Let ${\mathscr {G}} $ be a special parahoric group scheme of twisted type over the ring of formal power series over $\mathbb {C}$, excluding the absolutely special case of $A^{(2)}_{2\ell }$. Using the methods and results of Zhu, we prove a duality theorem for general ${\mathscr {G}} $: there is a duality between the level one twisted affine Demazure modules and the function rings of certain torus fixed point subschemes in affine Schubert varieties for ${\mathscr {G}} $. Along the way, we also establish the duality theorem for $E_6$. As a consequence, we determine the smooth locus of any affine Schubert variety in the affine Grassmannian of ${\mathscr {G}} $. In particular, this confirms a conjecture of Haines and Richarz.

We prove lower bounds on the density of regular minimal cones of dimension less than seven provided the complements of the cones are topologically nontrivial.

Given a symmetric monoidal category ${\mathcal C}$ with product $\sqcup $, where the neutral element for the product is an initial object, we consider the poset of $\sqcup $-complemented subobjects of a given object X. When this poset has finite height, we define decompositions and partial decompositions of X which are coherent with $\sqcup $, and order them by refinement. From these posets, we define complexes of frames and partial bases, augmented Bergman complexes and related ordered versions. We propose a unified approach to the study of their combinatorics and homotopy type, establishing various properties and relations between them. Via explicit homotopy formulas, we will be able to transfer structural properties, such as Cohen-Macaulayness.

In well-studied scenarios, the poset of $\sqcup $-complemented subobjects specializes to the poset of free factors of a free group, the subspace poset of a vector space, the poset of nondegenerate subspaces of a vector space with a nondegenerate form, and the lattice of flats of a matroid. The decomposition and partial decomposition posets, the complex of frames and partial bases together with the ordered versions, either coincide with well-known structures, generalize them, or yield new interesting objects. In these particular cases, we provide new results along with open questions and conjectures.

Nakamura and Tsuji recently obtained an integral inequality involving a Laplace transform of even functions that implies, at the limit, the Blaschke-Santaló inequality in its functional form. Inspired by their method, based on the Fokker-Planck semi-group, we extend the inequality to non-even functions. We consider a well-chosen centering procedure by studying the infimum over translations in a double Laplace transform. This requires a new look on the existing methods and leads to several observations of independent interest on the geometry of the Laplace transform. Application to reverse hypercontractivity is also given.

Following [1], given cardinals $\kappa <\lambda $, we say $\kappa $ is a club $\lambda $-Berkeley cardinal if for every transitive set N of size $<\lambda $ such that $\kappa \subseteq N$, there is a club $C\subseteq \kappa $ with the property that for every $\eta \in C$, there is an elementary embedding $j: N\rightarrow N$ with $\mathrm {crit }(j)=\eta $. We say $\kappa $ is $\nu $-club $\lambda $-Berkeley if $C\subseteq \kappa $ as above is a $\nu $-club. We say $\kappa $ is $\lambda $-Berkeley if C is unbounded in $\kappa $. We show that under $\textsf {AD}^{+}$, (1) every regular Suslin cardinal is $\omega $-club $\Theta $-Berkeley (see Theorem 7.1), (2) $\omega _1$ is club $\Theta $-Berkeley (see Theorem 3.1 and Theorem 7.1), and (3) the ’s are $\Theta $-Berkeley – in particular, $\omega _2$ is $\Theta $-Berkeley (see Remark 7.5).

Along the way, we represent regular Suslin cardinals in direct limits as cutpoint cardinals (see Theorem 5.1). This topic has been studied in [31] and [4], albeit from a different point of view. We also show that, assuming $V=L({\mathbb {R}})+{\textsf {AD}}$, $\omega _1$ is not $\Theta ^+$-Berkeley, so the result stated in the title is optimal (see Theorem 9.14 and Theorem 9.19).