In the proposed project "Entanglement-Assisted Thermoelectric Transport in Quantum Systems" (EATTS), the European Researcher Dr. Nahual Sobrino, together with Prof. Rosario Fazio at the International Centre for Theoretical Physics (ICTP), will collaborate with international research groups to unveil innovative thermoelectric properties that hold transformative potential for sustainable energy technologies, quantum information science, and the development of advanced quantum materials. The project's objectives will concentrate on the exploration of thermoelectric phenomena mediated by entangled electrons across a variety of quantum systems. A comprehensive theoretical and computational framework will be established to explore and enhance thermoelectric performance in systems where electron entanglement plays a pivotal role. Cutting-edge calculations will be performed using sophisticated theoretical frameworks, including Quantum Master Equations (QME), Hierarchical Equation of Motion (HEOM), and quantum information techniques. This will guide to elucidate the intricate relationship between entanglement, system-environment interactions, and thermoelectric performance in Double Quantum Dots (DQD) and Cooper Pair Splitters (CPS) systems. Moreover, the project aims to extend the scope of Density Functional Theory (DFT) to access to entanglement measures in transport scenarios through the development of extensions of iq-DFT. This will facilitate a more efficient computational description of thermoelectric phenomena under the influence of entangled electrons. Additionally, the thermoelectric and entanglement characteristics of multiple diatomic molecules will be investigated by mapping the system into effective Hamiltonian models. The computational framework developed will advance our understanding and serve as a guide for experimental endeavors by selecting promising materials and parameter regimes.

WaveNets aims to establish a novel theoretical paradigm for understanding quantum systems, centred on a network interpretation of many-body wave-functions. Ongoing experimental progress motivates the need for a new theoretical approach: in the field of quantum simulation and quantum computing, probing capabilities have reached unprecedented levels, with the ability to collect thousands of wave function snapshots with impressive accuracy. However, most of our theoretical understanding of such settings still relies on and relates to few-body observables. This has created a clear gap between experimental capabilities and theoretical tools and concepts available to understand physical phenomena. The overall goal of WaveNets is to bridge this gap by introducing a mathematical framework to describe wave-function snapshots based on network theory — wave function networks — that will enable a completely new set of tools to address open problems in the field of quantum matter. WaveNets' main objectives are: - to demonstrate that wave function snapshots of correlated systems are described by scale-free networks, and classify the robustness of quantum simulators according to such; - to formulate methods for quantifying the Kolmogorov complexity of many-body systems, and propose an information-theory-based characterization of topological matter and confinement in gauge theories; - to propose scalable methods for measuring entanglement in quantum simulators and computers, as well as for their validation. Achieving these objectives will (a) provide unique insights into the information structure of quantum matter, (b) enable methods of probing and controlling matter of direct experimental relevance thanks to the intrinsic scalability of network-type descriptions, and (c) establish a new, interdisciplinary bridge between quantum science, and network and data mining theory, that makes possible knowledge transfer between two mature, yet poorly connected disciplines.

In 1997 Juha Kinnunen proved that maximal operators satisfy a Sobolev bound if the Sobolev exponent p is strictly larger than 1. His article initiated the study of regularity of maximal functions, a field which has attracted several dozens of authors to this day. Geometric techniques have recently lead to a series of breakthrough endpoint regularity bounds for maximal operators in higher dimensions. This project pursues the novel strategy of combining these new geometric tools with already established extremization techniques in order to solve a wide range of open questions in the field. The goals of the project are organized around two themes: gradient bounds and sharp constants. The main goal from the first theme is to prove that the variation of the non-centered Hardy-Littlewood maximal function can be controlled by the variation of the function in all dimensions. This is the endpoint p=1 of Juha Kinnunen's original bound and is one of the main long standing open questions in the field. This project also aims to prove this variation bound for further maximal operators, along with the operator continuity of their gradient and bounds for higher derivatives. The main goal from the second theme is to prove that the centered Hardy-Littlewood maximal operator in one dimension does not increase the variation of a function. This bound would be sharp because examples show that in general, maximal operators do not strictly decrease the variation of a function. This project further aims to prove this sharp bound for convolution type maximal operators and to find the sharp constant in the variation bound for the dyadic maximal operator in all dimensions.