Research axes

The main goal of the axis is the study of strongly correlated quantum systems. These are a class of materials whose behaviour cannot be explained by the simple application of classical physics or single-particle quantum mechanics. Instead, their properties arise from the complex interactions between many particles, which give rise to novel phenomena such as high-temperature superconductivity, quantum phase transitions, many-body localisation, entanglement production and fractional topological order.

Quantum simulations are a powerful tool to study these systems because they allow emulating the behaviour of strongly correlated systems by engineering experimental setups with a high degree of flexibility and controllability. C²EMPI will use atomic, photonic, micro-mechanical and electronic systems to implement strongly correlated phases. In addition, the project will address questions in quantum information theory, quantum metrology and the development of novel quantum materials.

Many complex systems can neither be regarded as completely organized nor disordered, but rather possess a controllable degree of disorder. The times at which a suitable detector clicks when targeted by a low-intensity beam of photons or electrons, for instance, are definitely not deterministic, yet can exhibit various degrees of regularity; this observation is the cornerstone of optical coherence. Similarly to electrons, the eigenvalues of a random matrix also typically do not lump together. As a third example, controlled disorder can also be engineered into physical materials to achieve desirable propagation properties. The mathematical models that describe these situations of controlled disorder are typically interacting systems, where interactions between elements are responsible for the structured organization of the whole. However, an increasing level of interaction
progressively brings mathematical difficulties when it comes to analyse -- sometimes even rigorously define-- these interacting systems. We are an interdisciplinary community dedicated to understanding the theoretical properties of disordered and random systems, and their practical consequences.

This axis builds upon historical strengths at ULille, in stochastic geometry and its applications to physics and computer science. The CDP C²EMPI would give existing collaborations more visibility, as well as the means to take our ambitions to the next level. At the local scale, funded projects already involve participants from several labs on specific tasks, like ANR Random (LPP, IEMN) on point processes and metamaterials, or ERC Blackjack (CRIStAL, LPP) on Monte Carlo methods with repulsive point processes. We also run a workgroup that gathers 20 researchers every week across LPP, CRIStAL, and PhLAM on interacting point processes. More globally, we have been hosting and participating in national CNRS networks such as GDTs GeoSto, Ondes, Méga, as well as organizing international conferences on the themes of this axis, such as our 2023 conference on hyperuniformity and rigidity.

Topology is a branch of mathematics which studies the properties of objects that remain unchanged under continuous distortions. It mainly focuses on the description and classification of such properties through the concept of topological invariants. Thoroughly studied since the XIXth century, it has been possible to successfully define complete invariants for bundles and other mathematical objects in one and two dimensions, and also in dimensions four and higher, in which the large connectivity of the space simplifies the available topological classes. However, the description of three-dimensional manifolds using topological invariants is still a very active area of research in mathematics. The particular connectivity of this dimension
prevents the definition of complete invariants, and the topological classification of mathematical objects (knots, algebraic links, singularities, manifolds, etc.) requires a family by family study. A number of fundamental questions, which will be the subject of C²EMPI, are still open, like the solution of Ragsdale conjecture in specific cases and the definition of invariants in quantum field theory.

Inspired by these mathematical concepts, topology in physics has drawn a lot of attention since the 1980s. The first success of this approach was to explain the origin of the quantum Hall effect in electronic microstructures, a phenomenon at the origin of three different Nobel prizes since 1985. Indeed, the concept of topological invariant can be applied to the electronic and photonic bands of a material. Its value is always an integer and it is preserved under continuous deformations of the solid. More importantly, it allows predicting the existence of propagating channels at the edges of the material. These notions were used in 2005 to discover a new class of materials known as topological insulators, and in optics they have been used to engineer, among others, topological lasers in a photonic chip and to route photons around corners in integrated devices with an unprecedented resilience to backscattering.

C²EMPI aims at addressing some of the open questions in topology in mathematics, optics and, acoustics. One of the core activities of the consortium is the design and study of novel topological phases for photons, phonons and electrons both for fundamental interest and for eventual applications. They include topological phases in lattices under periodic modulation, in presence of nonlinearities, acoustic topological protection with multi-scale metamaterials, and the development of topological devices in the THz range for integrated communications.

Mastering ultrafast phenomena has become a key challenge for several applications ranging from high data rate telecommunications and signal processing to new generations of particle accelerators. THz science and technology, and ultra-fast physics has become one of the main technological enablers in a number of fields: radio astronomy, Earth observation, weather forecasting, security imaging, telecommunications, nondestructive device testing.

The University of Lille hosts several groups that have become leaders in complementary domains of ultra-fast science and communications, including the development of THz science and technology, novel ultra-fast systems for high speed optical and wireless communications, fundamental photonics, and the development of high capacity hollow core fibres. The Lille campus also hosts two major technological facilities on semiconductor, and photonic components (IEMN central and Fibertech), which can provide key components for these ultra-fast systems.

However, while this field has been pushed forward in the last 15 years, a lot of bottlenecks remain to unlock expected important
scientific and public benefits of this science, ranging from fundamental scientific research (radioastronomy, quantum optics) to highly applied and society-oriented subjects like 6G communications, medical imaging and climate monitoring. Another area of ultrafast science at which Lille groups excel is accelerator physics. In particular, these groups have reported novel ways to produce light (X-rays) in free-electron lasers, and novel approaches for the control of THz synchrotron radiation in collaboration with the FERMI free-electron laser (Italy), SOLEIL (France), and the Karlsruhe Institute of Technology (Germany).

This privileged standpoint in the development of ultra-fast science and data communication systems across different laboratories in the Lille campus, the IEMN tehcnological central and the Fibertech platform is the launchpad to take to new limits ultrafast and high-capacity communications, accelerator physics and ultra-fast photonics.

Breaking down large systems into smaller units is often an effective approach to manage their complexity. Nevertheless, many complex dynamical systems defy such simplification. These systems often feature components that exhibit strong, global interconnections, making it impossible to describe them through global modes. Moreover, these components interact in a nonlinearly, which precludes the use of linear superposition principles. Such dynamical systems display a wide range of intriguing emergent behaviors that do not follow directly from the laws and interactions governing individual components and elude simple approaches, from self-sustained oscillations to deterministic chaos and turbulence.

Yet, complex dynamical systems display an exquisite organisation which can be unveiled by using the appropriate mathematical concepts (for example, the multifractal formalism for chaotic attractors or the inverse scattering transform for integrable turbulence). What makes dynamical system theory a unifying formalism for various fields is the universality of the phenomena observed and their description by the same tools. As a result, this field offers many challenges but also many open mathematical problems, whose solutions can easily be transposed into another domain once they have been cast in a common language.

This axis explores the captivating realm of complex dynamical systems, bridging the gaps between physics, mathematics, electronics and biophysics, creating opportunities for disruptive applications. We will make advances in fields as diverse as integrable turbulence (leading to robust high-speed communications), the dynamics of relativistic electron bunches (with the generation of intense THz radiation), strongly nonlinear fiber optics (leading to nonlinear frequency combs for remote sensing or communications), nonlinear acoustics (leading to acoustic tweezers, or self-cleaning surfacs), dynamics of biological systems (with applications to therapeutic protocols), foliations and laminations in holomorphic dynamical systems, the dynamics of infinitedimensional linear and nonlinear systems and their relation to ergodic theory, the links between geometry, groups and dynamical systems, to mention some notable examples.

Symmetries are ubiquitous in mathematics and in everyday’s life. The basic concept of a periodic function over the real numbers studied in classical Fourier theory has been vastly generalized in the notion of an automorphic form (a function admitting multiple symmetries by a non-commutative group). Powerful objects that appear while studying automorphic forms are their L-functions which play a central role in the Langlands conjectures. The theories have even been extended to non-archimedean (p-adic or l-adic) numbers whose pro-finite and totally disconnected nature are more than relevant in the digital world we live in. Far more
fascinating are the hidden symmetries, such as those hidden in the Riemann zeta function or more generally in the L-function of an elliptic curve — the importance of unlocking the secrets of these objects is recognized by making them part of the Millennium Prize Problems.

A strong theme in this axis is the study of objects arising at the interface of algebraic geometry, the theory of automorphic forms and physics, whose common feature is that their moduli spaces are arithmetic quotients of complex symmetric domains. Those arithmetic questions are naturally approached by exploiting the geometry of algebraic varieties endowed with special symmetries, including Shimura varieties, K3 surfaces, hyperkähler varieties and Fano varieties. Such objects arise also naturally in physics, and their classification is related to fundamental open problems in Yang-Mills theory and mirror symmetry.
 

The increased availability of massive computing power over the last 5 decades constitutes both an opportunity and a challenge. Indeed, it makes the simulation of ever more complex phenomena possible, while requiring the development of adapted numerical techniques to provide efficient and robust algorithms to perform the computations needed reliably and quickly. This impacts modelling of physical and biophysical problems as it allows, for example, for numerical experiences in silico.

Researchers involved in this axis have a track record of cross-disciplinary work within the limits of the consortium, as well as with external partners. For example, researchers of the consortium have worked together in nonlinear optics using the FIBERTECH platform of the University of Lille. They also took part in the H2020 project EURAD and studied corrosion models in the context of nuclear waste storage, which is a key technology for energy transition.

We aim at simulating and studying models of physical and biophysical systems in various contexts. They include numerical Simulation of the physics of nonlinear dispersive equations, numerical simulation of corrosion in materials, effects of thermal fluctuations on mechanical phenomena, life sciences and ecology, and numerical simulation of eddy currents testing.