MIT PDE/Analysis Seminar

Abstract: Given a subset of Euclidean space, one may consider the space of harmonic functions vanishing on it. In two dimensions, this space is always either trivial or infinite-dimensional. In higher dimensions, this is no longer true. In this talk we will see that almost all cones defined by a quadratic homogeneous harmonic polynomial admit exactly two linearly independent harmonic functions vanishing on them.

This phenomenon also generalizes in a natural way to solutions of second order elliptic PDEs with smooth coefficients.

Abstract: Let $f(t)$ be a convex, positive, increasing nonlinearity. It is known that stable solutions of $-\Delta u =f(u)$ can be singular (i.e., unbounded) if the dimension $n \ge 10$.

Brezis conjectured that if $x=0$ is such a singular point, then $f'(u(x))$ blows-up like $|x|^{2-n}$. Villegas showed that such a strong statement fails for general nonlinearities.

In this talk, we prove — for all nonlinearities — a version of Brezis conjecture, which is essentially the best one can obtain in view of the counterexamples of Villegas. Building on this result we then show that the singular set has dimension n-10, at least for a large class of nonlinearities that includes the most relevant cases. This is a joint work with Alessio Figalli.

Abstract: A central problem in general relativity concerns the formation of naked singularities, a class of finite-time blowup solutions to Einstein-matter systems with starkly different properties than their black hole counterparts. In this talk we introduce the family of k-self-similar naked singularities, first constructed rigorously by Christodoulou, which are known to exhibit a blue-shift instability. In our main results, we quantify the strength of this blue-shift instability as the degree of concentration and the support of initial data are varied, identifying surprising regimes in which these spacetimes transition between stability and instability. We then discuss the consequences of these results for the weak cosmic censorship conjecture.

Abstract: In this talk, I will define various notions of the multi-fractal spectrum of harmonic measures and discuss finer features of the relationship between them and properties of the corresponding conformal maps. Furthermore, I will describe the role of multifractal formalism and dynamics in the universal counterparts. This is a developing story, based on a joint work with I. Binder.

Abstract: I will describe a general method to treat non-linear stability problems for quasilinear wave equations on asymptotically flat background spacetimes, provided the nonlinearities have good structure at infinity of the type present in many physically important examples. The method requires two elements as input, both of which refer only to the linearisation of the equation around the trivial solution: (i) an appropriate integrated local energy decay estimate and (ii) the existence of an energy current possessing certain weak coercivity properties. In addition to unifying many previous results, the method yields the first non-linear stability statement for quasilinear equations on Kerr black holes in the full subextremal range $|a|$ < $M$. This is joint work with Gustav Holzegel, Igor Rodnianski and Martin Taylor.

Abstract: 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 (so-called TBG in AA stacking). 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 and state our main results.

Abstract: In this talk, I will first present the Landis conjecture on exponential decay for solutions to second order elliptic equation in the Euclidean setting. While for complex-valued functions, the Landis conjecture was disproved by Meshkov in 1992, the question is still open for real-valued functions. I will present one way to tackle the conjecture, due to Bourgain and Kenig, that consists in establishing quantitative unique continuation results. Previous results in the two-dimensional setting by Kenig, Sylvestre, Wang in 2014 and more recently by Logunov, Malinnikova, Nadirashvili, Nazarov in 2020 will be recalled and explained. Then, the goal of the talk will be to prove that the qualitative and quantitative Landis conjecture hold for real-valued solutions of the Laplace operator, perturbed by lower order term, in the plane. The talk would be based on a joint work with Diego A. Souza (Universidad de Sevilla).

Abstract: Entire critical points of the abelian Higgs functional are known to blow down to generalized minimal submanifolds (of codimension 2). In this talk we prove an Allard type large-scale regularity result for the zero set of solutions. In the "multiplicity one" regime, we show the uniqueness of blow-downs and classify entire solutions in low dimensions and minimizers in all dimensions; thus obtaining an analogue of Savin's theorem in codimension two. This is based on a joint work with Guido de Philippis and Alessandro Pigati.

Abstract: In 2004 Bourgain proved a qualitative trilinear moment inequality for solutions to the Schrödinger equation on the circle and raised the question for quantitative estimates.

Here we show quantitative estimates. The proof combines decoupling iterations with semi-classical Strichartz estimates. Related arguments allow us to extend Bourgain’s -well-posedness result for the periodic KP-II equation to initial data with negative Sobolev regularity. One key ingredient are Strichartz estimates, which follow from a novel decoupling inequality due to Guth-Maldague-Oh. The latter part of the talk is based on joint work with Sebastian Herr and Nikolay Tzvetkov.