Variety, flexibility, and use of abstract concepts. A multiple grounded perspective.

The nature of concepts is a matter of intense debate in cognitive sciences. While traditional views claim that conceptual knowledge is represented in a unitary symbolic system, recent Embodied and Grounded Cognition theories (EGC) submit the idea that conceptual system is couched in our body and influenced by the environment (Barsalou, 2008). One of the major challenges for EGC is constituted by abstract concepts (ACs), like fantasy. Recently, some EGC proposals addressed this criticism, arguing that the ACs comprise multifaced exemplars that rely on different grounding sources beyond sensorimotor one, including interoception, emotions, language, and sociality (Borghi et al., 2018). However, little is known about how ACs representation varies as a function of life experiences and their use in communication. The theoretical arguments and empirical studies comprised in this dissertation aim to provide evidence on multiple grounding of ACs taking into account their varieties and flexibility. Study I analyzed multiple ratings on a large sample of ACs and identified four distinct subclusters. Study II validated this classification with an interference paradigm involving motor/manual, interoceptive, and linguistic systems during a difficulty rating task. Results confirm that different grounding sources are activated depending on ACs kind. Study III-IV investigate the variability of institutional concepts, showing that the higher the law expertise level, the stronger the concrete/emotional determinants in their representation. Study V introduced a novel interactive task in which abstract and concrete sentences serve as cues to simulate conversation. Analysis of language production revealed that the uncertainty and interactive exchanges increase with abstractness, leading to generating more questions/requests for clarifications with abstract than concrete sentences. Overall, results confirm that ACs are multidimensional, heterogeneous, and flexible constructs and that social and linguistic interactions are crucial to shaping their meanings. Investigating ACs in real-time dialogues may be a promising direction for future research.

Incompressible limit and well-posedness of PDE models of tissue growth

Both compressible and incompressible porous medium models are used in the literature to describe the mechanical aspects of living tissues. Using a stiff pressure law, it is possible to build a link between these two different representations. In the incompressible limit, compressible models generate free boundary problems where saturation holds in the moving domain. Our work aims at investigating the stiff pressure limit of reaction-advection-porous medium equations motivated by tumor development. Our first study concerns the analysis and numerical simulation of a model including the effect of nutrients. A coupled system of equations describes the cell density and the nutrient concentration and the derivation of the pressure equation in the stiff limit was an open problem for which the strong compactness of the pressure gradient is needed. To establish it, we use two new ideas: an L3-version of the celebrated Aronson-Bénilan estimate, and a sharp uniform L4-bound on the pressure gradient. We further investigate the sharpness of this bound through a finite difference upwind scheme, which we prove to be stable and asymptotic preserving. Our second study is centered around porous medium equations including convective effects. We are able to extend the techniques developed for the nutrient case, hence finding the complementarity relation on the limit pressure. Moreover, we provide an estimate of the convergence rate at the incompressible limit. Finally, we study a multi-species system. In particular, we account for phenotypic heterogeneity, including a structured variable into the problem. In this case, a cross-(degenerate)-diffusion system describes the evolution of the phenotypic distributions. Adapting methods recently developed in the context of two-species systems, we prove existence of weak solutions and we pass to the incompressible limit. Furthermore, we prove new regularity results on the total pressure, which is related to the total density by a power law of state.