Primer for the algebraic geometry of sandpiles

The Abelian Sandpile Model (ASM) is a game played on a graph realizing the dynamics implicit in the discrete Laplacian matrix of the graph. The purpose of this primer is to apply the theory of lattice ideals from algebraic geometry to the Laplacian matrix, drawing out connections with the ASM. An extended summary of the ASM and of the required algebraic geometry is provided. New results include a characterization of graphs whose Laplacian lattice ideals are complete intersection ideals; a new construction of arithmetically Gorenstein ideals; a generalization to directed multigraphs of a duality theorem between elements of the sandpile group of a graph and the graph's superstable configurations (parking functions); and a characterization of the top Betti number of the minimal free resolution of the Laplacian lattice ideal as the number of elements of the sandpile group of least degree. A characterization of all the Betti numbers is conjectured.Comment: 45 pages, 14 figures. v2: corrected typo

Constant mean curvature surfaces

In this article we survey recent developments in the theory of constant mean curvature surfaces in homogeneous 3-manifolds, as well as some related aspects on existence and descriptive results for $H$-laminations and CMC foliations of Riemannian $n$-manifolds.Comment: 102 pages, 17 figure

Huge networks, tiny faulty nodes

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2007.Includes bibliographical references (p. 87-91).Can one build, and efficiently use, networks of arbitrary size and topology using a "standard" node whose resources, in terms of memory and reliability, do not need to scale up with the complexity and size of the network? This thesis addresses two important aspects of this question. The first is whether one can achieve efficient connectivity despite the presence of a constant probability of faults per node/link. Efficient connectivity means (informally) having every pair of regions connected by a constant fraction of the independent, entirely non-faulty paths that would be present if the entire network were fault free - even at distances where each path has only a vanishingly small probability of being fault-free. The answer is yes, as long as some very mild topological conditions on the high level structure of the network are met - informally, if the network is not too "thin" and if it does not contain too many large "holes". The results go against some established "empyrical wisdom" in the networking community. The second issue addressed by this thesis is whether one can route efficiently on a network of arbitrary size and topology using only a constant number c of bits/node (even if c is less than the logarithm of the network's size!). Routing efficiently means (informally) that message delivery should only stretch the delivery path by a constant factor. The answer again is yes, as long as the volume of the network grows only polynomially with its radius (otherwise, we run into established lower bounds). This effectively captures every network one may build in a universe (like our own) with finite dimensionality using links of a fixed, maximum length and nodes with a fixed, minimum volume. The results extend the current results for compact routing, allowing one to route efficiently on a much larger class of networks than had previously been known, with many fewer bits.by Enoch Peserico.Ph.D

Riemann-Roch theory for sublattices of the root lattice An, graph automorphisms and counting cycles in graphs

This thesis consists of two independent parts. In the rst part of the thesis, we develop a Riemann-Roch theory for sublattices of the root lattice An extending the work of Baker and Norine (Advances in Mathematics, 215(2): 766-788, 2007) and study questions that arise from this theory. Our theory is based on the study of critical points of a certain simplicial distance function on a lattice and establishes connections between the Riemann-Roch theory and the Voronoi diagrams of lattices under certain simplicial distance functions. In particular, we provide a new geometric approach for the study of the Laplacian of graphs. As a consequence, we obtain a geometric proof of the Riemann-Roch theorem for graphs and generalise the result to other sub-lattices of An. Furthermore, we use the geometric approach to study the problem of computing the rank of a divisor on a nite multigraph G to obtain an algorithm that runs in polynomial time for a xed number of vertices, in particular with running time 2O(n log n)poly(size(G)) where n is the number of vertices of G. Motivated by this theory, we study a dimensionality reduction approach to the graph automorphism problem and we also obtain an algorithm for the related problem of counting automorphisms of graphs that is based on exponential sums. In the second part of the thesis, we develop an approach, based on complex-valued hash functions, to count cycles in graphs in the data streaming model. Our algorithm is based on the idea of computing instances of complex-valued random variables over the given stream and improves drastically upon the naive sampling algorithm.Diese Dissertation besteht aus zwei unabhaengigen Teilen. Im ersten Teil entwickeln wir auf der Arbeit von Baker und Norine (Advances in Mathematics, 215(2): 766-788, 2007) aufbauend eine Riemann-Roch Theorie fuer Untergitter (sublattices) des Wurzelgitter (root lattice) An und untersuchen die Fragestellungen, die sich daraus ergeben. Unsere Theorie basiert auf der Untersuchung kritischer Punkte einer bestimmten simplizialen (simplicial) Metrik (distance function) auf Gitter und zeigt Verbindungen zwischen der Riemann-Roch Theorie und Voronoi-Diagrammen von Gittern unter einer gewissen simplizialen Metrik. Insbesondere liefern wir einen neuen geometrischen Beweis des Riemann-Roch Theorems fuer Graphen und generalisieren das Resultat fuer andere Untergitter von An. Des Weiteren verwenden wir den geometrischen Ansatz um das Problem der Berechnung des Rang (rank) eines Teilers (divisor) auf einem endlichen Multigraphen G und erhalten einen Algorithmus, der fuer eine xe Anzahl von Knoten in Polynomialzeit, genauer in Zeit 2O(n log n)poly(size(G)) mit n ist die Anzahl der Knoten in G, laeuft. Von dieser Theorie ausgehend untersuchen wir einen Anzatz fuer das Graphautomorphismusproblem ueber eine Dimensionalitaetsreduktion und erhalten ebenfalls einen Algorithmus fuer das verwandte Problem des Zaehlens von Automorphismen eines Graphen, der auf exponentiellen Summen basiert. Im zweiten Teil der Dissertation entwickeln wir einen auf komplexwertigen Hashfunktionen basierenden Ansatz um in einem Streaming-Modell die Zyklen eines Graphen zu zaehlen. Unser Algorithmus basiert auf der Idee Instanzen von komplexwertigen Zufallsvariablen ueber dem gegebenen Stream zu berechnen und stellt eine drastische Verbesserung ueber den naiven Sampling-Algorithmus dar. Im zweiten Teil der Dissertation entwickeln wir einen auf komplexwertigen Hashfunktionen basierenden Ansatz um in einem Streaming-Modell die Zyklen eines Graphen zu zaehlen. Unser Algorithmus basiert auf der Idee Instanzen von komplexwertigen Zufallsvariablen ueber dem gegebenen Stream zu berechnen und stellt eine drastische Verbesserung ueber den naiven Sampling-Algorithmus dar