Discrete Geometry

A number of important recent developments in various branches of discrete geometry were presented at the workshop. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics or algebraic geometry. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry

Algebraic Methods for Dynamical Systems and Optimisation

This thesis develops various aspects of Algebraic Geometry and its applications in different fields of science. In Chapter 2 we characterise the feasible set of an optimisation problem relevant in chemical process engineering. We consider the polynomial dynamical system associated with mass-action kinetics of a chemical reaction network. Given an initial point, the attainable region of that point is the smallest convex and forward closed set that contains the trajectory. We show that this region is a spectrahedral shadow for a class of linear dynamical systems. As a step towards representing attainable regions we develop algorithms to compute the convex hulls of trajectories. We present an implementation of this algorithm which works in dimensions 2,3 and 4. These algorithms are based on a theory that approximates the boundary of the convex hull of curves by a family of polytopes. If the convex hull is represented as the output of our algorithms we can also check whether it is forward closed or not. Chapter 3 has two parts. In this first part, we do a case study of planar curves of degree 6. It is known that there are 64 rigid isotopy types of these curves. We construct explicit polynomial representatives with integer coefficients for each of these types using different techniques in the literature. We present an algorithm, and its implementation in software Mathematica, for determining the isotopy type of a given sextic. Using the representatives various sextics for each type were sampled. On those samples we explored the number of real bitangents, inflection points and eigenvectors. We also computed the tensor rank of the representatives by numerical methods. We show that the locus of all real lines that do not meet a given sextic is a union of up to 46 convex regions that is bounded by its dual curve. In the second part of Chapter 3 we consider a problem arising in molecular biology. In a system where molecules bind to a target molecule with multiple binding sites, cooperativity measures how the already bound molecules affect the chances of other molecules binding. We address an optimisation problem that arises while quantifying cooperativity. We compute cooperativity for the real data of molecules binding to hemoglobin and its variants. In Chapter 4, given a variety X in n-dimensional projective space we look at its image under the map that takes each point in X to its coordinate-wise r-th power. We compute the degree of the image. We also study their defining equations, particularly for hypersurfaces and linear spaces. We exhibit the set-theoretic equations of the coordinate-wise square of a linear space L of dimension k embedded in a high dimensional ambient space. We also establish a link between coordinate-wise squares of linear spaces and the study of real symmetric matrices with degenerate eigenspectrum

Triangulations

The earliest work in topology was often based on explicit combinatorial models – usually triangulations – for the spaces being studied. Although algebraic methods in topology gradually replaced combinatorial ones in the mid-1900s, the emergence of computers later revitalized the study of triangulations. By now there are several distinct mathematical communities actively doing work on different aspects of triangulations. The goal of this workshop was to bring the researchers from these various communities together to stimulate interaction and to benefit from the exchange of ideas and methods

Collection of abstracts of the 24th European Workshop on Computational Geometry

International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop

Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System

Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum