Seminars 2014

In this talk we formulate total variation based denoising algorithms to recover numerically-simulated gravitational wave signals. We shall give a brief introduction to the variational denoising and compressing models to better understand the purpose of this research work. We also introduce the fundamentals of the theory of gravitational wave signals generated by high energy astrophysical events and the interest to detect these signals. We present some numerical results for two types of waveforms, namely, "bursts" and "chirps" for which catalogs are available.

The Kuramoto model is a system of ordinary differential equations for describing synchronization phenomena defined as a coupled phase oscillators. In this talk, an infinite dimensional Kuramoto model is considered, and Kuramoto's conjecture on a bifurcation diagram of the system will be proved.
A linear operator obtained from the linearization of the Kuramoto model has the continuous spectrum on the imaginary axis, so that the usual spectrum theory does not determine the dynamics of the system. To handle such continuous spectra, a new spectral theory of linear operators based on Gelfand triplets is developed. In particular, a generalized eigenvalue is defined. It is proved that a generalized eigenvalue determines the stability and bifurcation of the system.

I will discuss the relationship between entanglement (or geometric) entropy with statistical mechanical entropy of horizon degrees of freedom when described in the framework of isolated horizons in loop quantum gravity. We show that, once the relevant degrees of freedom are identified, the two notions coincide. The key ingredient linking the two notions is the structure of quantum geometry at Planck scale implied by loop quantum gravity, where correlations between the inside and outside of the black hole are mediated by eigenstates of the horizon area operator.

In biology and medicine we may observe a wide spectrum of formation of patterns, usually due to self-organization phenomena. Patterns are usually explained in terms of a collective behavior driven by "forces", either external and/or internal, acting upon individuals (cells or organisms). In most of these organization phenomena, randomness plays a major role; here we wish to address the issue of the relevance of randomness as a key feature for producing nontrivial geometric patterns in biological structures. As working examples we offer a review of a couple of important case studies involving angiogenesis, i.e. tumor-driven angiogenesis [1], and retina angiogenesis [2]. In both cases the reactants responsible for pattern formation are the cells organizing as a capillary network of vessels, and a family of underlying fields driving the organization, such as nutrients, growth factors and alike. The strong coupling of the kinetic parameters of the relevant stochastic branching-and-growth of the capillary network, with the family of interacting underlying fields is a major source of complexity from a mathematical and computational point of view. Thus our main goal is to address the mathematical problem of reduction of the complexity of such systems by taking advantage of its intrinsic multiscale structure; the (stochastic) dynamics of cells will be described at their natural scale (the microscale), while the dynamics of the underlying fields will be described at a larger scale (the macroscale) via deterministic averaged concentrations, by applying suitable "laws of large numbers" at a mesoscale. The intrinsic randomness of the phenomena is responsible of the building up of a realistic stochastic network of vessels.

[1] ‍Capasso, V., Morale, D.: Stochastic Modelling of Tumour-induced Angiogenesis. J. Math. Biol., 58, 219-33 (2009)
[2] ‍Capasso, V., Morale D., Facchetti, G.: The Role of Stochasticity for a Model of Retinal Angiogenesis, IMA J. Appl. Math. (2012); 19 pages; doi:10.1093/imamat/hxs050

The crossover or Kovacs effect is basically the non-monotonic relaxation of a physical quantity to its equilibrium value, from an initial non-equilibrium state. We investigate it in two cases (i) glassy systems (ii) granular gases. In the former, the time evolution of the energy passes through a maximum because the system needs to pass through a more disordered state to reach a typical equilibrium configuration. Within a very simple model, we discuss the relevance of recent linear response results for understanding the Kovacs effect. The granular gas case is quite different, since it is an intrinsically out-of-equilibrium system due to the continuous loss of energy in collisions. By introducing a simple energy input mechanism, the granular gas reaches an out-of-equilibrium steady state. A simple à la Kovacs protocol is investigated, for which a crossover effect is also displayed. Interestingly, it becomes anomalous for large enough inelasticity: the granular temperature (kinetic energy) shows a minimum instead of a maximum.