**Research Themes and Projects**

*Theme 1: Electronic Structure in Chemistry and Physics: *

The group has developed new approximate energy density functionals inspired to the exact mathematical structure of the strong-interaction limit of Density Functional Theory (DFT) and tested them in a controlled way on prototypical problems in chemistry and physics.

### On-going and Future Projects:

- Exact properties, exact forms of the kinetic correlation part
- Include the effects of spin correlations
- Functionals based on the mathematical structure of the strong-interaction limit of DFT
- Include in a rigorous way the effects of dispersion interactions
- Implementation of these functionals
- Simplified forms and their performances
- The strong-interaction limit of Hartree-Fock theory

*Theme 2: Quantum Matter*

We have tested our new approximations on electrons confined in low-dimensional geometries, providing the first exchange-correlation (XC) functional able to capture the correct physics at all correlation regimes, from the Fermi liquid case, to the formation of Wigner molecules

The group has extended the formalism to bosonic and fermionic ultracold quantum gases with dipolar and ionic interactions.

### On-going and future projects:

- Hybrid Green’s functions methods
- Time-dependent formalism
- Bosonic and fermionic ultracold quantum gases with long-range interactions in the presence of disorder

*Theme 3: Mathematical Aspects *

Crossing traditional disciplinary boundaries, the group collaborates with mathematicians from mass transportation theory (or optimal transport), an important field of mathematics and economics, reformulating the strong-interaction limit of DFT in these terms. This reformulation paved the way to a cross-fertilization between two very different research areas, leading to new formal results and new algorithms coming from the optimal transport community. The group also focused on other exact conditions in Density Functional Theory, such as the Lieb-Oxford inequality, providing rigorous lower bounds for the optimal constant.

### On-going and future projects

- Rigorous study of the next leading term in the strong-interaction limit
- Structure of the optimal maps for the multimarginal problem with Coulomb cost
- Optimal constant in the Lieb-Oxford inequality