The moduli space of matroids

In the first part of the paper, we clarify the connections between several algebraic objects appearing in matroid theory: both partial fields and hyperfields are fuzzy rings, fuzzy rings are tracts, and these relations are compatible with the respective matroid theories. Moreover, fuzzy rings are ordered blueprints and lie in the intersection of tracts with ordered blueprints; we call the objects of this intersection pastures. In the second part, we construct moduli spaces for matroids over pastures. We show that, for any non-empty finite set $E$, the functor taking a pasture $F$ to the set of isomorphism classes of rank-$r$ $F$-matroids on $E$ is representable by an ordered blue scheme $Mat(r,E)$, the moduli space of rank-$r$ matroids on $E$. In the third part, we draw conclusions on matroid theory. A classical rank-$r$ matroid $M$ on $E$ corresponds to a $\mathbb{K}$-valued point of $Mat(r,E)$ where $\mathbb{K}$ is the Krasner hyperfield. Such a point defines a residue pasture $k_M$, which we call the universal pasture of $M$. We show that for every pasture $F$, morphisms $k_M\to F$ are canonically in bijection with $F$-matroid structures on $M$. An analogous weak universal pasture $k_M^w$ classifies weak $F$-matroid structures on $M$. The unit group of $k_M^w$ can be canonically identified with the Tutte group of $M$. We call the sub-pasture $k_M^f$ of $k_M^w$ generated by ``cross-ratios' the foundation of $M$,. It parametrizes rescaling classes of weak $F$-matroid structures on $M$, and its unit group is coincides with the inner Tutte group of $M$. We show that a matroid $M$ is regular if and only if its foundation is the regular partial field, and a non-regular matroid $M$ is binary if and only if its foundation is the field with two elements. This yields a new proof of the fact that a matroid is regular if and only if it is both binary and orientable.Comment: 83 page

Mecanismos articulados para trazar curvas como recurso educativo digital para la didáctica de las matemáticas en Secundaria y Bachillerato

Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Formación de Profesorado y Educación. Departamento de Didáctica y Teoría de la Educación. Fecha de lectura: 12-07-201

Constraint-Based Graphic Statics - A geometrical support for computer-aided structural equilibrium design

This thesis introduces “constraint-based graphic statics”, a geometrical support for computer-aided structural design. This support increases the freedom with which the designer interacts with the plane static equilibriums being shaped. Constraint-based graphic statics takes full advantage of geometry, both its visual expressiveness and its capacity to solve complex problems in simple terms. Accordingly, the approach builds on the two diagrams of classical graphic statics: a form diagram describing the geometry of a strut-and-tie network and a force diagram vectorially representing its inner static quilibrium. Two new devices improve the control of these diagrams: (1) nodes — considered as the only variables — are constrained within Boolean combinations of graphical regions; and (2) the user modifies these diagrams by means of successive operations whose geometric properties do not at any time jeopardise the static equilibrium of the strut-and-tie network. These two devices offer useful features, such as the ability to describe, constrain and modify any static equilibrium using purely geometric grammar, the ability to compute and handle multiple solutions to a problem at the same time, the ability to switch the hierarchy of constraint dependencies, the ability to execute dynamic conditional statements graphically, the ability to compute full interdependency and therefore the ability to remove significantly the limitations of compass-and-straightedge constructions and, finally the ability to propagate some solution domains symbolically. As a result, constraint-based graphic statics encourages the emergence of new structural design approaches that are highly interactive, precognitive and chronology-free: highly interactive because forces and geometries are simultaneously and dynamically steered by the designer; precognitive because the graphical region constraining each points marks out the set of available solutions before they are even explored by the user; and chronology-free because the deductive process undertaken by the designer can be switched whenever desired

Qualitative Reasoning about Relative Directions : Computational Complexity and Practical Algorithm

Qualitative spatial reasoning (QSR) enables cognitive agents to reason about space using abstract symbols. Among several aspects of space (e.g., topology, direction, distance) directional information is useful for agents navigating in space. Observers typically describe their environment by specifying the relative directions in which they see other objects or other people from their point of view. As such, qualitative reasoning about relative directions, i.e., determining whether a given statement involving relative directions is true, can be advantageously used for applications, for example, robot navigation, computer-aided design and geographical information systems. Unfortunately, despite the apparent importance of reasoning about relative directions, QSR-research so far could not provide efficient decision procedures for qualitative reasoning about relative directions. Accordingly, the question about how to devise an efficient decision procedure for qualitative reasoning about relative directions has meanwhile turned to the question about whether an efficient decision procedure exists at all. Answering the latter existential question, which requires a formal analysis of relative directions from a computational complexity point of view, has remained an open problem in the field of QSR. The present thesis solves the open problem by proving that there is no efficient decision procedure for qualitative reasoning about relative directions, even if only left or right relations are involved. This is surprising as it contradicts the early premise of QSR believed by many researchers in and outside the field, that is, abstracting from an infinite domain to a finite set of relations naturally leads to efficient reasoning. As a consequence of this rather negative result, efficient reasoning with any of the well-known relative direction calculi (OPRAm, DCC, DRA, LR) is impossible. Indeed, the present thesis shows that all the relative direction calculi belong to one and the same class of ∃R-complete problems, which are the problems that can be reduced to the NP-hard decision problem of the existential theory of the reals, and vice versa. Nevertheless, in practice, many interesting computationally hard AI problems can be tackled by means of approximative algorithms and heuristics. In the same vein, the present thesis shows that qualitative reasoning about relative directions can also be tackled with approximative algorithms. In the thesis we develop the qualitative calculus SVm which allows for a practical algorithm for qualitative reasoning about relative directions. SVm also provides an effective semi-decision procedure for the OPRAm calculus, the most versatile one among the relative direction calculi. In this thesis we substantiate the usefulness of SVm by applying it in the marine navigation domain

Incorporating sedimentological observations, hydrogeophysics and conceptual knowledge to constrain 3D numerical heterogeneity models of alluvial systems

Coarse, braided river deposits are highly heterogeneous in terms of hydraulic properties and make up many groundwater reservoirs worldwide and more than two-thirds of the exploited aquifers in Switzerland. The management of these resources often requires the understanding of the subsurface flow processes and therefore, of the subsurface heterogeneity. While coarse, braided river deposits were the focus of many studies, the relationship between the braided river dynamics and the resulting sedimentary structures is still poorly known. A better knowledge of this relationship is the key to geologically more realistic modelling of the subsurface heterogeneity by accounting for the former controls on the fluvial system (slope of the floodplain, terraces, lateral confinement, bedrock steps, etc.). This thesis aims to (i) investigate the link between braided river morphodynamics and subsurface heterogeneity in a hydrogeological context and (ii) develop a numerical model of subsurface heterogeneity that accounts for this link. The sedimentological knowledge on coarse, braided river deposits was mainly gained from the observation of analogue coarse deposits in Switzerland. The Tagliamento River (northeast Italy) was chosen as a field laboratory to thoroughly study the geomorphology (surface) and near-surface sedimentology of coarse, braided rivers. The Tagliamento River is considered to be one of the last large semi-natural rivers of the Alps that has retained much of its natural sediment and discharge dynamics. The observations focused on a single reach sharing similar characteristics with partly confined valleys such as in the alpine foreland. The sedimentary structures of coarse, braided river deposits can be comprehensively described by a small limited number of sedimentary structures that have specific hydraulic properties. A handful of depositional elements were identified. Among them, the cross bedded trough fills can significantly influence the flow field because their highly-permeable cross beds act as fast flow conduit. Such trough fills most probably form from confluence scour holes. The geomorphological analysis of the Cimano-Pinzano reach is based on a LiDAR-derived digital elevation model (LiDAR-derived DEM), aerial and satellite photographs, a water-stage time series and regular field observations. Complex aggradation/degradation dynamics resulted in the formation of higher-lying zones incised by a drainage gully network and surrounded by zones that are often reworked by the river. The main geomorphological elements are identified in terms of their topographic signature and genesis, setting apart the trichotomy water–vegetation–bar. Two morphologies mark the active zones: a low-discharge morphology (low-discharge incisions and channels, slip-face lobes, etc.) superimposed on a high-discharge morphology (gravel sheets, scours, etc.). Based on the observations of vertical exposure of coarse, braided river deposits, each morphological element is associated with a depositional element. The preservation potential of the depositional elements is evaluated as a function of the river-bed aggradation dynamics and the resulting subsurface heterogeneity is discussed in terms of its impact on the subsurface flow. Ground-penetrating radar (GPR) surveys performed on the active zones of the Cimano-Pinzano reach imaged many cross bedded trough fills as observed in ancient deposits. Finding a link between these structures and the evolution of the morphology is challenging. Nevertheless, some hypotheses about the formation of the trough structures are advanced. All the observations suggest the importance of the gravel sheets in the formation of cross bedded trough fills. An object-based model was developed that mimics the dominant processes of floodplain dynamics. Contrary to existing models, this object-based model possesses the following properties: (i) it is consistent with field observations (outcrops, ground-penetrating radar data, etc.), (ii) it allows different sedimentological dynamics to be modeled that result in different subsurface heterogeneity patterns, and (iii) it is light in memory and computationally fast. To demonstrate its applicability, the object-based model is conditioned to interpreted two-dimensional data and the uncertainty on the three-dimensional subsurface heterogeneity is quantified with Monte Carlo sampling. The impact of an isolated trough fill complex on subsurface flow mixing is evaluated in terms of advective mixing. The trough fill complex is modelled with the object-based model that is fitted to GPR data. Hydraulic properties are assigned to the model cells and a subsurface flow through the model is simulated. The advective mixing is quantified with particle tracking. The results indicate strong advective mixing as well as a large flow deviation induced by the asymmetry of the trough fills with regard to the main flow direction. These findings depict possible advective mixing found in natural environments and can guide the interpretation of ecological processes such as in the hyporheic zone. The geomorphological and sedimentological characterisations of the coarse, braided Cimano-Pinzano reach of the Tagliamento River contribute to a better understanding of the morphodynamics of coarse, braided river in relation to the subsurface heterogeneity. The object-based model allows the simulation of various geological settings and the methodology developed for the stereological study can be adapted to other types of data without many changes. Preliminary results of subsurface flow simulations through coarse, braided river deposits show a strong subsurface flow mixing

Combinatorics and degenerations in algebraic geometry: Hurwitz numbers, Mustafin varieties and tropical geometry

Degenerationstechniken sind mächtige Werkzeuge in der Untersuchung verschiedener Objekte der algebraischen Geometrie. Sie ermöglichen es geometrische Objekte mit Hilfe kombinatorischer Methoden zu untersuchen. Diesem großen Nutzen geht jedoch häufig die große Herausforderung voraus den Zusammenhang zur Kombinatorik zu konkretisieren. Das junge Gebiet der tropischen Geometrie bietet einen konzeptionellen Rahmen, um diesen Zusammenhang herzustellen. In dieser Arbeit untersuchen wir verschiedene Probleme der algebraischen Geometrie mit Hilfe degenerativer und kombinatorischer Methoden. Diese Probleme lassen sich wiederum in drei Themengebiete unterteilen: Enumeration von Überlagerungen (Hurwitz Zahlen), Degenerationen von projektiven Räumen (Mustafin varietäten), Modulräume tropischer Kurven und treue Tropikalisierungen

Raising public awareness of mathematics

This book arose from the presentations given at the international workshop held in Óbidos, 26–29 September 2010, as a result of a joint initiative of the Centro Internacional de Matemática and the Raising Public Awareness (RPA) committee of the European Mathematical Society (EMS). The objective was to provide a forum for general reflection with an international mix of experts on building the image of mathematics, ten years after the World Mathematical Year 2000 (WMY 2000). Óbidos, a charming town situated one hour by car to the north of Lisbon, Portugal, was also the site of the re-creation in the year 2000 of the international mathematics exhibition “Beyond the Third Dimension” (http://alem3d.obidos.org/en/) and a meeting of the EMS WMY2000 Committee. The opening of the workshop was also a public “mathematical afternoon” organised by the Portuguese Mathematical Society (SPM) in cooperation with the town of Óbidos. At this event mathematical films and lectures to the general public were presented. The first lecture was given by H. Leitão, from the University of Lisbon, on mathematics in the “Age of Discoveries”, and the second one by G.-M. Greuel, the current president of ERCOM (the EMS committee of the European Research Centres on Mathematics), on the topic “Mathematics between Research, Application and Communication”, which text is included in this book.info:eu-repo/semantics/publishedVersio

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum