Georgia Institute of Technology College of Sciences

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.

From a 3-body problem (CR3BP)
to modeling periodic trajectories with algebraic curves
to minimal problems (related to liaison navigation)
solved via computer algebra (python, Macaulay2).