Davide Ambrosi
The mechanics of cell migration
Cell migration is a very attractive mechanobiological system for mathematical modelling. The reason of such a fascination resides in the challenging contrast between the simplicity of the basic ingredients of the system (two phases that exchange mass), versus the rich observed dynamics: steady and pulsatile motion, excitability, polarization, material reorganization, transition between rest and excited states. In this talk I will provide examples of mathematical questions that stem when modelling a migrating cell as mechanical system, obeying balance laws. In some cases energetic arguments can support a purely mechanical explanation for the observed behavior.
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Antonio DiCarlo
Yonder beyond Cauchy-Born: a constructive criticism
Cauchy's aim was much worth of pursuit, but his rule was too crude and his results disappointing. In 1853 Maxwell wrote: “There are few parts of mechanics in which theory has differed more from experiment than in the theory of elastic solids.” Forty years later, Voigt’s decisive experimental results allowed Love to draw the following firm conclusion: “Modern Physics is perfectly capable of deducing a theory of elasticity from the known laws of energy, without the aid of a subsidiary hypothesis about inter-molecular force, and, being in that position, it is bound to discard the hypothesis.” Twenty years later, however, the first results from X-ray crystallography vindicated Cauchy’s corpuscular hypothesis and drew Born to revisit it. By fiat, the Cauchy-Born rule inhibits all the microscopic degrees of freedom averaged out in the macroscopic deformation, thus suppressing microscopic fluctuations. This is why it just applies to elastic perfect crystals, and with mixed success even there. Already in 1852, Lamé foresaw the existence, if not the significance, of microscopic fluctuations: “[L]e temps n’est-il pas venu de se demander si l’état moléculaire des corps dont le repos nous paraît le mieux établi, est bien réellement un état statique ; s’il n’est pas, au contraire, le résultat de vibrations très rapides, et qui ne s’arrêtent jamais ? Tout porte à penser, en effet, que le repos relatif des molécules d’un corps n’est qu’un cas très-exceptionnel, une pure abstraction, une chimère peut-être.” I argue that we are now in a position to realise Cauchy’s aim in a more supple and general way, capitalising on Lamé’s intuition.
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Alfredo Marzocchi
Analytical and thermodynamical issues for second-gradient fluids
We consider a general second-gradient fluid and by an analysis of the dissipation we get conditions on the coefficients for isotropic fluids. Subsequently, we analyze the IBVP problem for such a fluid in an unbounded domain.
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Paolo Podio-Guidugli
Analytical, statistical, and microscopically-informed, continuum thermodynamics
I'll spend my time in decoding my title. In fact, while I expect the typical member of my audience to have a fair idea of what either Statistical or Continuum Thermodynamics or both are about, this person might wonder what Analytical Thermodynamics would be and why Continuum Thermodynamics should be microscopically informed, let alone how.
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Mario Putti
Complexity, Optimality and Robustness: vibes from optimal transportation theory
Complex networks arising in natural systems typically display small clustering and large average path length. It is recognized that while optimality leads to tree-like configurations, the presence of loops increases robustness and resiliency. Recently, the group of Amos Maritan analyzed this behavior and introduced the concept of explorability to ''measure of the ability of an interacting system to adapt to newly intervening changes''. They showed that sparse networks optimize both explorability and dynamical robustness.
A number of natural networks can be related to the problem of optimal re-allocation of resources. In this talk we explore these concepts within the viewpoint of Optimal Transportation (OT) theory. Starting from the PDE-based L1 OT problem proposed by Evans & Gangbo in 1999, we look at its recent dynamic extension and its ability to generate ramified OT structures closely resembling complex networks. We analyze the optimization process underlying OT as a gradient flow of an energy functional for both the L1 and ramified versions of our dynamic OT. This energy functional is the sum of the energy dissipated during the transport process and a term containing
the total transport flux intensity. In the case of L1-OT, we report rigorous results showing the existence of this gradient flow and the equivalence of its asymptotic solution with the OT density of Evans-Gangbo original problem.
The situation becomes mathematically more difficult in the ramified case as the energy functional becomes non-convex. However, numerical results show that the extension of the gradient flow approach still leads to interesting structures, with network-like asymptotic solutions that are stable in time. Numerical experiments show that these solutions are invariably tree-like if the transported mass (the forcings of the system) is stationary. On the contrary, if the forcings vary in time with a zero mean stable loops appear in the network structures.
These results seem to suggest the idea that reliability is driven by time-varying external forcings. Several numerical results will be presented in support of this conjecture.
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Giovanni Zanzotto
Bursty deformation of crystalline materials: some experimental, empirical, and modelling results
We describe some recent experimental and empirical studies of the bursty strain behavior of memory alloys undergoing reversible martensitic transformations, highlighting the role of kinematic compatibility also on these avalanching phenomena. We present some related crystallographically-informed modeling, with results on strain intermittency.
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Marta Zoppello
The mechanics of swimming
What does it mean swimming? How can mathematics treat this problem? What is the best strategy to move in a certain direction? The modeling and control of artificial devices that mimic the motion of real microorganisms has become a fascinating branch in mathematical physics, especially for the future realization of micro-robots for localized surgery or drug delivery. I will present models for deformable bodies immersed either in a viscous or ideal irrotational fluid which are suitable for control applications.