Cette thèse se compose de plusieurs travaux portant sur l'énumération et le comportement asymptotique de structures combinatoires apparentées aux partitions d'entiers. Un premier travail s'intéresse aux partitions d'entiers bipartites, qui constituent une généralisation bidimensionnelle des partitions d'entiers. Des équivalents du nombre de partitions sont obtenus dans le régime critique où l'un des entiers est de l'ordre du carré de l'autre entier et au delà de ce régime critique. Ceci complète les résultats établis dans les années cinquante par Auluck, Nanda et Wright. Le deuxième travail traite des chaînes polygonales à sommets entiers dans le plan. Pour un modèle statistique introduit par Sinaï, une représentation intégrale exacte de la fonction de partition est donnée. Ceci conduit à un équivalent du nombre de chaînes joignant deux points distants qui fait intervenir les zéros non triviaux de la fonction zêta de Riemann. Une analyse combinatoire détaillée des chaînes convexes est présentée. Elle permet de montrer l'existence d'une forme limite pour les chaînes convexes aléatoires ayant peu de sommets, répondant ainsi à une question ouverte de Vershik. Un troisième travail porte sur les zonotopes à sommets entiers en dimension supérieure. Un équivalent simple est donné pour le logarithme du nombre de zonotopes contenus dans un cône convexe et dont les extrémités sont fixées. Une loi des grands nombres est établie et la forme limite est caractérisée par la transformée de Laplace du cône. / This thesis consists of several works dealing with the enumeration and the asymptotic behaviour of combinatorial structures related to integer partitions. A first work concerns partitions of large bipartite integers, which are a bidimensional generalization of integer partitions. Asymptotic formulæ are obtained in the critical regime where one of the numbers is of the order of magnitude of the square of the other number, and beyond this critical regime. This completes the results established in the fifties by Auluck, Nanda, and Wright. The second work deals with lattice convex chains in the plane. In a statistical model introduced by Sinaï, an exact integral representation of the partition function is given. This leads to an asymptotic formula for the number of chains joining two distant points, which involves the non trivial zeros of the Riemann zeta function. A detailed combinatorial analysis of convex chains is presented. It makes it possible to prove the existence of a limit shape for random convex chains with few vertices, answering an open question of Vershik. A third work focuses on lattice zonotopes in higher dimensions. An asymptotic equality is given for the logarithm of the number of zonotopes contained in a convex cone and such that the endings of the zonotope are fixed. A law of large numbers is established and the limit shape is characterized by the Laplace transform of the cone.

Plusieurs familles de fonctions spéciales de plusieurs variables, appelées fonctions d'orbites, sont définies dans le contexte des groupes de Weyl de groupes de Lie simples compacts/d'algèbres de Lie simples. Ces fonctions sont étudiées depuis près d'un siècle en raison de leur lien avec les caractères des représentations irréductibles des algèbres de Lie simples, mais également de par leurs symétries et orthogonalités. Nous sommes principalement intéressés par la description des relations d'orthogonalité discrète et des transformations discrètes correspondantes, transformations qui permettent l'utilisation des fonctions d'orbites dans le traitement de données multidimensionnelles. Cette description est donnée pour les groupes de Weyl dont les racines ont deux longueurs différentes, en particulier pour les groupes de rang $2$ dans le cas des fonctions d'orbites du type $E$ et pour les groupes de rang $3$ dans le cas de toutes les autres fonctions d'orbites. / Several families of multivariable special functions, called orbit functions, are defined in the context of Weyl groups of compact simple Lie groups/Lie algebras. These functions have been studied for almost a century now because of their relation to characters of irreducible representations of Lie algebras, their symmetries and orthogonalities. Our main interest is the description of discrete orthogonality relations and their corresponding discrete transforms which allow the applications of orbit functions in the processing of multidimensional data. This description is provided for the Weyl group of different lengths of root, in particular groups of rank 2 for so-called $E-$orbit functions and of rank 3 for all the other families of special functions.

L'approximation des convexes lisses par des polytopes pour la distance de Hausdorff a connu de nombreux résultats théoriques grâce à l'apport de la géométrie riemannienne. Nous rappelons ces résultats portant principalement sur le comportement asymptotique et montrons leur utilité pour certains cas pratiques. Puis nous établissons notre résultat principal, à savoir que ce problème d'approximation d'un convexe est, en un sens bien précis, équivalent à celui de l'approximation d'une norme par une autre. Nous établissons ensuite les propriétés d'un produit d'approximations de normes, ce qui nous permet de construire par récurrence sur la dimension des polytopes approchant certains convexes lisses, ainsi que des approximations optimales des normes Lp. Enfin nous montrons à travers différentes applications à la géométrie algorithmique en quoi une approximation de norme permet de transformer un algorithme de résolution exacte en un algorithme de résolution approchée mais moins coûteux.

The main subject of this thesis is the development of new geometric tools and techniques for solving classic problems within the geometry of numbers and convex geometry. At a high level, the problems considered in this thesis concern the varied interplay between the continuous and the discrete, an important theme within computer science and operations research. The first subject we consider is the study of cutting planes for non-linear integer programs. Cutting planes have been implemented to great effect for linear integer programs, and so understanding their properties in more general settings is an important subject of study. As our contribution to this area, we show that Chvatal-Gomory closure of any compact convex set is a rational polytope. As a consequence, we resolve an open problem of Schrijver (Ann. Disc. Math. `80) regarding the same question for irrational polytopes. The second subject of study is that of ellipsoidal approximation of convex bodies. Different such notions have been important to the development of fundamental geometric algorithms: e.g. the ellipsoid method for convex optimization (enclosing ellipsoids), or random walk methods for volume estimation (inertial ellipsoids). Here we consider the construction of an ellipsoid with good "covering" properties with respect to a convex body, known in convex geometry as the M-ellipsoid. As our contribution, we give two algorithms for constructing M-ellipsoids, and provide an application to near-optimal deterministic volume estimation in the oracle model. Equipped with this new geometric tool, we move to the study of classic lattice problems in the geometry of numbers, namely the Shortest (SVP) and Closest Vector Problems (CVP). Here we use M-ellipsoid coverings, combined with an algorithm of Micciancio and Voulgaris for CVP in the ℓ₂ norm (STOC `10), to obtain the first deterministic 2^O(ⁿ) time algorithm for the SVP in general norms. Combining this algorithm with a novel lattice sparsification technique, we derive the first deterministic 2^O(ⁿ)(1+1/ϵ)ⁿ time algorithm for (1+ϵ)-approximate CVP in general norms. For the next subject of study, we analyze the geometry of general integer programs. A central structural result in this area is Kinchine's flatness theorem, which states that every lattice free convex body has integer width bounded by a function of dimension. As our contribution, we build on the work Banaszczyk, using tools from lattice based cryptography, to give a new and tighter proof of the flatness theorem. Lastly, combining all the above techniques, we consider the study of algorithms for the Integer Programming Problem (IP). As our main contribution, we give a new 2^O(ⁿ)nⁿ time algorithm for IP, which yields the fastest currently known algorithm for IP and improves on the classic works of Lenstra (MOR `83) and Kannan (MOR `87).

Un système intégrable semi-torique sur une variété symplectique de dimension 2n est un système intégrable dont le flot de n − 1 composantes de l'application moment est 2 -périodique. On obtient donc une action hamiltonienne du tore Tn−1. En outre, on demande que tous les points critiques du système soient non-dégénérés et sans composante hyperbolique. En dimension 4, San V˜u Ngo.c et Álvaro Pelayo ont étendu à ces systèmes semi-toriques les résultats célèbres d'Atiyah, Guillemin, Sternberg et Delzant concernant la classification des systèmes toriques. Dans cette thèse nous proposons une extension de ces résultats en dimension quelconque, à commencer par la dimension 6. Les techniques utilisées relèvent de l'analyse comme de la géométrie symplectique, ainsi que de la théorie de Morse dans des espaces différentiels stratifiés. Nous donnons d'abord une description de l'image de l'application moment d'un point de vue local, en étudiant les asymptotiques des coordonnées actionangle au voisinage d'une singularité foyer-foyer, avec le phénomène de monodromie du feuilletage qui en résulte. Nous passons ensuite à une description plus globale dans la veine des polytopes d'Atiyah, Guillemin et Sternberg. Ces résultats sont basés sur une étude systématique de la stratification donnée par les fibres de l'application moment. Avec ces résultats, nous établissons la connexité des fibres des systèmes intégrables semi-toriques de dimension 6 et indiquons comment nous comptons démontrer ce résultat en dimension quelconque.

Plusieurs familles de fonctions spéciales de plusieurs variables, appelées fonctions d'orbites, sont définies dans le contexte des groupes de Weyl de groupes de Lie simples compacts/d'algèbres de Lie simples. Ces fonctions sont étudiées depuis près d'un siècle en raison de leur lien avec les caractères des représentations irréductibles des algèbres de Lie simples, mais également de par leurs symétries et orthogonalités. Nous sommes principalement intéressés par la description des relations d'orthogonalité discrète et des transformations discrètes correspondantes, transformations qui permettent l'utilisation des fonctions d'orbites dans le traitement de données multidimensionnelles. Cette description est donnée pour les groupes de Weyl dont les racines ont deux longueurs différentes, en particulier pour les groupes de rang $2$ dans le cas des fonctions d'orbites du type $E$ et pour les groupes de rang $3$ dans le cas de toutes les autres fonctions d'orbites. / Several families of multivariable special functions, called orbit functions, are defined in the context of Weyl groups of compact simple Lie groups/Lie algebras. These functions have been studied for almost a century now because of their relation to characters of irreducible representations of Lie algebras, their symmetries and orthogonalities. Our main interest is the description of discrete orthogonality relations and their corresponding discrete transforms which allow the applications of orbit functions in the processing of multidimensional data. This description is provided for the Weyl group of different lengths of root, in particular groups of rank 2 for so-called $E-$orbit functions and of rank 3 for all the other families of special functions.

No description available.

A range space (P, S) consists of a set P of n elements and a collection S = {S1,...,Sm} of subsets of P , referred to as ranges. A hitting set for this range space refers to a subset H of P such that every Si in S contains at least one element of H. The hitting set problem is studied for many geometric range spaces where P is a set of n points in Rd and the ranges are defined by the intersection of geometric objects with P . The algorithmic question of finding the minimum-sized hitting set for a given range space is well studied and is NP-Complete even for geometric range spaces defined by unit disks. The dual of the hitting set problem is the equally well studied set cover problem. A set cover is a sub-collection C⊆S such that every element in P is contained in at least one range in C. A classic problem which is related to the minimum set cover problem is the maximum coverage problem. In this problem, given a range space (P, S) and an integer k, we have to find k ranges from S such that the number of elements in P that are covered by these k ranges are maximized. In this thesis, we study combinatorial questions on a similar variant of hitting set problem for specific geometric range spaces where the size of the hitting set is fixed as a small constant. We study the small hitting set problem mainly for two broad classes of range spaces. We first consider the Dense range space (P, S) where P is a set of n points in Rd and S is defined by “dense” geometric objects i.e, geometric objects that contain more than a constant fraction, say �, of points from P . We fix the size of the hitting set as a small constant k and investigate bounds on the value of � such that all ranges that contain more than �n points from P are hit. Next we consider the Induced range space (P, I) where P isa setof n points in R2 and the ranges are all geometric objects that are induced(spanned) by P i.e., the ranges are defined by geometric objects that have a distinct tuple of points from P on its boundary. For Induced range spaces, we prove bounds on the maximum number of ranges that can be hit using a single point. We also prove combinatorial bounds on the size of the hitting set for various families of induced objects. Now, we describe the problems that we study in this thesis and summarize the results obtained. 1. Strong centerpoints: Here we study the small hitting set question for dense range spaces when the size of hitting set is one. This is related to a classic result in geometry called Centerpoint Theorem. A point x ∈ Rd is said to be the centerpoint of P if x is contained in all convex objects that contain more than dn points from P . Centerpoint Theorem states d+1 that a centerpoint always exists for any point set P . A centerpoint need not be an input point. A natural question to ask is the following: Does there exist a strong centerpoint? i.e., is it true that for any given point set P there exists a point p ∈ P such that p is contained in every convex object that contains more than a constant fraction, say �, of points of P ? It can be easily seen that a strong centerpoint does not exist even for geometric range spaces defined by half spaces. We study the existence and the corresponding bounds for strong centerpoints for some range spaces. In particular, we prove the existence of strong centerpoint and show tight bounds for the following range spaces. Convex polytopes defined by a fixed set of orientations : Geometric range spaces like those induced by axis-parallel boxes, skylines and downward facing equilateral triangles belong to this family of convex polytopes. Hyperplanes in Rd Range spaces with discrete intersection 2. Small Strong Epsilon Nets: This can be considered as an extension of strong centerpoints. This question is related to a well studied area called �-nets. N ⊂ P is called a (strong) �-net of P with respect to S if N ∩ S =�∅ for all objects S ∈S that contain more than �n points of P . We study the following question. Let �S∈ [0, 1] represent the smallest real number such that, for any given point set P , there exists Q ⊂ P of size i which is an �S-net with respect to S. Thus a strong centerpoint will be an �S1 -net. We prove bounds on �Si for small values of i where S is the family of axis-parallel rectangles, halfspaces and disks. 3. Strong First Selection Lemma: Here we consider the hitting question for induced range spaces when the size of the hitting set is one. In other words, given an induced range space, we prove bounds on the maximum number of ranges that can be hit using a single input point. Such questions are referred to as First Selection Lemma and are well studied. We consider the strong version of the First Selection Lemma where the “heavily covered” point is required to be an input point. We study the strong first selection lemma for induced rectangles, induced special rectangles and induced disks. We prove an exact result for the strong variant of the first selection lemma for axis-parallel rectangles. We also prove exact results for the strong variant of the first selection lemma for some subclasses of axis-parallel rectangles like orthants and slabs. We prove strong first selection lemma with almost tight bounds for skylines, another sub-class of axis-parallel rectangles. We prove bounds for first selection lemma for disks in the plane and exact results for a special case when the discs are induced by a centrally symmetric point set. 2 Hitting all Induced Objects: Here we discuss and prove combinatorial bounds on the size of the minimum hitting set for induced range spaces. We prove tight bounds on the hitting set size when induced objects are special rectangles, disks and downward facing equilateral triangles.

In this thesis, we undertake a combinatorial study of certain aspects of triangulations of cartesian products of simplices, particularly in relation to their relevance in toric algebra and to their underlying product structure. The first chapter reports joint work with Samu Potka. The object of study is a class of homogeneous toric ideals called cut ideals of graphs, that were introduced by Sturmfels and Sullivant 2006. Apart from their inherent appeal to combinatorial commutative algebra, these ideals also generalize graph statistical models for binary data and are related to some statistical models for phylogenetic trees. Specifically, we consider minimal free resolutions for the cut ideals of trees. We propose a method to combinatorially estimate the Betti numbers of the ideals in this class. Using this method, we derive upper bounds for some of the Betti numbers, given by formulas exponential in the number of vertices of the tree. Our method is based on a common technique in commutative algebra whereby arbitrary homogeneous ideals are deformed to initial monomial ideals, which are easier to analyze while conserving some of the information of the original ideals. The cut ideal of a tree on n vertices turns out to be isomorphic to the Segre product of the cut ideals of its n-1 edges (in particular, its algebraic properties do not depend on its shape). We exploit this product structure to deform the cut ideal of a tree to an initial monomial ideal with a simple combinatorial description: it coincides with the edge ideal of the incomparability graph of the power set of the edges of the tree. The vertices of the incomparability graph are subsets of the edges of the tree, and two subsets form an edge whenever they are incomparable. In order to obtain algebraic information about these edge ideals, we apply an idea introduced by Dochtermann and Engström in 2009 that consists in regarding the edge ideal of a graph as the (monomial) Stanley-Reisner ideal of the independence complex of the graph. Using Hochster\''s formula for computting Betti numbers of Stanley-Reisner ideals by means of simplicial homology, the computation of the Betti numbers of these monomial ideals is turned to the enumeration of induced subgraphs of the incomparability graph. That the resulting values give upper bounds for the Betti numbers of the cut ideals of trees is an important well-known result in commutative algebra. In the second chapter, we focus on some combinatorial features of triangulations of the point configuration obtained as the cartesian product of two standard simplices. These were explored in collaboration with César Ceballos and Arnau Padrol, and had a two-fold motivation. On the one hand, we intended to understand the influence of the product structure on the set of triangulations of the cartesian product of two point configurations; on the other hand, the set of all triangulations of the product of two simplices is an intricate and interesting object that has attracted attention both in discrete geometry and in other fields of mathematics such as commutative algebra, algebraic geometry, enumerative geometry or tropical geometry. Our approach to both objectives is to examine the circumstances under which a triangulation of the polyhedral complex given by the the product of an (n-1)-simplex times the (k-1)-skeleton of a (d-1)-simplex extends to a triangulation of an (n-1)-simplex times a (d-1)-simplex. We refer to the former as a partial triangulation of the product of two simplices. Our main result says that if d >= k > n, a partial triangulation always extends to a uniquely determined triangulation of the product of two simplices. A somewhat unexpected interpretation of this result is as a finiteness statement: it asserts that if d is sufficiently larger than n, then all partial triangulations are uniquely determined by the (compatible) triangulations of its faces of the form “(n-1)-simplex times n-simplex”. Consequently, one can say that in this situation ‘\''triangulations of an (n-1)-simplex times a (d-1)-simplex are not much more complicated than triangulations of an (n-1)-simplex times an n-simplex\''\''. The uniqueness assertion of our main result holds already when d>=k>=n. However, the same is not true for the existence assertion; namely, there are non extendable triangulations of an (n-1)-simplex times the boundary of an n-simplex that we explicitly construct. A key ingredient towards this construction is a triangulation of the product of two (n-1)-simplices that can be seen as its ``second simplest triangulation\''\'' (the simplest being its staircase triangulation). It seems to be knew, and we call it the Dyck path triangulation. This triangulation displays symmetry under the cyclic group of order n that acts by simultaneously cycling the indices of the points in both factors of the product. Next, we exhibit a natural extension of the Dyck path triangulation to a triangulation of an (n-1)-simplex times an n-simplex that, in a sense, enjoys some sort of ‘\''rigidity\''\'' (it also seems new). Performing a ‘\''local modification\''\'' on the restriction of this extended triangulation to the polyhedral complex given by (n-1)-simplex times the boundary of an n-simplex yields the non-extendable partial triangulation. The thesis includes two appendices on basic commutative algebra and triangulations of point configuration, included to make it slightly self-contained.

Dans ce travail, nous nous intéressons à l'estimation d'ensembles convexes dans l'espace Euclidien R^d, en nous penchant sur deux modèles. Dans le premier modèle, nous avons à notre disposition un échantillon de n points aléatoires, indépendants et de même loi, uniforme sur un ensemble convexe inconnu. Le second modèle est un modèle additif de régression, avec bruit sous-gaussien, et dont la fonction de régression est l'indicatrice d'Euler d'un ensemble convexe ici aussi inconnu. Dans le premier modèle, notre objectif est de construire un estimateur du support de la densité des observations, qui soit optimal au sens minimax. Dans le second modèle, l'objectif est double. Il s'agit de construire un estimateur du support de la fonction de régression, ainsi que de décider si le support en question est non vide, c'est-'a-dire si la fonction de régression est effectivement non nulle, ou si le signal observé n'est que du bruit. Dans ces deux modèles, nous nous intéressons plus particulièrement au cas où l'ensemble inconnu est un polytope convexe, dont le nombre de sommets est connu. Si ce nombre est inconnu, nous montrons qu'une procédure adaptative permet de construire un estimateur atteignant la même vitesse asymptotique que dans le cas précédent. Enfin, nous démontrons que ce m$eme estimateur pallie à l'erreur de spécification du modèle, consistant à penser à tort que l'ensemble convexe inconnu est un polytope. Nous démontrons une inégalité de déviation pour le volume de l'enveloppe convexe des observations dans le premier modèle. Nous montrons aussi que cette inégalité implique des bornes optimales sur les moments du volume manquant de cette enveloppe convexe, ainsi que sur les moments du nombre de ses sommets. Enfin, dans le cas unidimensionnel, pour le second modèle, nous donnons la taille asymptotique minimale que doit faire l'ensemble inconnu afin de pouvoir être détecté, et nous proposons une règle de décision, permettant un test consistant du caractère non vide de cet ensemble.