Graphene is an exciting new two-dimensional material. Though it was considered theoretical for a long time, it was isolated about 20 years ago. Since then, it has drawn much attention due to its numerous exciting properties. More recently, it was discovered that when twisting two layers of graphene with respect to each other, at certain angles called ``magic angles", exotic transport properties emerge. The primary tool for studying this thus far is the famous Bistritzer-MacDonald model, which relies on several approximations. This work aims to build the first steps in studying magic angles without using this model. Thus, we study a model for TBG without the approximations mentioned above in the continuum setting, using two copies of potential with the symmetries of graphene, sharing a common origin and twisted with respect to each other (with a possible shift between the lattices). We describe the angles for which the two twisted lattices are commensurate and prove the existence of Dirac cones for such angles. Furthermore, we show that for small potentials, the slope of the Dirac cones is small for commensurate angles that are close to incommensurate angles. This work is the first in a series of works to build a more fundamental understanding of the phenomenon of magic angles. In this talk, I will introduce the main phenomena of twisted bilayer graphene, state our main results, and discuss some new results enabling us to extend this analysis to the so-called AB stacking twisted graphene.

To a Z^d-periodic weighted graph with vertices V, we may associated a periodic graph operator that acts on ℓ₂(V). After Floquet transform, we obtain its Bloch variety, which is an algebraic hypersurface in T^d×R whose projection to R is the spectrum of the operator. Features on Bloch varieties such as Dirac (double) points, critical points, and their level sets (Fermi varieties) reflect spectral properties of the operator.
    With students Faust and Robinison, and using the Brazos cluster, we investigated many thousands of small periodic graphs, recording invariants and features of their Bloch anf Fermi varieties. In this talk, I will briefly discuss the background and present some examples of interesting behavior of Bloch varieties that we uncovered.

We establish for the 1D Soler model with power nonlinearities $f(s)=s|s|^{p-1}$, $p>0$, that the upper-right block operator~$L_0$ of the linearized operator satisfies: its ground states $-2\omega$ and $0$ are its only two eigenvalues in the gap of its essential spectrum if only if $p\geq1$. Our second main result is the simplicity of generalized eigenfunction at the threshold of the essential spectrum for Dirac operators with potential. These results apply in particular to lower-left block operator~$L_\mu$.

We consider scalar field theories on the line with Ginzburg-Landau (double-well) self-interaction potentials. Prime examples include the \\phi^4 model and the sine-Gordon model. These models feature simple examples of topological solitons called kinks. The study of their asymptotic stability leads to a rich class of problems owing to the combination of weak dispersion in one space dimension, low power nonlinearities, and intriguing spectral features of the linearized operators such as threshold resonances or internal modes. We present a perturbative proof of the full asymptotic stability of the sine-Gordon kink outside symmetry under small perturbations in weighted Sobolev norms. The strategy of our proof combines a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects with modulation techniques to take into account the invariance under Lorentz transformations and under spatial translations. A major difficulty is the slow local decay of the radiation term caused by the threshold resonances of the non-selfadjoint linearized matrix operator around the modulated kink. Our analysis hinges on two remarkable null structures that we uncover in the quadratic nonlinearities of the evolution equation for the radiation term as well as of the modulation equations. The entire framework of our proof, including the systematic development of the distorted Fourier theory, is not specific to the sine-Gordon model and extends to many other asymptotic stability problems for moving kinks and other Klein-Gordon solitons. We conclude with a discussion of potential applications in the generic setting (no threshold resonances) and with a discussion of the outstanding challenges posed by internal modes such as in the well-known \\phi^4 model. This is joint work with Gong Chen (GeorgiaTech).

For shear flows in a 2D channel, we define resonances near regular values of the shear profile for the Rayleigh equation under an analyticity assumption. This is done via complex deformation of the interval on which the Rayleigh equation is considered. We show such resonances are inviscid limits of the eigenvalues of the corresponding Orr–Sommerfeld equation. We present examples for which the viscous perturbations of resonances are unstable. Joint work with Malo Jézéquel.