Proof Theory of Graph Minors and Tree Embeddings

This thesis explores metamathematical properties of theorems appearing in the Graph Minors series. A number of these theorems have been known to have very high proof-theoretic strength, but an upper bound on many of them, including the graph minor theorem, had never been proved. We give such upper bounds, by showing that any proofs in the Graph Minors series can be carried out within a system of Pi^1_1-comprehension augmented with induction and bar-induction principles for certain classes of formulas. This establishes a narrow corridor for the possible proof-theoretic strength of many strong combinatorial principles, including the graph minor theorem, immersion theorem, theorems about patchwork containment, and various restrictions, extensions and labelled versions of these theorems. We also determine the precise proof-theoretic strength of some restrictions of the graph minor theorem, and show that they are equivalent to other restricted versions that had been considered before. Finally, we present a combinatorial theorem employing ordinal labelled trees ordered by embedding with gap-condition that may additionally have well-quasi-ordered labels on the vertices, which turns out not to be provable in the theory Pi^1_1-CA. This result suggests a potential for raising the lower bounds of the immersion theorem, and the thesis concludes by outlining this possibility and other avenues for further research

Assouad-Nagata dimension of minor-closed metrics

Assouad-Nagata dimension addresses both large and small scale behaviors of metric spaces and is a refinement of Gromov's asymptotic dimension. A metric space $M$ is a minor-closed metric if there exists an (edge-)weighted graph $G$ in a fixed minor-closed family such that the underlying space of $M$ is the vertex-set of $G$, and the metric of $M$ is the distance function in $G$. Minor-closed metrics naturally arise when removing redundant edges of the underlying graphs by using edge-deletion and edge-contraction. In this paper, we determine the Assouad-Nagata dimension of every minor-closed metric. It is a common generalization of known results for the asymptotic dimension of $H$-minor free unweighted graphs and the Assouad-Nagata dimension of some 2-dimensional continuous spaces (e.g.\ complete Riemannian surfaces with finite Euler genus) and their corollaries.Comment: arXiv admin note: text overlap with arXiv:2007.0877

Detecting resilient structures in stochastic networks: A two-stage stochastic optimization approach

We propose a two-stage stochastic programming framework for designing or identifying "resilient," or "reparable" structures in graphs whose topology may undergo a stochastic transformation. The reparability of a subgraph satisfying a given property is defined in terms of a budget constraint, which allows for a prescribed number of vertices to be added to or removed from the subgraph so as to restore its structural properties after the observation of random changes to the graph's set of edges. A two-stage stochastic programming model is formulated and is shown to be N P -complete for a broad range of graph-theoretical properties that the resilient subgraph is required to satisfy. A general combinatorial branch-and-bound algorithm is developed, and its computational performance is illustrated on the example of a two-stage stochastic maximum clique problem. © 2016 Wiley Periodicals, Inc. NETWORKS, 201

Embedded Q-Resolutions and Yomdin-Lê Surface Singularities

El objetivo principal de la tesis doctoral es estudiar invariantes topológicos y algebraicos de las singularidades de Yomdin-Lê utilizando Q-resoluciones encajadas. La diferencia esencial de esta clase de resoluciones reside en permitir que el espacio ambiente contenga singularidades cocientes, simplificando de esta manera la complejidad combinatoria del divisor excepcional. Para ello hemos tenido que fijar un marco teórico en el que se han generalizado resultados clásicos de teoría de singularidades y geometría algebraica de espacios lisos al caso en el que se admiten singularidades cocientes. Destacamos el estudio de espacios proyectivos y explosiones ponderadas, teoría de intersección racional, fórmula de A\'Campo con singularidades cocientes y estudio de la estructura de Hodge mixta de la cohomología de la fibra de Milnor mediante la adaptación de la sucesión espectral de Steenbrink al caso cociente. El marco teórico obtenido se aplica al estudio de las singularidades superaisladas y de Yomdin-Lê (ponderadas y clásicas). Finalmente, se atacan contenidos más algorítmicos ligados al polinomio de Bernstein-Sato de una singularidad mediante bases de Gröbner en anillos no conmutativos

Implicitization of rational maps

Motivated by the interest in computing explicit formulas for resultants and discriminants initiated by B\'ezout, Cayley and Sylvester in the eighteenth and nineteenth centuries, and emphasized in the latest years due to the increase of computing power, we focus on the implicitization of hypersurfaces in several contexts. Our approach is based on the use of linear syzygies by means of approximation complexes, following [Bus\'e Jouanolou 03], where they develop the theory for a rational map $f:P^{n-1}\dashrightarrow P^n$. Approximation complexes were first introduced by Herzog, Simis and Vasconcelos in [Herzog Simis Vasconcelos 82] almost 30 years ago. The main obstruction for this approximation complex-based method comes from the bad behavior of the base locus of $f$. Thus, it is natural to try different compatifications of $A^{n-1}$, that are better suited to the map $f$, in order to avoid unwanted base points. With this purpose, in this thesis we study toric compactifications $T$ for $A^{n-1}$. We provide resolutions $Z.$ for $Sym_I(A)$, such that $\det((Z.)_\nu)$ gives a multiple of the implicit equation, for a graded strand $\nu\gg 0$. Precisely, we give specific bounds $\nu$ on all these settings which depend on the regularity of \SIA. Starting from the homogeneous structure of the Cox ring of a toric variety, graded by the divisor class group of $T$, we give a general definition of Castelnuovo-Mumford regularity for a polynomial ring $R$ over a commutative ring $k$, graded by a finitely generated abelian group $G$, in terms of the support of some local cohomology modules. As in the standard case, for a $G$-graded $R$-module $M$ and an homogeneous ideal $B$ of $R$, we relate the support of $H_B^i(M)$ with the support of $Tor_j^R(M,k)$.Comment: PhD. Thesis of the author, from Universit\'e de Paris VI and Univesidad de Buenos Aires. Advisors: Marc Chardin and Alicia Dickenstein. Defended the 29th september 2010. 163 pages 15 figure