The purpose of this thesis is to focus on the aspect of passive devices allowing for WDM, routing, switching and filtering of optical signals, investigating novel routing concepts based on micro optical side coupled resonators to achieve large bandwidth by multiple cascading and/or multiple coupling (low group velocity) periodicity effects. We will describe some technical aspects necessary for the design and fabrication of some passive circuitry, and usually neglected in purely theoretical approaches, including optical routers based on racetrack resonators and novel SCISSOR and CROW devices.

Scientists are continuously fascinated by the high degree of sophistication found in natural materials, arising from evolutionary optimisation. In living organisms, nature provides a wide variety of materials, architectures, systems and functions, often based on weak constituents at the lower scales. One of the most extensively studied natural materials is spider silk, renowned for its outstanding mechanical properties, which include exceptional strength and toughness. Owing to its wide range of properties, which vary depending on factors such as the type of silk (up to seven) that each spider can produce, and the species of spider, it can be considered a class of semi-crystalline polymeric material. Indeed, spider silk cleverly combines, depending on the application required, the great deformability of an amorphous phase with the stiffness and strength conferred by pseudo-crystals consisting of specific secondary structures of some of the proteins constituting the material. Based on the countless studies conducted on spider silk, it is now also clear that its remarkable performance are the result of a sophisticated optimisation of the material's hierarchical structure. Nevertheless, many of the multiscale mechanisms that give rise to the striking macroscopic properties are still unclear. Many open problems are also related to the relevant effects of environmental conditions and in particular on temperature and humidity, strongly conditioning the mechanical performances. In this thesis, aimed at unveiling some of these open problems, we introduce a multiscale model for the thermo-hygro-mechanical response, starting with the influence of water molecules modifying the microstructure, up to the effects at the macroscopic scale, including softening, increase in elongation at break and supercontraction, i.e. the shortening (up to half the initial length) of the spider threads in wet environments. Thereafter, we describe how the supercontraction effect can be adopted to obtain humidity-driven actuators, and in particular, we determine the maximum actuation force depending on the silk properties at the molecular scale and on the constraining system representing other silk threads or the actuated device. The spider silk actuation properties turned out to be extraordinary, making spider silk potentially the best performing humidity-driven actuator known to date in terms of work density. As observed in many natural materials, spider silks are characterized by a strong variability in both chemical and structural organization, as for example described in the recently published experimental database of properties at different scales of about a thousand different spider silks, where evident correlations among quantities are scarce. This large variability makes the theoretical understanding of the observed material behavior, in relation of the complex hierarchical structure, particularly intriguing. To address this novel amount of experimental data without losing sight of theoretical analytical modelling, we propose a new data modelling methodology to obtain simple and interpretable relationships linking quantities at different scales. In particular, we employ a symbolic regression technique, known as 'Evolutionary Polynomial Regression', which integrates regression capabilities with the Genetic Programming paradigm, enabling the derivation of explicit analytical formulas, finally delivering a deeper comprehension of the analysed physical phenomenon. Eventually, we provide insights to improve our multiscale theoretical model accounting for the humidity effects on spider silks. This approach may represent a proof of concept for modelling in fields governed by multiscale, hierarchical differential equations. We believe that the analytical description of the macroscopic behaviour from microscale properties is of great value both for the full understanding of biological materials, as well as in the perspective of bioinspired materials and structures.

Immunotherapy, by enhancing the endogenous anti-tumor immune responses, is showing promising results for the treatment of numerous cancers refractory to conventional therapies. However, its effectiveness for advanced castration-resistant prostate cancer remains unsatisfactory and new therapeutic strategies need to be developed. To this end, mathematical modeling provides a quantitative framework for testing in silico the efficacy of new treatments and combination therapies, as well as understanding unknown biological mechanisms. In this dissertation we present two mathematical models of prostate cancer immunotherapy defined as systems of ordinary differential equations. The first work, introduced in Chapter 2, provides a mathematical model of prostate cancer immunotherapy which has been calibrated using data from pre-clinical experiments in mice. This model describes the evolution of prostate cancer, key components of the immune system, and seven treatments. Numerous combination therapies were evaluated considering both the degree of tumor inhibition and the predicted synergistic effects, integrated into a decision tree. Our simulations predicted cancer vaccine combined with immune checkpoint blockade as the most effective dual-drug combination immunotherapy for subjects treated with androgen-deprivation therapy that developed resistance. Overall, this model serves as a computational framework to support drug development, by generating hypotheses that can be tested experimentally in pre-clinical models. The Chapter 3 is devoted to the description of a human prostate cancer mathematical model. The potential effect of immunotherapies on castration-resistant form has been analyzed. In particular, the model includes the dendritic vaccine sipuleucel-T, the only currently available immunotherapy option for advanced prostate cancer, and the ipilimumab, a drug targeting the cytotoxic T-lymphocyte antigen 4 , exposed on the CTLs membrane, currently under Phase II clinical trial. From a mathematical analysis of a simplified model, it seems likely that, under continuous administration of ipilimumab, the system lies in a bistable situation where both the no-tumor equilibrium and the high-tumor equilibrium are attractive. The schedule of periodic treatments could then determine the outcome, and mathematical models could help in deciding an optimal schedule.

The present work analyzes and describes a method for the direct numerical solution of the Maxwell's equations of classical electromagnetism. This is the FDTD (Finite-Difference Time-Domain) method, along with its implementation in an "in-house" computing code for large parallelized simulations. Both are then applied to the modelization of photonic and plasmonic structures interacting with light. These systems are often too complex, either geometrically and materially, in order to be mathematically tractable and an exact analytic solution in closed form, or as a series expansion, cannot be obtained. The only way to gain insight on their physical behavior is thus to try to get a numerical approximated, although convergent, solution. This is a current trend in modern physics because, apart from perturbative methods and asymptotic analysis, which represent, where applicable, the typical instruments to deal with complex physico-mathematical problems, the only general way to approach such problems is based on the direct approximated numerical solution of the governing equations. Today this last choice is made possible through the enormous and widespread computational capabilities offered by modern computers, in particular High Performance Computing (HPC) done using parallel machines with a large number of CPUs working concurrently. Computer simulations are now a sort of virtual laboratories, which can be rapidly and costless setup to investigate various physical phenomena. Thus computational physics has become a sort of third way between the experimental and theoretical branches. The plasmonics application of the present work concerns the scattering and absorption analysis from single and arrayed metal nanoparticles, when surface plasmons are excited by an impinging beam of light, to study the radiation distribution inside a silicon substrate behind them. This has potential applications in improving the eciency of photovoltaic cells. The photonics application of the present work concerns the analysis of the optical reflectance and transmittance properties of an opal crystal. This is a regular and ordered lattice of macroscopic particles which can stops light propagation in certain wavelenght bands, and whose study has potential applications in the realization of low threshold laser, optical waveguides and sensors. For these latters, in fact, the crystal response is tuned to its structure parameters and symmetry and varies by varying them. The present work about the FDTD method represents an enhacement of a previous one made for my MSc Degree Thesis in Physics, which has also now geared toward the visible and neighboring parts of the electromagnetic spectrum. It is organized in the following fashion. Part I provides an exposition of the basic concepts of electromagnetism which constitute the minimum, although partial, theoretical background useful to formulate the physics of the systems here analyzed or to be analyzed in possible further developments of the work. It summarizes Maxwell's equations in matter and the time domain description of temporally dispersive media. It addresses also the plane wave representation of an electromagnetic field distribution, mainly the far field one. The Kirchhoff formula is described and deduced, to calculate the angular radiation distribution around a scatterer. Gaussian beams in the paraxial approximation are also slightly treated, along with their focalization by means of an approximated diraction formula useful for their numericall FDTD representation. Finally, a thorough description of planarly multilayered media is included, which can play an important ancillary role in the homogenization procedure of a photonic crystal, as described in Part III, but also in other optical analyses. Part II properly concerns the FDTD numerical method description and implementation. Various aspects of the method are treated which globally contribute to a working and robust overall algorithm. Particular emphasis is given to those arguments representing an enhancement of previous work.These are: the analysis from existing literature of a new class of absorbing boundary conditions, the so called Convolutional-Perfectly Matched Layer, and their implementation; the analysis from existing literature and implementation of the Auxiliary Differential Equation Method for the inclusion of frequency dependent electric permittivity media, according to various and general polarization models; the description and implementation of a "plane wave injector" for representing impinging beam of lights propagating in an arbitrary direction, and which can be used to represent, by superposition, focalized beams; the parallelization of the FDTD numerical method by means of the Message Passing Interface (MPI) which, by using the here proposed, suitable, user dened MPI data structures, results in a robust and scalable code, running on massively parallel High Performance Computing Machines like the IBM/BlueGeneQ with a core number of order 2X10^5. Finally, Part III gives the details of the specific plasmonics and photonics applications made with the "in-house" developed FDTD algorithm, to demonstrate its effectiveness. After Chapter 10, devoted to the validation of the FDTD code implementation against a known solution, Chapter 11 is about plasmonics, with the analytical and numerical study of single and arrayed metal nanoparticles of different shapes and sizes, when surface plasmon are excited on them by a light beam. The presence of a passivating embedding silica layer and a silicon substrate are also included. The next Chapter 12 is about the FDTD modelization of a face-cubic centered (FCC) opal photonic crystal sample, with a comparison between the numerical and experimental transmittance/reflectance behavior. An homogenization procedure is suggested of the lattice discontinuous crystal structure, by means of an averaging procedure and a planarly multilayered media analysis, through which better understand the reflecting characteristic of the crystal sample. Finally, a procedure for the numerical reconstruction of the crystal dispersion banded omega-k curve inside the first Brillouin zone is proposed. Three Appendices providing details about specific arguments dealt with during the exposition conclude the work.

Specific non-relativistic quantum mechanical one-dimensional systems, interacting via point interactions, are discussed within the resolvent algebra setting.

Le mappe comomento omotopiche sono una generalizzazione della nozione di mappa momento introdotta al fine di estendere il concetto di azione hamiltoniana al contesto della geometria multisimplettica. L'obiettivo di questa tesi è fornire nuove costruzioni esplicite ed esempi concreti di azioni di gruppi di Lie su varietà multisimplettiche che ammettono delle mappe comomento. Il primo risultato è una classificazione completa delle azioni di gruppi compatti su sfere multisimplettiche. In questo caso, l'esistenza di mappe comomento omotopiche dipende dalla dimensione della sfera e dalla transitività dell'azione di gruppo. Il secondo risultato è la costruzione esplicita di un analogo multisimplettico dell’inclusione dell'algebra di Poisson di una varietà simplettica dentro il corrispondente algebroide di Lie twistato. E’ possibile dimostrare che questa inclusione soddisfa una relazione di compatibilità nel caso di varietà multisimplettiche gauge-correlate in presenza di un'azione di gruppo Hamiltoniana. Tale costruzione potrebbe giocare un ruolo nella formulazione di un analogo multisimplettico della procedura di quantizzazione geometrica. L’ultimo risultato è una costruzione concreta di una mappa comomento omotopica relativa all'azione multisimplettica del gruppo di diffeomorfismi che preservano la forma volume dello spazio Euclideo. Questa mappa ammette naturalmente un’interpretazione idrodinamica, nello specifico trasgredisce alla mappa comomento idrodinamica introdotta da Arnol'd, Marsden, Weinstein e altri. La mappa comomento così costruita può essere inoltre messa in relazione alla teoria dei nodi avvalendosi dell’approccio ai link nel formalismo dei vortici. Questo punto di apre la strada a un'interpretazione semiclassica del polinomio HOMFLYPT nel linguaggio della quantizzazione geometrica. / Homotopy comomentum maps are a higher generalization of the notion of moment map introduced to extend the concept of Hamiltonian actions to the framework of multisymplectic geometry. Loosely speaking, higher means passing from considering symplectic $2$-form to consider differential forms in higher degrees. The goal of this thesis is to provide new explicit constructions and concrete examples related to group actions on multisymplectic manifolds admitting homotopy comomentum maps. The first result is a complete classification of compact group actions on multisymplectic spheres. The existence of a homotopy comomentum maps pertaining to the latter depends on the dimension of the sphere and the transitivity of the group action. Several concrete examples of such actions are also provided. The second novel result is the explicit construction of the higher analogue of the embedding of the Poisson algebra of a given symplectic manifold into the corresponding twisted Lie algebroid. It is also proved a compatibility condition for such embedding for gauge-related multisymplectic manifolds in presence of a compatible Hamiltonian group action. The latter construction could play a role in determining the multisymplectic analogue of the geometric quantization procedure. Finally a concrete construction of a homotopy comomentum map for the action of the group of volume-preserving diffeomorphisms on the multisymplectic 3-dimensional Euclidean space is proposed. This map can be naturally related to hydrodynamics. For instance, it transgresses to the standard hydrodynamical co-momentum map of Arnol'd, Marsden and Weinstein and others. A slight generalization of this construction to a special class of Riemannian manifolds is also provided. The explicitly constructed homotopy comomentum map can be also related to knot theory by virtue of the aforementioned hydrodynamical interpretation. Namely, it allows for a reinterpretation of (higher-order) linking numbers in terms of multisymplectic conserved quantities. As an aside, it also paves the road for a semiclassical interpretation of the HOMFLYPT polynomial in the language of geometric quantization.

In this thesis we develop a mathematically rigorous framework of the so-called ''classical limit'' of quantum systems and their semi-classical properties. Our methods are based on the theory of strict, also called C*- algebraic deformation quantization. Since this C*-algebraic approach encapsulates both quantum as classical theory in one single framework, it provides, in particular, an excellent setting for studying natural emergent phenomena like spontaneous symmetry breaking (SSB) and phase transitions typically showing up in the classical limit of quantum theories. To this end, several techniques from functional analysis and operator algebras have been exploited and specialised to the context of Schrödinger operators and quantum spin systems. Their semi-classical properties including the possible occurrence of SSB have been investigated and illustrated with various physical models. Furthermore, it has been shown that the application of perturbation theory sheds new light on symmetry breaking in Nature, i.e. in real, hence finite materials. A large number of physically relevant results have been obtained and presented by means of diverse research papers.

The main bulk of this Thesis is focused on the response of cell membranes due to chemical and mechanical stimuli. Henceforth, it is mainly devoted to deduce how the key aspect of the cell response activated by chemical signaling can be predicted by a simplified energetics, making use of both theoretical models and numerical simulations. The a ention is focused on cell membranes embedding G protein-coupled receptors (GPRCs). By analyzing the behavior of cell mem- branes, one can isolate three main contributions in order to model their respon- se: (1) diffusion of receptors and transporters embedded in the lipid membrane; (2) conformational changes of the receptors; (3) membrane elasticity. Moreover, the interplay between TM confomational changes and lateral pressure of the lipid membrane against such TMs is introduced. The chemical potential of the receptor-ligand compound, deduced as the variational derivative of such energy, is compared with the one calculated by accounting for the work done by the lateral pressure. The result yields a relationship between the conformational field, the mechanical field (interpreted as either the thickness change or the areal change) and the distribution of the compounds receptor-ligand. The analysis of such resulting constitutive equation among those three quantities shows that, essentially, the reason why ligand-GPRCs compounds prefer to live on lipid ra is a necessity involving the interplay between the work performed by the lateral pressure and the need of TMs to change their conformation during ligand binding. Henceforth, mechanobiology gives a justification to the experimental findings of Kobilka and Lei ovitz, Chemistry Nobel Prizes 2012.