental group of a curve over ℚ. We develop new methods to prove finiteness of the intersection of the Selmer varieties with the set of local points (and hence of the set of rational points) and new methods to implement this explicitly, giving the first examples of explicit nonabelian Chabauty theory for rational points on projective curves.

A key property of projective varieties is the very ampleness of line bundles as this provides embeddings into projective space and allows us to express the variety in equational terms. In this thesis we study the general version of this property which is k- very ampleness of line bundles. We introduce the notation of critical k-very ampleness and compute it for abelian surfaces. The property of k-very ampleness is usually discussed using tools from divisor theory but we take a different approach and use methods from derived algebraic geometry as part of program to use properties of the derived category of a variety to access the geometry of the variety. In particular, we use the Fourier-Mukai transform, moduli spaces of sheaves and properties of Bridgeland stability. We compute walls for certain Bridgeland stable spaces and certain Chern characters and to complete the picture we study the moduli spaces of torsion sheaves with minimal first Chern class and we go on to compute the walls for these as well building on tools developed earlier in the thesis.

Cette thèse est consacrée à l'étude d'une famille des flots géométriques associés au problème de la courbure scalaire prescrite sur une variété riemannienne compacte. Plus précisément, si on désigne par (M,g0) une variété riemannienne compacte de dimension n≥3, et si F∈C∞ (M) est une fonction donnée, le problème de la courbure scalaire prescrite consiste à trouver une métrique g conforme à g0 telle que F soit sa courbure scalaire. Ce problème est équivalent à la résolution de l'EDP suivante :-4 (n-1)/(n-2) ∆u+R0 u=Fu((n+2)/(n-2 )) , u>0 , (E), Où R0 est la courbure scalaire de la métrique initiale g0 et ∆ est le laplacien associé à g0. Il s'agit d'une équation elliptique non-linéaire dont la difficulté principale provient du terme u((n+2)/(n-2 )). Hormis le cas de la sphère standard Sn , tous les travaux consacrés à l'étude de l'équation (E) sont basés sur la méthode variationnelle. Dans cette thèse, on développe une autre approche basée sur l'étude d'une famille de flots géométriques qui permet, entre autres, de résoudre l'équation (E). La question dépend bien entendu de la métrique initiale g0 et en particulier du signe de sa courbure scalaire R0. Les flots introduits sont des flots de gradient associés à deux fonctionnelles distinctes dépendant du signe de R0. La première partie de cette thèse est consacrée au cas R0<0 et dans la deuxième partie on traite le cas R0>0. Dans les deux cas, on démontre l'existence globale du flot et on étudie son comportement asymptotique à l'infini. / This thesis is devoted to the study of a family of geometric flows associated with the prescribed scalar curvature problem. More precisely, if we denote by (M,g0) a compact riemannian manifold with dimension n≥3, and if F∈C∞ (M) is a given function, the prescribed scalar curvature problem consists of finding a conformal metric g to g0 such that F is its scalar curvature. This problem is equivalent to the resolution of the following PDE : -4 (n-1)/(n-2) ∆u+R0 u=Fu((n+2)/(n-2 )) , u>0 , (E), Where R0 is the scalar curvature of the initial metric g0 and ∆ is the laplacian associated with g0.It is a nonlinear elliptic equation, whose the main difficulty comes from the term u((n+2)/(n-2 )). Apart from the case of the standard sphere Sn all the works that study the equation (E) are based on the variational method. In this thesis, we develop another approach based on the study of a family of geometric flows which allows to solve equation (E).The flows introduced are gradient flows associated with two distinct functional functions depending on the sign of R0.The first part of this thesis is devoted to the case R0<0 and in the second part we treat the case R0>0. In both cases, our aim is to proof the global existence of the flow and study its asymptotic behavior at infinity.

In this thesis we study the problem of determining the possible cobordism types of r-fold self-intersection manifolds associated to self-transverse immersions f: M^{n-k} -> \R^n for certain values of n, k, and r. Namely, we study the double-point self-intersection manifolds of immersions M^{n+2} -> \R^{2n+2} and M^{n+4} -> \R^{2n+4}, focusing on the case when $n$ is even. In the case of self-transverse immersions f : M^{n+2} -> \R^{2n+2}, we see that when n is even the double-point self-intersection manifold is a boundary, which is a result originally due to Szucs. In the case of self-transverse immersions f : M^{n+4} -> \R^{2n+4}, we show than when n is even the double-point self-intersection manifold is either a boundary or cobordant to RP^2 x RP^2, which is a new result. We then show that for even n such that the binary expansion of n+4 contains 5 or more 1s, the double-point self-intersection manifold of a self-transverse immersion M^{n+4} -> \R^{2n+4} is necessarily a boundary. We also survey the case when n is odd. We also set up and study the complex versions of the above problems: self-transverse immersions f : M^{2k+2} -> \R^{4k+2} and f : M^{2k+4} -> \R^{4k+4} of stably complex manifolds with a given complex structure on the normal bundle of f$. In these cases, the double-point self-intersection manifold L associated to the immersion inherits a stably complex structure, and we attempt to determine which complex cobordism classes of stably complex manifolds may arise in this way. This is all new work. In the case of self-transverse complex immersions f : M^{2k+2} -> \R^{4k+2}, we show that the first normal Chern number of the double-point self-intersection manifold is a multiple of 2^{\lambda_{k+1}} for some integer \lambda_{k+1}, and provide upper and lower bounds for the value of \lambda_{k+1}. We also determine the exact value of \lambda_{k+1} in certain cases. In the case of self-transverse complex immersions f : M^{2k+4} -> \R^{4k+4}, we identify a large class of stably complex manifolds that may arise as the double-point self-intersection manifold of such an immersion and also identify a class of manifolds that may not. Additionally, in both cases we identify a necessary (and sometimes sufficient) condition for a stably complex manifold of the appropriate dimension to admit a complex immersion of the appropriate codimension.

This thesis studies derived equivalences between total spaces of vector bundles and dg-quivers. A dg-quiver is a graded quiver whose path algebra is a dg-algebra. A quiver with superpotential is a dg-quiver whose differential is determined by a "function" Φ. It is known that the bounded derived category of representations of quivers with superpotential with finite dimensional cohomology is a Calabi- Yau triangulated category. Hence quivers with superpotential can be viewed as noncommutative Calabi- Yau manifolds. One might then ask if there are derived equivalences between Calabi-Yau manifolds and quivers with superpotential. In this thesis, we answer this question and, generalizing Bridgeland [15], give a recipe on how to construct such derived equivalences.

The purpose of this thesis is to define a “local” version of Ozsváth and Szabó’s Heegaard Floer homology HFL^ for links in the 3-sphere, i.e. a Heegaard Floer homology HFT^ for tangles in the 3-ball. The decategorification of HFL^ is the classical Alexander polynomial for links; likewise, the decategorification of HFT^ gives a local version ∇ˢ of the Alexander polynomial. In the first chapter of this thesis, we give a purely combinatorial definition of this polynomial invariant ∇ˢ via Kauffman states and Alexander codes and investigate some of its properties. As an application, we show that the multivariate Alexander polynomial is mutation invariant. In the second chapter, we define HFT^ in two slightly different, but equivalent ways: One is via Juhász’s sutured Floer homology, the other by imitating the construction of HFL^. We then state a glueing theorem in terms of Zarev’s bordered sutured Floer homology, which endows HFT^ with additional structure. As an application, we show that any two links related by mutation about a (2,−3)-pretzel tangle have the same δ-graded link Floer homology. This result relies on a computer calculation. In the third and last chapter, we specialise to 4-ended tangles. In this case, we give a reformulation of HFT^ with a glueing structure in terms of (what we call) peculiar modules. Together with a glueing theorem, we can easily recover oriented and unoriented skein relations for HFL^. Our peculiar modules also enjoy some symmetry relations, which support a conjecture about δ-graded mutation invariance of HFL^. However, stronger symmetries would be needed to actually prove this conjecture. Finally, we explore the relationship between peculiar modules and twisted complexes in the wrapped Fukaya category of the 4-punctured sphere. There are four appendices, some of which might be of independent interest: In the first appendix, we describe a general construction of dg categories which unifies all algebraic structures used in this thesis, in particular type A and type D modules from bordered theory. In the second appendix, we prove a generalised version of Kauffman’s clock theorem, which plays a major role for our decategorified invariants. The last two appendices are manuals for two Mathematica programs. The first is a tool for computing the generators of HFT^ and the decategorified tangle invariant ∇ˢ. The second allows us to compute bordered sutured Floer homology using nice diagrams.