Georgia Institute of Technology College of Sciences

The Plücker embedding exhibits the Grassmannian Gr(r, n) as a closed subvariety of projective space. A theorem of Hodge shows that its homogeneous ideal has as a quadratic Gröbner basis the so-called multiple-exchange relations between Plücker coordinates. Since the set of these polynomials is quite large and unwieldy, it is often preferable to work with a smaller set of single-exchange Plücker relations. An even smaller set of polynomials is the collection of local (or 3-term) exchange relations. We will recall and clarify the relationships between these three. We go on to examine the situation over hyperfields. In their pioneering work, Baker and Bowler showed that the theories of matroids, oriented matroids, valuated matroids etc. can be collectively understood under a common banner as the theory of Grassmanians over hyperfields. Their work gives a good accounting of the relationship between single- and local-exchange relations in this generalized setting. We will discuss what can be said about the multiple-exchange relations. This leads to considerations of elementary linear-algebraic facts in the hyperfield setting. All results may be suitably extended to the flag setting---which we will discuss, time permitting. The talk is based on joint work with Nathan Bowler and Changxin Ding.

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.