Browsing by Author "Hoffmann, Franca"

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    Emergence in biology and social sciences
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach gGmbH, 2022) Hoffmann, Franca; Merino-Aceituno, Sara
    Mathematics is the key to linking scientific knowledge at different scales: from microscopic to macroscopic dynamics. This link gives us understanding on the emergence of observable patterns like flocking of birds, leaf venation, opinion dynamics, and network formation, to name a few. In this article, we explore how mathematics is able to traverse scales, and in particular its application in modelling collective motion of bacteria driven by chemical signalling.
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    Variational Methods for Evolution
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2023) Hoffmann, Franca; Mielke, Alexander; Peletier, Mark; Slepcev, Dejan
    Variational principles for evolutionary systems arise in many settings, both in those describing the physical world and in man-made algorithms for data science and optimization tasks. Variational principles are available for Hamiltonian systems in classical mechanics, gradient flows for dissipative systems, as well as in time-incremental minimization techniques for more general evolutionary problems. Additional challenges arise via the interplay of two or more functionals (e.g. a free energy and a dissipation potential), thus encompassing a large variety of applications in the modeling of materials and fluids, in biology, and in multi-agent systems. Variational principles and associated evolutions are also at the core of the modern approaches to machine learning tasks, since many of them are posed as minimizing an objective functional that models the problem. The discrete and random nature of these problems and the need for accurate computation in high dimension present a set of challenges that require new mathematical insights. Variational methods for evolution allow for the usage of the rich toolbox provided by the calculus of variations, metric-space geometry, partial differential equations, and other branches of applied analysis. The variational methods for evolution have seen a rapid growth over the last two decades. This workshop continued the successful line of meetings (2011, 2014, 2017, and 2020), while evolving and branching into new directions. We have brought together a wide scope of mathematical researchers from calculus of variations, partial differential equations, numerical analysis, and stochastics, as well as researchers from data science and machine learning, to exchange ideas, foster interaction, develop new avenues, and generally bring these communities closer together.