Georgia Institute of Technology College of Sciences

It is well-known that a spacetime which expands sufficiently fast can stabilize the fluid for relativistic/Einstein-fluid systems. One may wonder whether the expansion of the fluid, instead of the background spacetime geometry, is also able to achieve a similar stabilizing effect. As an attempt to address this question, we consider the free boundary relativistic Euler equations in Minkowski background M1+3 equipped with a physical vacuum boundary, which models the motion of relativistic gas. For the class of isentropic, barotropic, and polytropic gas, we construct an open class of initial data which launch future-global solutions. Such solutions are spherically symmetric, have small initial density, and expand asymptotically linearly in time. In particular, the asymptotic rate of expansion is allowed to be arbitrarily close to the speed of light. Therefore, our main result is far from a perturbation of existing results concerning the classical Euler counterparts. This is joint work with Marcelo Disconzi and Chenyun Luo.

Rescaling limits were first introduced by Jan Kiwi to study degenerations of rational maps of degree at least two. Building on the work of Luo and Favre–Gong, we explain how rescaling limits can serve as a substitute for a good compactification of $Rat_d$, the moduli space of degree d rational maps. In particular, this framework allows one to promote pointwise results to uniform statements in a systematic way. 

Proving positivity of the top Lyapunov exponent ($\lambda_1$​) and obtaining parameter-dependent lower bounds is an interesting and challenging problem for SDEs (stochastic differential equations). We outline methods to obtain lower bounds and establish positivity of $\lambda_1$​ for certain SDEs, combining the coordinate rescaling framework of Pinsky–Wihstutz (1988) for nilpotent linear It\^{o} systems with Fisher information formulas for Lyapunov exponents introduced by J. Bedrossian, A. Blumenthal, and S. Punshon-Smith (2022). This approach uses hypoellipticity and regularity of 2nd order linear PDEs.

We apply these techniques to a 2-D toy SDE to obtain positive lower bounds and small-noise scaling (in terms of noise parameter $\sigma$) for $\lambda_1$​ as $\sigma \to 0$. These techniques avoid computing the stationary density explicitly, using only qualitative regularity of the limiting stationary density coming from hypoellipticity. We also present how a similar approach yields shear-induced chaos for a stochastically driven limit cycle closely related to the Hopf normal form with additive noise, by proving $\lambda_1 > 0$. Finally, we briefly discuss additional SDEs where we believe variants of these ideas may yield positive lower bounds on $\lambda_1$. This work is part of ongoing joint work with Samuel Punshon-Smith.
 

The study of the electronic properties of twisted bilayer graphene (TBG) has garnered much attention from the condensed matter community recently. TBG is obtained by stacking two graphene monolayers on top of each other, and rotating one of them with respect to the other. Theoretical and experimental analyses have found that the electronic properties of TBG depend very strongly on the angle between the layers. In fact, a handful of “magic” angles have been predicted at which TBG becomes a superconductor, and this has even been verified experimentally. The model commonly used to study TBG is an effective one, and was derived by Bistritzer and MacDonald. In this talk, I will present recent results on developing a framework to study TBG from first principles. To be more exact, we consider a tight-binding model for the electrons, but make no further approximations. Using a renormalization group technique, we construct a perturbative expansion to study TBG that is convergent when the twisting angle satisfies certain diophantine conditions. This is joint work with V. Mastropietro.