In this thesis we study the self-Floer theory of a monotone Lagrangian submanifold $L$ of a closed symplectic manifold $X$ in the presence of various kinds of symmetry. First we consider the group $\mathrm{Symp}(X, L)$ of symplectomorphisms of $X$ preserving $L$ setwise, and extend its action on the Oh spectral sequence to coefficients of arbitrary characteristic, working over an enriched Novikov ring. This imposes constraints on the differentials in the spectral sequence which force them to vanish in certain situations. We then specialise to the case where $L$ is $K$-homogeneous for a compact Lie group $K$, meaning roughly that $X$ is Kaehler, $K$ acts on $X$ by holomorphic automorphisms, and $L$ is a Lagrangian orbit. By studying holomorphic discs with boundary on $L$ we compute the image of low codimension $K$-invariant subvarieties of $X$ under the length zero closed-open string map. This places restrictions on the self-Floer cohomology of $L$ which generalise and refine the Auroux-Kontsevich-Seidel criterion. These often result in the need to work over fields of specific positive characteristics in order to obtain non-zero cohomology. The disc analysis is then developed further, with the introduction of the notion of poles and a reflection mechanism for completing holomorphic discs into spheres. This theory is applied to two main families of examples. The first is the collection of four Platonic Lagrangians in quasihomogeneous threefolds of $\mathrm{SL}(2, \mathbb{C})$, starting with the Chiang Lagrangian in $\mathbb{CP}^3$. These were previously studied by Evans and Lekili, who computed the self-Floer cohomology of the latter. We simplify their argument, which is based on an explicit construction of the Biran-Cornea pearl complex, and deal with the remaining three cases. The second is a family of $\mathrm{PSU}(n)$-homogeneous Lagrangians in products of projective spaces. Here the presence of both discrete and continuous symmetries leads to some unusual properties: in particular we obtain non-displaceable monotone Lagrangians which are narrow in a strong sense. We also discuss related examples including applications of Perutz's symplectic Gysin sequence and quilt functors. The thesis concludes with a discussion of directions for further research and a collection of technical appendices.

We give congruences modulo powers of 2 for the Fourier coefficients of certain level 2 modular functions with poles only at 0, answering a question posed by Andersen and Jenkins. The congruences involve a modulus that depends on the binary expansion of the modular form's order of vanishing at infinity. We also demonstrate congruences for Fourier coefficients of some level 4 modular functions.

Soit f un germe de fonction holomorphe dans deux variables qui s'annule à l'origine. L'ensemble zéro de cette fonction définit un germe de courbe analytique. Bien que la classification topologique d'un tel germe est bien connue depuis les travaux de Zariski, la classification analytique est encore largement ouverte. En 2012, Hefez et Hernandes ont résolu le cas irréductible et ont annoncé le cas de deux components. En 2015, Genzmer et Paul ont résolu le cas des fonctions topologiquement quasi-homogènes. L'objectif principal de cette thèse est d'étudier la première classe topologique de fonctions non quasi-homogènes. Dans le deuxième chapitre, nous décrivons l'espace local des modules des feuillages de cette classe et nous donnons une famille universelle de formes normales analytiques. Dans le même chapitre, nous prouvons l'unicité globale de ces formes normales. Dans le troisième chapitre, nous étudions l'espace des modules de courbes, qui est l'espace des modules des feuillages à une équivalence analytique des séparatrices associées près. En particulier, nous présentons un algorithme pour calculer sa dimension générique. Le quatrième chapitre présente une autre famille universelle de formes normales analytiques, qui est globalement unique aussi. En effet, il n'ya pas de modèle canonique pour la distribution de l'ensemble des paramètres sur les branches. Ainsi, avec cette famille, nous pouvons voir que la famille précédente n'est pas la seule et qu'il est possible de construire des formes normales en considérant une autre distribution des paramètres. Enfin, pour la globalisation, nous discutons dans le cinquième chapitre une stratégie basée sur la théorie géométrique des invariants et nous expliquons pourquoi elle ne fonctionne pas jusqu'à présent. / Let f be a germ of holomorphic function in two variables which vanishes at the origin. The zero set of this function defines a germ of analytic curve. Although the topological classification of such a germ is well known since the work of Zariski, the analytical classification is still widely open. In 2012, Hefez and Hernandes solved the irreducible case and announced the two components case. In 2015, Genzmer and Paul solved the case of topologically quasi-homogeneous functions. The main purpose of this thesis is to study the first topological class of non quasi-homogeneous functions. In chapter 2, we describe the local moduli space of the foliations in this class and give a universal family of analytic normal forms. In the same chapter, we prove the global uniqueness of these normal forms. In chapter 3, we study the moduli space of curves which is the moduli space of foliations up to the analytic equivalence of the associated separatrices. In particular, we present an algorithm to compute its generic dimension. Chapter 4 presents another universal family of analytic normal forms which is globally unique as well. Indeed, there is no canonical model for the distribution of the set of parameters on the branches. So, with this family, we can see that the previous family is not the only one and that it is possible to construct normal forms by considering another distribution of the parameters. Finally, concerning the globalization, we discuss in chapter 5 a strategy based on geometric invariant theory and explain why it does not work so far.

In this paper we study a class of non-linear singular partial differential equation in complex domain Csub(t) x C n sub(x). Under certain assumptions, we prove the existence and uniqueness of holomorphic solution near origin of Csub(t) x C n sub(x).

The paper deals with a non-linear singular partial differential equation: (E) t∂/∂t = F(t, x, u, ∂u/∂x) in the holomorphic category. When (E) is of Fuchsian type, the existence of the unique holomorphic solution was established by Gérard-Tahara [2]. In this paper, under the assumption that (E) is of totally characteristic type, the authors give a sufficient condition for (E) to have a unique holomorphic solution. The result is extended to higher order case.

The Cauchy–Riemann equations admit a bilinear multiplication of solutions, since the product of two holomorphic functions is again holomorphic. This multiplication plays the role of a nonlinear superposition principle for solutions, allowing for construction of new solutions from already known ones, and it leads to the exceptional property of the Cauchy–Riemann equations that all solutions can locally be built from power series of a single solution z = x + iy ∈ C. In this thesis we have found a differential algebraic characterization of linear first order systems of partial differential equations admitting a bilinear ∗-multiplication of solutions, and we have determined large new classes of systems having this property. Among them are the already known quasi-Cauchy–Riemann equations, characterizing integrable Newton equations, and the gradient equations ∇f = M∇g with constant matrices M. A systematic description of linear systems of PDEs with variable coefficients have been given for systems with few independent and few dependent variables. An important property of the ∗-multiplication is that infinite families of solutions can be constructed algebraically as power series of known solutions. For the equation ∇f = M∇g it has been proved that the general solution, found by Jodeit and Olver, can be locally represented as convergent power series of a single simple solution similarly as for solutions of the Cauchy–Riemann equations.

This thesis is devoted to the study of foliations that come from dynamical systems. In the first part we study foliations of Stein manifolds locally given by vector fields. The leaves of such foliations are Riemann surfaces. We prove that for a generic foliation all leaves except for not more than a countable number are homeomorphic to disks, the rest are homeomorphic to cylinders. We also prove that a generic foliation is complex Kupka-Smale. In the second part of the thesis we study complex H\'non maps. The sets of points $U^+$ and $U^-$ that have unbounded forward and backwards orbits correspondingly, is naturally endowed with holomorphic foliations $^+$ and $^-$. We describe the critical locus -- the set of tangencies between these foliations -- for H\'{e}non maps that are small perturbations of quadratic polynomials with disconnected Julia set.

This thesis is devoted to the study of foliations that come from dynamical systems. In the first part we study foliations of Stein manifolds locally given by vector fields. The leaves of such foliations are Riemann surfaces. We prove that for a generic foliation all leaves except for not more than a countable number are homeomorphic to disks, the rest are homeomorphic to cylinders. We also prove that a generic foliation is complex Kupka-Smale. In the second part of the thesis we study complex H\'non maps. The sets of points $U^+$ and $U^-$ that have unbounded forward and backwards orbits correspondingly, is naturally endowed with holomorphic foliations $^+$ and $^-$. We describe the critical locus -- the set of tangencies between these foliations -- for H\'{e}non maps that are small perturbations of quadratic polynomials with disconnected Julia set.

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Thesis (M. Phil.)--University of Hong Kong, 2002. / Includes bibliographical references (leaves 115-118).