Relative Quot schemes over families of smooth and nodal curves

The present work studies relative Quot schemes parameterizing locally free sheaf quotients with zero--dimensional support on the fibers of a morphism. We focus on families of smooth curves with at most nodal singularities, as well as families of smooth, higher dimensional varieties. The thesis is organized around three projects, which explore both geometric and combinatorial aspects of these relative Quot schemes. The first project, carried out in collaboration with Barbara Fantechi and...

Aspects of entanglement in holography, harmonic lattices and non-relativistic quantum field theories

This thesis presents three studies elucidating aspects of entnaglement entropy in holography, harmonic lattices and non-realtivistic quantum field theory. In the first part we review the geodesic bit thread forumlation for holographic entanglement entropy and show that in presence of an event horizon we can recover the holographic thermal entropy. In the second part we study the entanglement hamiltonian for a two-interval system in a harmonic lattice, in both the massless and super massiv...

Entanglement and Asymmetry in Quantum Field Theory

This thesis investigates the use of concepts from quantum information theory to analyze and quantify symmetry breaking in extended quantum systems, particularly focusing on the key measure known as Entanglement Asymmetry. The core of the work introduces and applies Entanglement Asymmetry. This quantity is defined based on the difference in information content between a quantum state and a version of that state where the symmetry has been enforced (or "symmetrized"). It represents an altern...

Multiscale phenomena in quantum and classical models

In this thesis we present several works dealing with multiscale phenomena. First, we treat a model of pattern formation: the Ising model with competing short-range and long-range interactions. We compute the energy asymptotic of excitations above the periodic ground states. In a second work we study a model of Quantum Hall Effect with a single edge-mode and subject to quasi-periodic disorder: through a combination of rigorous Renormalization Group techniques and \textsc{kam}-like estimates we...

Many-body physics of cavity embedded quantum matter

Controlling the properties of quantum matter is a central goal of condensed matter physics. In recent years, cavity embedding—that is, shaping the electromagnetic environment surrounding a material by placing it inside a cavity—has emerged as a new potential tuning knob. While such concepts were first successfully explored with ultracold atomic clouds, more recent experiments have demonstrated their feasibility in solid-state systems. Despite the differences between these platforms, they shar...

Pro-inflammatory small extracellular vesicles: neuroinflammation propagation and target spinal cord cells in explant cultures

Within the vast landscape of intercellular messengers, extracellular vesicles, namely membrane-enclosed nanosized-particles released by virtually all cells, are increasingly tied to central nervous system transcellular communication, yet their roles in propagating neuro-inflammation, in kindling astrocytic reactivity and neuronal dysfunctions, as well as the relationship among vesicle source cells and the recipient ones, remain largely unclear. To dissect extracellular vesicles’ dynamics and ...

PrPC Absence Dysregulates The Synaptic Machinery: Electrophysiological and Molecular Profile of Prion Protein-Deficient Cortical Neurons

The Analogue of the Shapiro - Tater Conjecture for the Painlevé IV Equation

In this thesis we study an intrinsic connection relating the fourth Painlevé equation and the even sextic anharmonic oscillator with a pole. We compare 1. zeros of generalized Hermite polynomials and Okamoto polynomials, coming as building blocks for rational solutions of the Painlevé equation, 2. zeros of certain resultants, coming as the condition of degeneration of algebraic spectrum of the sextic oscillator. Similar comparison for Painlevé II and the quartic oscillator is know as Sha...

Large-Scale Response Theory for Gapless Lattice Fermi Systems in Low Dimensions

This thesis develops a rigorous large-scale response theory for gapless lattice Fermi systems in low dimensions, focusing on the mathematical validity of linear response -- the Kubo formula -- for transport in gapless systems, where the absence of a spectral gap prevents the direct use of standard adiabatic theorems. After reviewing fundamental results on many-body lattice fermions, we first analyse the transport properties of non-interacting, gapless, one-dimensional quantum systems and o...