An upper bound on the number of rational points of arbitrary projective varieties over finite fields

We give an upper bound on the number of rational points of an arbitrary Zariski closed subset of a projective space over a finite field. This bound depends only on the dimensions and degrees of the irreducible components and holds for very general varieties, even reducible and non equidimensional. As a consequence, we prove a conjecture of Ghorpade and Lachaud on the maximal number of rational points of an equidimensional projective variety

Cellular Automata on Group Sets

We introduce and study cellular automata whose cell spaces are left-homogeneous spaces. Examples of left-homogeneous spaces are spheres, Euclidean spaces, as well as hyperbolic spaces acted on by isometries; uniform tilings acted on by symmetries; vertex-transitive graphs, in particular, Cayley graphs, acted on by automorphisms; groups acting on themselves by multiplication; and integer lattices acted on by translations. For such automata and spaces, we prove, in particular, generalisations of topological and uniform variants of the Curtis-Hedlund-Lyndon theorem, of the Tarski-F{\o}lner theorem, and of the Garden-of-Eden theorem on the full shift and certain subshifts. Moreover, we introduce signal machines that can handle accumulations of events and using such machines we present a time-optimal quasi-solution of the firing mob synchronisation problem on finite and connected graphs.Comment: This is my doctoral dissertation. It consists of extended versions of the articles arXiv:1603.07271 [math.GR], arXiv:1603.06460 [math.GR], arXiv:1603.07272 [math.GR], arXiv:1701.02108 [math.GR], arXiv:1706.05827 [math.GR], and arXiv:1706.05893 [cs.FL

Discrete Geometry

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50 years of first passage percolation

We celebrate the 50th anniversary of one the most classical models in probability theory. In this survey, we describe the main results of first passage percolation, paying special attention to the recent burst of advances of the past 5 years. The purpose of these notes is twofold. In the first chapters, we give self-contained proofs of seminal results obtained in the '80s and '90s on limit shapes and geodesics, while covering the state of the art of these questions. Second, aside from these classical results, we discuss recent perspectives and directions including (1) the connection between Busemann functions and geodesics, (2) the proof of sublinear variance under 2+log moments of passage times and (3) the role of growth and competition models. We also provide a collection of (old and new) open questions, hoping to solve them before the 100th birthday.Comment: 160 pages, 17 figures. This version has updated chapters 3-5, with expanded and additional material. Small typos corrected throughou

For a "nanomusic" - Sound nanomachines and elastic space-time: A transversal approach to music composition

This thesis describes my transversal approach to composition and underlines how structuralist psychoanalys is, Deleuze and Guattari’s philosophy, and areas of thinking in the nanosciences played a central role on my conceptions of space and time. My most original contribution lies in the endeavour to elaborate a malleable and evanescent musical matter in which micro-fluctuations exert their forces on macro-entities creating a musical space-time as elastic and mobile as that of the psychic unconscious. The pieces are conceived as cartographies of flux, fleeting constellations of sound particles whose migrations either interweave filaments in elastic and heterochronic textures or converge on an object. Small modular pendulums are the elementary figures of this ramified network. They connect to one another and permeate the musical matter. Time is multiple and pluridirectional. Jumps and rebounds, temporal gaps, annunciative anticipation and retroactive reverberation (Green), keep combining fragmentary elements which echo with one another and which allow the listener to perceive the work of time as a "compound of splinters", an assemblage of "quanta of memory", or a pure flow. Repetition has a paradoxical function, both favouring the identification of abstract gestures and their step-by-step mutation. Sound trajectories are envisaged as "pulsounds" and chains of "sonifiers", in reference to concepts of pulsional phenomena (Freud) or signifiers (Lacan). This intersection between psychic topology and sound topography also investigates the structural potential of relationships between humans and urban environments. The addition of electronics, the use of both instrumental and mechanical or urban sounds, materialize a transitional space (Winnicott) with ambiguous boundaries between inside and outside, a hybrid sound territory, a zone of indiscernibility (Deleuze) between mind and machine, scream and noise, the organic and the mechanical. These evolving sound diagrams refer to nanotechnology’s atom-by-atom engineering with its increasing crossing-over of animate and inanimate matters. Sound processes operate as quasi microscopic units grouped in transitory configurations under the action of forces,and with this perspective I have coined the term "nanomusic" to describe my musical project

Computerised Modelling for Developmental Biology

Many studies in developmental biology rely on the construction and analysis of models. This research presents a broad view of modelling approaches for developmental biology, with a focus on computational methods. An overview of modelling techniques is given, followed by several case studies. Using 3D reconstructions, the heart development of the turtle is examined, with special attention to heart looping and the development of the outflow tract. Subsequently, an ontology system is presented in which anatomical, developmental and physiological information on the vertebrate heart is modelled. Finally, two Petri net models are discussed, which model the developmental process of gradient formation, both in a qualitative and quantitative manner.LEI Universiteit LeidenImagin