**Pablo Jarillo-Herrero** – Quantum electronic transport and thermodynamic measurements in magic angle twisted bilayer and trilayer graphene.

**Jennifer Cano** – Topological quantum chemistry and symmetry indicators: the lectures will cover the role of symmetry eigenvalues in diagnosing band topology in topological crystalline insulators. We will then introduce symmetry-protected semimetals (i.e., Dirac semimetals and “new fermions”) and discuss their topology and bulk-boundary correspondence.

**Oskar Vafek** – Correlations and topology in the magic angle twisted bilayer graphene narrow bands. We will cover RG treatment of the projection onto the narrow bands and the flow towards the chiral limit, results from DMRG, non-Abelian Dirac node braiding and theory of cascade transitions.

**Rebeca Ribeiro-Palau** – Topological valley states in graphene/BN systems. The relative angular alignment between 2D layers of a van der Waals (vdW) heterostructure can dramatically alter its fundamental properties. A striking example is the recent observation of strongly correlated states and intrinsic superconductivity in twisted bilayer graphene. Another remarkable effect of angular layer alignment, predicted for certain vdW heterostructures, is the emergence of phases of matter with non-trivial topological properties, where charge carriers ow without dissipation, being protected against perturbations. In graphene aligned with boron nitride (BN), such a phase has been predicted, with topological protection linked not to the spin, as commonly observed, but rather to the valley degree of freedom. In this talk we will introduce the main concepts to understand the emergence of this topological effects and the we will review the latest experimental observations.

**Titus Neupert**– Topology Bands from homotopy theory. The lectures will introduce Wilson loops and homotopy invariants, such as (mirror) Chern numbers, winding numbers etc., to define and identify topological bands in crystalline solids. The relation between Wilson loops and Wannier functions can be used to discern stable, fragile, and delicate topological phases. Wilson loops can further be used to prove the bulk-boundary correspondence of topological phases. We will discuss explicitly the 1D Su-Schrieffer-Heeger chain, (mirror) Chern insulators, and higher-order topological insulators. Finally, I will comment on the topological classification of non-Hermitian Hamiltonians.

**Cécile Repellin**– Interaction-driven insulators in flat-band moire superlattices

This set of lecture will focus on the correlated insulators arising in multilayer moire graphene materials, when the valence or conduction band (the active band) is sufficiently flat. Examples of such materials include magic angle twisted bilayer graphene, ABC trilayer graphene, twisted double bilayer graphne, twisted monolayer-bilayer graphene, and more. I will distinguish scenarios corresponding to insulators with three different topological natures. First, we will consider an integer filling of the active band, where a quantized anomalous Hall effect and emergent orbital ferromagnetism have been experimentally observed in several materials, and can be attributed to spontaneous spin and valley polarization due to Coulomb repulsion (flat-band ferromagnetism). Secondly, we will discuss the insulating phases that can arise through the interplay of topology and strong interactions: topological charge density waves, which appear at fractional filling of the active band, with an integer quantized Hall conductance, and fractional Chern insulators, whose filling and Hall conductance are fractional, and have anyonic excitations. We will review experimental evidence for these phases, the band parameters favoring them, as well as the theoretical methods (numerical and analytical) permitting the understanding of their conditions of emergence.

**Cristiane Morais-Smith** – Thermodynamic description of topological insulators: the search for universal behavior. Topological insulators are states of matter distinguished by the presence of symmetry protected metallic boundary states. These edge modes have been characterised in terms of transport and spectroscopic measurements, but a thermodynamic description has been lacking. Recently, we have shown that using Hill thermodynamics, it is possible to separate the edge and bulk thermodynamic potential, and describe the topological phase transition using thermodynamic observables, such as heat capacity or density of states [1]. Subsequently, we extended this approach to different topological models in various dimensions (the Kitaev chain and Su-Schrieffer-Heeger (SSH) model in one dimension, the Kane-Mele model in two dimensions and the Bernevig-Hughes-Zhang model in two and three dimensions) at zero temperature. Surprisingly, all models exhibit the same universal behavior in the order of the topological-phase transition, depending on the dimension. Moreover, we derived the topological phase diagram at finite temperatures using this thermodynamic description, and showed that it displays a good agreement with the one calculated from the Uhlmann phase [2]. More recently, we investigated also the long-range Kitaev chain [3], higher-order topological insulators (HOTI) [4], and the non-Hermitian SSH model [5]. Our work reveals unexpected universalities and opens the path to a thermodynamic description of systems with a non-local order parameter.

[1] A. Quelle, E. Cobanera, and C. Morais Smith, Phys. Rev. B 94, 075133 (2016).

[2] S. N. Kempkes, A. Quelle, and C. Morais Smith, Nature Scientific Reports 6, 38530 (2016).

[3] P. Cats, A. Quelle, O. Viyuela, M. A. Martin-Delgado, and C. Morais Smith, Phys. Rev. B 97, 121106 (R) (2018).

[4] R. Arouca, S. N. Kempkes, and C. Morais Smith, Phys. Rev. Research 2, 023097 (2020).

[5] R. Arouca, C.H. Lee, and C. Morais Smith, Phys. Rev. B 102, 245145 (2020).

**Sami Mitra** – Roles of journals in general and PRL in particular in disseminating physics results through a cascading sequence involving journal editors, referees, conference chairs, journalists, department chairs, deans, funding agencies, and others.