Georgia Institute of Technology College of Sciences

We discuss a new problem by Dmitri Burago and Anton Petrunin. Consider a $C^2$-smooth surface embedded in $\RR^3$ with principal curvatures $\leq 1$. Does such a surface necessarily enclose a volume at least that of the unit ball? We address results from two opposite perspectives: On the one hand, we can construct a counterexample by deforming Lagunov's fishbowl (a fattened Bing's house); on the other hand, there are positive results with additional conditions. Many problems remain open.

We develop an extension of the Stein-Malliavin calculus which allows to measure the Wasserstein distance between the probability distributions of $ (X, Y)$ and $(Z,Y)$, where $X,Y$ are arbitrary random vectors and $ Z\sim N(0, \sigma ^{2})$ is independent of $Y$. In particular, this method allows to quantify the asymptotic independence between sequences of random variables and vectors. We will discuss some particular applications of this method to various limit theorems.

The multipermutohedral Chow ring was introduced in a series of papers by Clader, Damiolini, Eur, Huang, Li, and Ramadas to study moduli spaces with cyclic symmetry. It generalizes Chow rings of permutohedral varieties and type-B Coxeter arrangements. In this talk, we establish the combinatorial structure of the multipermutohedral Chow ring through an explicit Gröbner basis, yielding a Feichtner-Yuzvinsky-type monomial basis and a formula for the Hilbert series. Using this formula, we refine the palindromicity of the Hilbert series. From a representation-theoretic perspective, we also compute the equivariant Hilbert series under two natural group actions and construct combinatorial maps that establish equivariant unimodality and palindromicity.